Exponential Asymptotic Stability for the Klein Gordon Equation on Non-compact Riemannian Manifolds

Abstract

The Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold $(\mathcal {M}^n,\mathbf {g})$ without boundary is considered. Let us assume that the dissipative effects are effective in $(\mathcal {M}\backslash \Omega ) \cup (\Omega \backslash V)$, where $\Omega$ is an arbitrary open bounded set with smooth boundary. In the present article we introduce a new class of non compact Riemannian manifolds, namely, manifolds which admit a smooth function f, such that the Hessian of f satisfies the pinching conditions (locally in $\Omega$), for those ones, there exist a finite number of disjoint open subsets $V_k$ free of dissipative effects such that $\bigcup _k V_k \subset V$ and for all $\varepsilon >0$, $meas(V)\ge meas(\Omega )-\varepsilon$, or, in other words, the dissipative effect inside $\Omega$ possesses measure arbitrarily small. It is important to be mentioned that if the function f satisfies the pinching conditions everywhere, then it is not necessary to consider dissipative effects inside $\Omega$.

1 Introduction

This paper addresses the well-posedness as well as sharp uniform decay rate estimates of the energy related to the Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold $(\mathcal {M}^n,\mathbf {g})$ without boundary:

$$\begin{aligned} \left\{ \begin{aligned}&u_{tt} - \Delta u + u+ a(x) g(u_t)=0, \hbox { in } \mathcal {M} \times (0,+\infty ),\\&u(x,0)=u_0(x);~u_t(x,0)=u_1(x),~x\in \mathcal {M}, \end{aligned} \right. \end{aligned}$$

(1.1)

where $\Delta$ denotes the Laplace–Beltrami operator. The non-negative and essentially bounded function $a=a(x)$, responsible by the nonlinear and localized dissipative effect, lies properly in $\mathcal {M}\backslash \overline{\Omega }$, where $\Omega$ is an arbitrary open and bounded set in $\mathcal {M}$ with smooth boundary $\partial \Omega$, that is, $a(x)\ge a_0 >0$ a.e. in $\mathcal {M}\backslash \overline{\Omega }$.

The counterpart of the above problem (1.1) in the Euclidean setting is precisely when $\mathcal {M}={\mathbb {R}}^n$ endowed with the usual Euclidean topology. In this case, it is sufficient to consider dissipative effects in ${\mathbb {R}}^n \backslash \overline{\Omega }$ as considered in the Fig. 1. See, for instance, the following references [17, 44].

Fig. 1

When $g(s)=s$, the exponential decay is expected

This is well-known since in the Euclidean setting the bicharacteristics (which are the graphs of unit-speed geodesics) or also called rays of the geometric optics, are straight lines so that, roughly speaking, every ray of the geometric optics that intercepts the region $\Omega$ never remains in $\Omega$. However, if one considers a non compact Riemannian manifold $({\mathbb {R}}^n, \mathbf {g})$, where $\mathbf {g}$ is a generic Riemannian metric, we have to be very careful because of the existence of complete geodesics contained in $\Omega$, which severely violates the law of the geometric optics due to Bardos et al. [5, 33, 34], namely: there exists a time $T_0$ such that any geodesic of length $\le T_0$ meets the open set $\{x;a(x)>0\}$. In this case it has been established by Rauch and Taylor [34] that the energy $E(u,t)=\frac{1}{2} \int _{\mathcal {M}} \left( |\nabla _x u|^2 + |u_t|^2\right) \,dx$ associated to problem (1.1) (for $g(s)=s$ and compact manifolds without boundary) decays exponentially, and this result was extended to the case of compact manifolds with boundary by Barbos, Lebeau and Rauch [5]. Consequently the existence of complete geodesics contained in $\Omega$ implies that no exponential decay is expected. In other words: the existence of trapped geodesics breaks the exponential stability (see Fig. 2).

Fig. 2

The existence of trapped geodesics breaks the exponential stability

From the above comments and in order to obtain the exponential stability to problem (1.1) it is strongly necessary to consider dissipative effects inside the set $\Omega$. The best way of doing this is to introduce damping as little as possible. Indeed, the first main goal of the present article is to show that there exists a region $V \subset \Omega$ free of dissipative effects such that $\hbox {meas} (V) \ge \hbox {meas} (\Omega ) - \varepsilon$, for an arbitrary positive number $\varepsilon$. This result can be considered sharp in the sense that the region with damping possesses measure arbitrarily small, however totally distributed (see Fig. 3). We also show that there are no geodesics trapped in the interior of each region free of dissipative effects (see Appendix).

Fig. 3

The region in black inside $\Omega$ possesses dissipative effects and measure arbitrarily small, while the white region contained in $\Omega$ does not have dissipative effects and measure arbitrarily large. However, both regions are totally distributed

The second main task of the present paper is to prove that we can also avoid to put damping in open subsets inside $\Omega$ for non compact manifolds $({\mathcal {M}},g)$ which satisfy the condition $k_1\le sec_g\le k_2<0$, for some negative constants $k_1$ and $k_2$, where $sec_g$ denotes the sectional curvature of $({\mathcal {M}},g)$, according to Fig. 4. In order to achieve our goal, we combine ideas from the article [13] of the first and the third authors with new tools we shall describe in the sequel. We proceed as follows:

• (1) We prove that for every $x\in \Omega$, there exist a neighborhood that can be left without damping;

• (2) We prove that a very precise portion of open subsets inside $\Omega$ can be left without damping;

• (3) Let $\varepsilon >0$ and $V_1, \ldots , V_k$ be open subsets inside $\Omega$ as in (1) and (2) which closures are pairwise disjoint. We prove that there exist a $V\supset \cup _{i=1}^k V_i$ that can be left without damping and such that $meas(V)\ge meas(\Omega )-\varepsilon$.

Fig. 4

No damping is necessary in open subsets of $\Omega$ when the sectional curvature of $({\mathcal {M}},g)$ satisfies $k_1\le sec_g\le k_2<0$

For this purpose, we will construct an intrinsic multiplier that will play an important role when establishing the desired uniform decay rates of the energy, in this sense the compactness of $\Omega$ is crucial. Fix $\varepsilon > 0$. This multiplier is given by $\left<\nabla f,\nabla u\right>$, where $f:\overline{\Omega } \rightarrow \mathbb {R}$ is a smooth function such that its Hessian, Hess(f) (sometimes also denoted by $\nabla ^2 f$), is closely related to $\mathbf g$ on an open subset $V \subset \Omega$ that satisfies $meas(V)\ge meas (\Omega )-\varepsilon$, that is, assuming that $k_1\le sec_g\le k_2<0$ we deduce that the Hess(f) satisfies the “pinching conditions’ in each component $V_i$ of V or, in other words, $A \le \hbox {Hess}(f)\le B$ in $V_i$ for some positive positive constants $0<A <B$. It is important to be mentioned that if f satisfies the pinching conditions everywhere, then we can let $\Omega$ free of dissipative effects. This construction will be clarified during the proof.

It is worth observing that the scenarios presented in Figs. 1 and 4 are distinct, i.é., the Fig. 1 refers to the Euclidean environment and Fig. 4 to a manifold with a generic Riemannian metric.

While in [12, 13] the authors work with compact manifolds here we have to deal with non compact manifolds. Another technical difficulty found in the present work was the following: while in [12, 13] a regular solution u to problem (1.1) lies globally in $H^2(\mathcal {M})$, in the present case, it simply exists locally in $H^2(\Omega )$ for all $\Omega \subset \mathcal {M}$. Fortunately it is enough for our purpose. However, the main technical difficulty found in the present paper was to deal with the boundary terms (see the proof of Theorem 4.1). Indeed, in [12, 13] the homogeneous Dirichlet boundary condition is assumed so that the boundary terms that appear in the computations and that do not vanish are easier to be handled, when compared with the present situation. Note that in the present case we do not have any control on the terms defined on the boundary $\partial \Omega$. In the Euclidean setting $\Omega$ is assumed to be a ball centered in the origin with radius R so that its boundary possesses nice geometric properties as considered, for instance, in [44]. We employ a similar idea as considered in [44] to deal with boundary terms, however our approach is much more delicate since $\Omega$ is an arbitrary open set contained in $\mathcal {M}$. Since we are interested in the impact of the geometry in the decay rate estimates we shall assume that there exist positive constants k, K such that $k|s|^2 \le g(s) s \le K |s|^2$ for all $s\in {\mathbb {R}}$, where the real function g is assumed to be monotone increasing. It is worth mentioning that some ideas used in this work are based on works due to Yao, see for example [42].

We would like to emphasize that the main novelty in the present article is to introduce a new class of non compact Riemannian manifolds, namely, manifolds which admit a smooth function f, such that the Hessian $\nabla ^2 f$ satisfies the “pinching conditions”, for those ones, there exist open disjoint subsets without damping as mentioned before. In addition, analogously what has been proved in [13], for compact manifolds, it is possible, as well, by using the present approach, to let free of damping radially symmetric disjoint regions for non compact manifolds, assuming, as in [13], that we are endowing $(\mathcal {M},\mathbf {g})$ by a radial metric. As a consequence, the present work is an expansion of [13]. It is worth mentioning that in the non compact case, we have examples of existence of functions satisfying the pinching condition, see the class of Warped Products (Sect. 5).

In the particular case when one has the non compact manifold $({{\mathbb {R}}}^n, g_{\varphi })$, endowed with the radial metric described in polar coordinates $(r,\theta )\in \big [0,+\infty \big [\times \mathbb {S}^{n-1}$ by the formula:

$$\begin{aligned} g_\varphi =\mathrm dr^2+\varphi (r)^2\mathrm d\theta ^2, \end{aligned}$$

where $\mathrm d\theta ^2$ is the standard round metric (of radius 1) of $\mathbb S^{n-1}$ and $\varphi :\big ]0,+\infty \big [\rightarrow \mathbb {R}^+$ is a smooth function satisfying $\varphi ^{(2k)}(0)=0$ for all $k\ge 0$, $\varphi '(0)=1$, we can avoid to put damping in every open ball $\Omega$ of radius $R>0$ (see Fig. 1) according to the properties of the function $\varphi$. Indeed, if $\varphi '(r)\ge c>0$ for all r, then $(\mathbb {R}^n,g_\varphi )$ admits a proper, strongly convex function which is bounded from below. This means that there exists a positive constant c such that:

$$\begin{aligned} \mathrm {Hess}(f)(X,X)\ge c\,g(X,X),\qquad \forall \,X\in TM. \end{aligned}$$

(1.2)

The above property plays an essential role when establishing the sharp decay rate estimates above mentioned. Indeed, in a general setting, that is, for non compact complete manifolds $(\mathcal {M}^n, \mathbf {g})$ we first prove that (1.2) is satisfied locally in $\Omega$ and from the compactness of $\overline{\Omega }$ we obtain the property globally in $\Omega$ by gluing all the finite connected components $V_k$ of V above mentioned and putting damping between them in a region of measure arbitrary small. Note that connected components $V_k$ can be extremely small, however the measure of $\cup V_k$ is arbitrarily large (see Figs. 3, 4). Now, for the particular case when one has $({\mathbb {R}}^n, g_{\varphi })$ and assuming that $\varphi$ has nice properties, inequality (1.2) is now valid globally and we can avoid putting damping in the whole set $\Omega$. This recovers the previous results in the literature in the Euclidean setting with the usual metric. Our result remains valid for the semi-linear problem $u_{tt}- \Delta u + f(u) + a(x) g(u_t)=0$, as well, provided that: (i) we assume that $(M^n,\mathbf {g})$ possesses bounded curvature and global injectivity radius (so that we can recover the Sobolev imbedding); (ii) we assume that there exists a unique continuation property to the linear problem with potential $v_{tt} - \Delta v + V(x,t)v=0$ where $V(x,t)\equiv f'(u(x,t))$ grows polynomially. The second condition is true for certain nonlinearities as proved in Triggiani and Yao’s paper [37].

Another interesting case is the following: $(\mathbb {R}^n,g_\varphi )$ has finite volume if and only if $\displaystyle \int _0^{+\infty }\varphi (r)^{n-1}\,\mathrm dr<+\infty$. In this case, we can relax the assumptions imposed on g, namely, it is sufficient to assume that $ks^2 \le g(s) s \le Ks^2$ for $|s|>1$ and it is possible obtain a wide assortment of decay rate estimates different of exponential. For this purpose Jessen’s inequality combined with arguments of Lasiecka and Tataru [27] play an essential role when establishing the above mentioned decay rates (see also Cavalcanti et. al [11]).

Our paper is organized as follows. Section 2 we present the preliminaries in Riemannian Geometry, in Sect. 3 is proven to well-posedness of the problem, in Sect. 4 is stated and proved the main result. The Sect. 5 is destined examples of manifolds that admit open subsets without damping and finally the Appendix aims to justify the application of unique continuation principle used in Sect. 4.

2 Preliminaries: Geometric Riemannian Tools

Let $({\mathcal {M}}^n,\mathbf {g})$ be a n-dimensional complete Riemanniana manifold, $n\ge 2$ orientable, connect and without boundary, induced by the Riemannian metric $\mathbf {g}(\cdot ,\cdot )= \langle \cdot ,\cdot \rangle$, of class $C^\infty$. We shall denote by $(g_{ij})_{n\times n}$ the matrix $n\times n$ in connection with the metric $\mathbf {g}$. The tangent space at ${\mathcal {M}}$ em $p \in {\mathcal {M}}$ will be denoted by $T_p{\mathcal {M}}\equiv {\mathbb {R}}^n$.

Let $f\in C^2({\mathcal {M}})$, and let us define the Laplace–Beltrami operator of f, as

$$\begin{aligned} \Delta f = div(\nabla f), \end{aligned}$$

(2.1)

where $\nabla f$ denotes the gradient of f in the metric $\mathbf {g}$, that is, for all vector field X in ${\mathcal {M}}$

$$\begin{aligned} \langle \nabla f , X \rangle = X(f), \end{aligned}$$

(2.2)

and div denotes the divergent operator, namely, if X is a vector field in ${\mathcal {M}}$, $div X(p) :=$ trace of the linear map $Y(p)\mapsto \nabla _YX(p)$, $p\in {\mathcal {M}}$.

From the definitions and notations above we have the following lemma:

Lemma 2.1

Let $p \in {\mathcal {M}}$. Let us consider $f \in C^1({\mathcal {M}})$ and H a vector field in ${\mathcal {M}}$. Then, the following identity hold (see [28] p. 21):

$$\begin{aligned} \langle \nabla f,\nabla (H(f))\rangle =\nabla H(\nabla f,\nabla f)+\frac{1}{2}\left[ div(|\nabla f|^2 H)-|\nabla f|^2 divH\right] , \end{aligned}$$

where $\nabla H$ is the differential covariant derivative defined by $\nabla H(X,Y)=\langle \nabla _{X}H,Y\rangle$.

Finally we shall define the Hessian of $f \in C^2({\mathcal {M}})$ as the symmetric tensor of order two in ${\mathcal {M}}$, namely,

$$\begin{aligned} \mathrm {Hess}(f)(X,Y)=\nabla ^2f(X,Y):=\nabla (\nabla f)(X,Y)=\langle \nabla _Y(\nabla f),X\rangle , \end{aligned}$$

(2.3)

for all X and Y vector fields in ${\mathcal {M}}$.

Remark 1

In order to simplify the notation, we denote the $L^2$-norm, without distinguishing whether the argument of the norm is a function or tensor field of type (0, m).

Let $k \in \mathbb {N}$ e $p\ge 1$. We define the space $C_k^p({\mathcal {M}})$ as

$$\begin{aligned} C_k^p({\mathcal {M}})= \left\{ u\in C^\infty ({\mathcal {M}}); \int _{{\mathcal {M}}}|\nabla ^ju|^p \ d{\mathcal {M}}< \infty , \forall j=0,1, \ldots , k \right\} , \end{aligned}$$

(2.4)

where $\nabla ^ju$ denotes the jth differential covariant derivative of $u, (\nabla ^0u=u, \nabla ^1u=\nabla u$).

Thus, we define the Sobolev space $H_k^p({\mathcal {M}})$ as the closure of $C_k^p({\mathcal {M}})$ with respect to the topology

$$\begin{aligned} \Vert u\Vert _{H_k^p({\mathcal {M}})}^p= \sum _{j=0}^k\int _{{\mathcal {M}}}|\nabla ^ju|^p\ d{\mathcal {M}}. \end{aligned}$$

(2.5)

From the above, we deduce:

• (i)   $L^2({\mathcal {M}}):=H_0^2({\mathcal {M}})$ is the closure of $C_0^2({\mathcal {M}})$ with respect to the tolopogy

  $$\begin{aligned} \Vert u\Vert _{L^2(M)}^2= \int _{{\mathcal {M}}}|u|^2\ d{\mathcal {M}}. \end{aligned}$$

  (2.6)

• (ii)   $H^1({\mathcal {M}}):=H_1^2({\mathcal {M}})$ is the closure of $C_1^2({\mathcal {M}})$ with respect to the topology

  $$\begin{aligned} \Vert u\Vert _{H^1({\mathcal {M}})}^2= \int _{{\mathcal {M}}}|\nabla u|^2\ d{\mathcal {M}}+\int _{{\mathcal {M}}}|u|^2\ d{\mathcal {M}}. \end{aligned}$$

  (2.7)

• (iii)   $H^2({\mathcal {M}}):=H_2^2({\mathcal {M}})$ is the closure of $C_2^2({\mathcal {M}})$ with respect to the topology

  $$\begin{aligned} \Vert u\Vert _{H^2({\mathcal {M}})}^2= \int _{{\mathcal {M}}}|\nabla ^2 u|^2\ d{\mathcal {M}}+\int _{{\mathcal {M}}}|\nabla u|^2\ d{\mathcal {M}}+\int _{{\mathcal {M}}}|u|^2\ d{\mathcal {M}}. \end{aligned}$$

  (2.8)

Remark 2

From the above definitions we have the following chain of continuous embbeding

$$\begin{aligned} H^2({\mathcal {M}})\hookrightarrow H^1({\mathcal {M}})\hookrightarrow L^2({\mathcal {M}}). \end{aligned}$$

(2.9)

Furthermore, by Hebey ([23], Theorem 2.7, p.13), it follows that $H_0^1({\mathcal {M}})=H^1({\mathcal {M}})$, where $H_0^1({\mathcal {M}}):= \overline{{\mathcal {D}}({\mathcal {M}})}^{H^1({\mathcal {M}})}$, in other words, the space of infinitely differentiable functions with compact support is dense in $H^1({\mathcal {M}})$.

So, from the above and making use of density arguments we can extend the formulas presented previously to Sobolev spaces. In the sequel, we shall announce three theorems that will play an important rule in the present work.

Theorem 2.1

(Gauss Divergent Theorem) Let ${\mathcal {M}}^n$ a Riemannian manifold, orientable, with smooth boundary $\partial {\mathcal {M}}$ , $X \in [H^1({\mathcal {M}})]^n$ a vector filed and $\nu$ the normal unitary vector field point towards $\partial {\mathcal {M}}$, thus

$$\begin{aligned} \int _{{\mathcal {M}}} divX \, d{\mathcal {M}}= \int _{\partial {\mathcal {M}}}\langle X,\nu \rangle \, d\partial {\mathcal {M}}. \end{aligned}$$

(2.10)

Theorem 2.2

(Green Theorem 1) Let ${\mathcal {M}}^n$ a Riemannian orientable manifold , with smooth boundary $\partial {\mathcal {M}}$ , $X \in [H^1({\mathcal {M}})]^n$ a vector field, $q \in H^1({\mathcal {M}})$ and $\nu$ the normal unitary vector field point towards $\partial {\mathcal {M}}$, then

$$\begin{aligned} \int _{{\mathcal {M}}}(divX)q \, d{\mathcal {M}}= -\int _{{\mathcal {M}}}\langle X,\nabla q\rangle \, d{\mathcal {M}}+ \int _{\partial {\mathcal {M}}}(\langle X,\nu \rangle )q \, d\partial {\mathcal {M}}. \end{aligned}$$

(2.11)

Theorem 2.3

(Green Theorem 2) Let ${\mathcal {M}}^n$ a orientable Riemannian Manifold, with smooth boundary $\partial {\mathcal {M}}$, $f \in [H^2({\mathcal {M}})]$, $q \in H^1({\mathcal {M}})$ and $\nu$ the normal unitary vector field point towards $\partial {\mathcal {M}}$, then

$$\begin{aligned} \int _{{\mathcal {M}}}(\Delta f)q \, d{\mathcal {M}}= -\int _{{\mathcal {M}}}\langle \nabla f,\nabla q\rangle \, d{\mathcal {M}}+ \int _{\partial {\mathcal {M}}}(\partial _{\nu } f)q \, d\partial {\mathcal {M}}. \end{aligned}$$

(2.12)

3 Existence and Uniqueness of Solutions

In what follows we shall omit some variables in order to make easier the notation and we will denote the Laplace–Beltrami operator simply by $\Delta$ . We shall study the existence and uniqueness of the following damped problem:

$$\begin{aligned} \left\{ \begin{array}{ll} {u_{tt}-\Delta u +u +a(x)g(u_{t})=0} &{} {in\,\,\,{\mathcal {M}}\times (0,\infty )} \\ {u(0)=u_{0}\,\ ,\,\,u_{t}(0)=u_1} &{} {in\,\,\,{\mathcal {M}}} \\ \end{array}\right. \end{aligned}$$

(3.1)

where $({\mathcal {M}}^n,\mathbf {g})$ a n-dimensional Rimannian manifold, $n\ge 2$, simply connected, orientable and without boundary endowed by a Riemannian metric $\mathbf {g}(\cdot ,\cdot )= \langle \cdot ,\cdot \rangle$ complete, of class $C^\infty$.

Assumption 3.1

Hypotheses on the function $g:\mathbb {R}\rightarrow \mathbb {R}:$

(i):

g(s) is continuous, monotone increasing;

(ii):

$g(s)s>0$ para $s\ne 0$;

(iii):

$k|s|\le |g(s)|\le K|s|,$   $\forall s \in \mathbb {R}$, where k e K are two positive constants.

Assumption 3.2

Hypotheses on the function $a:{\mathcal {M}}\rightarrow \mathbb {R}:$

(i):

$a(x)\in L^{\infty }({\mathcal {M}})$ is non negative function;

(ii):

Let $\Omega \subset {\mathcal {M}}$ be an open set and bounded with smooth boundary $\partial \Omega$. We suppose that $a(x)\ge a_0>0$ in ${\mathcal {M}}\backslash \overline{\Omega }$ and on an open proper subset ${\mathcal {M}}_*$ of ${\mathcal {M}}$ namely:

Let $\Omega ^{*}$ be an open and bounded set with smooth boundary such that $\Omega \subset \subset \Omega ^{*}$. For $\varepsilon >0$ we shall ensure that there exist an open subset $V \subset \overline{\Omega ^{*}}$ and smooth functions $\alpha$ and $f:\overline{\Omega ^{*}} \rightarrow \mathbb {R}$ such that $\hbox {meas}(V) \ge \hbox {meas} (\overline{ \Omega ^{*}})- \varepsilon$, $\hbox {meas}(V \cap \partial \overline{\Omega ^{*}} ) \ge \hbox {meas} (\partial \overline{ \Omega ^{*}})-\varepsilon$, $\nabla \alpha |_V \equiv 0$ and such that $\alpha$ and f satisfy

$$\begin{aligned} C\int _0^T\int _V u_t^2 +|\nabla u|^2 \ d{\mathcal {M}}dt\le & {} \int _0^T\int _V\left( \frac{\Delta f}{2}-\alpha \right) u_t^2 \ d{\mathcal {M}}dt\\+ & {} \int _0^T\int _V \nabla ^2 f(\nabla u,\nabla u)+\left( \alpha -\frac{\Delta f}{2}\right) |\nabla u|^2 \ d{\mathcal {M}}dt, \end{aligned}$$

for some positive constant C.

Moreover, if $V_1, \ldots , V_k$ are radially symmetric or, more generally, for a wide class as in the Sect. 8, with pairwise disjoint closures, we can choose V in such a way that $V \supset (\bigcup ^k_{i=1}V_i)$.

The open subset ${\mathcal {M}}_*$ is such that $(\overline{\Omega ^{*}}\backslash V) \subset \subset {\mathcal {M}}_* .$

Remark 3

The geometric idea related to ${\mathcal {M}}_*$, is that, ${\mathcal {M}}_*$ is an open subset of ${\mathcal {M}}$ immediately larger than the black region that is inside the $\Omega$ as show the Fig. 3. It is worth noting that the “net” is built on $\Omega ^*\supset \supset \Omega$, but out of $\Omega$ the dissipative effect is effective everywhere, so the region of greatest importance is that in the interior of $\Omega$.

The energy associated to problem (3.1) is defined by:

$$\begin{aligned} E(t){:=}\frac{1}{2}\int _{{\mathcal {M}}}u_t^2(x,t)+\left| \nabla u(x,t)\right| ^2+u^2(x,t)\ d{\mathcal {M}}dt. \end{aligned}$$

(3.2)

Denoting $U=\bigg (\begin{array}{c}u\\ u_t\end{array}\bigg )$, $U_0=\bigg (\begin{array}{c}u_0\\ u_1\end{array}\bigg )$, $\mathcal {H}:= H^1({\mathcal {M}})\times L^2({\mathcal {M}})$,

$$\begin{aligned} F:&{\mathcal {H}}&\longrightarrow {\mathcal {H}}\\ {}&\bigg (\begin{array}{c}u\\ v\end{array}\bigg )&\longmapsto \bigg (\begin{array}{c}0\\ a(x)g(v)\end{array}\bigg ) \end{aligned}$$

and,

$$\begin{aligned} A:&D(A)\subset {\mathcal {H}}&\longrightarrow {\mathcal {H}}\\ {}&\bigg (\begin{array}{c}u\\ v\end{array}\bigg )&\longmapsto A\bigg (\begin{array}{c}u\\ v \end{array}\bigg )=\bigg (\begin{array}{c} -v\\ -\Delta u +u\end{array}\bigg ), \end{aligned}$$

then problem (3.1) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{dU}{dt}(t)+(A+F)U(t)=0\\ \\ U(0)=U_0\end{array}\right. \end{aligned}$$

(3.3)

It is possible to prove that A is a maximal monotone operator, F is monotone, bounded and hemicontinuous. Employing Barbu ([4], Corol. 1.1, p. 39) it follows that $A+F$ is maximal monotone in ${\mathcal {H}}$, and making use of standard semigroup arguments we have the following result:

Theorem 3.1

• (i) Under the conditions above, the problem (3.1) is well posed in the space $D(A)=\{u\in H^1({\mathcal {M}}); \Delta u \in L^2({\mathcal {M}}) \}\times H^1({\mathcal {M}})$, that is, for any initial data $\{u_0,u_1\}\in D(A)=\{u\in H^1({\mathcal {M}}); \Delta u \in L^2({\mathcal {M}}) \}\times H^1({\mathcal {M}})$ exists a unique map $u:[0,\infty )\longrightarrow H^1({\mathcal {M}})$, which is regular solution to problem (3.1), belonging to the class

  $$\begin{aligned}&u \in C^1([0,\infty );L^2({\mathcal {M}}))\cap C([0,\infty );H^1({\mathcal {M}})); \end{aligned}$$

  (3.4)
  $$\begin{aligned}&u_t \in W^{1,\infty }(0,\infty ;L^2({\mathcal {M}}))\cap L^\infty (0,\infty );H^1({\mathcal {M}})). \end{aligned}$$

  (3.5)

• (ii) Under the conditions above, the problem (3.1) is well posed in the space $H^1({\mathcal {M}}) \times L^2({\mathcal {M}})$, that is, for any initial data $\{u_0,u_1\}\in H^1({\mathcal {M}}) \times L^2({\mathcal {M}})$ exists a map $u:[0,\infty )\longrightarrow H^1({\mathcal {M}})$ which is the unique weak solution to problem (3.1) belonging to the class

  $$\begin{aligned} u \in C^1([0,\infty );L^2({\mathcal {M}}))\cap C([0,\infty );H^1({\mathcal {M}})). \end{aligned}$$

  (3.6)

In addition u satisfies the additional local regularity:

Proposition 3.1

Let $\Omega \subset {\mathcal {M}}$ be an open and bounded set with smooth boundary. Then $u(t)\in H^2(\Omega ),$ $\forall \ t\ge 0$, where u is the regular solution to problem (3.1).

Proof

Let us fix $t\ge 0$. Defining $v:=u(t)$ and $f:=-u_{tt}(t)-a(\cdot )g(u_t(t))\in L^2({\mathcal {M}})$, we deduce the following identity which is valid in $L^2({\mathcal {M}})$

$$\begin{aligned} -\Delta v + v = f. \end{aligned}$$

(3.7)

Let $\Omega ^*\subset {\mathcal {M}}$, be an open and bounded set with smooth boundary such that $\Omega \subset \subset \Omega ^*$ e $\psi \in {\mathcal {D}}({\mathcal {M}})$ satisfying:

(i):

$\psi \equiv 1 \,\ \text{ in } \,\ \overline{\Omega }$

(ii):

$\psi \equiv 0 \,\ \text{ in } \,\ \overline{\Omega ^*}\backslash \Omega$

(iii):

$0\le \psi \le 1$

Multiplying equation (3.7) by $\psi$ we infer

$$\begin{aligned} -\Delta v \psi + v\psi = f\psi . \end{aligned}$$

We observe that

$$\begin{aligned} \Delta (v\psi )=v\Delta \psi + \psi \Delta v + 2\langle \nabla v, \nabla \psi \rangle , \end{aligned}$$

(3.8)

thus

$$\begin{aligned} -\Delta (v\psi )+v\psi = f \psi - v \Delta \psi - 2\langle \nabla v, \nabla \psi \rangle \,\ \text{ in } \,\ L^2({\mathcal {M}}). \end{aligned}$$

In particular,

$$\begin{aligned} -\Delta (v\psi )+v\psi = f \psi - v \Delta \psi - 2\langle \nabla v, \nabla \psi \rangle \,\ \text{ in } \,\ L^2(\Omega ^*). \end{aligned}$$

Furthermore,

$$\begin{aligned} v\psi \mid _{\partial \Omega ^*}=v\mid _{\partial \Omega ^*}\psi \mid _{\partial \Omega ^*}=0. \end{aligned}$$

From standard elliptic regularity results we deduce that $v\psi \in H^2(\Omega ^*)$, from which we obtain $v\psi \in H^2(\Omega )$. Since $\psi \equiv 1$ in $\Omega$ we deduce that $v=u(t)\in H^2(\Omega )$. $\square$

4 Stability Result

Before stating the main theorem of stability we shall consider an useful identity called identity of energy that reads as follows. Let u be a regular solution to problem (3.1), then multiplying the equation by $u_t$ and performing an integration by parts, we infer:

$$\begin{aligned} E(t_2)-E(t_1)=-\int _{t_1}^{t_2}\int _{{\mathcal {M}}}a(x)g(u_t)u_t \ d{\mathcal {M}}dt, \end{aligned}$$

(4.1)

for all $t_2 > t_1\ge 0.$

It is worth mentioning that the identity of energy (4.1) remains true for every weak solution to problem (3.1) by using standard arguments of density.

Theorem 4.1

Let u be a weak solution to problem (3.1), with the energy defined as in (3.2). Then, under Assumption 3.1 and Assumption 3.2 there exist positive constants $T_0$, $C_0$ and $\lambda _0$ such that

$$\begin{aligned} E(t)\le C_0 \textit{e}^{- \lambda _0 t}E(0); \ \ \forall t\ge T_0. \end{aligned}$$

Our main task is to prove the following inequality inequality:

$$\begin{aligned} \int _0^TE(t)dt\le C_1E(T)+C_2\int _0^T\int _{{\mathcal {M}}}a(x)[u_t^2(x,t)+g^2(u_t(x,t))]\ d{\mathcal {M}}dt, \end{aligned}$$

(4.2)

where $C_1$ and $C_2$ are positive constants and $C_1$ doe not depend on T. As we have already mentioned, it is sufficient to work with regular solutions to problem (3.1), since the exponential decay rate can be recovered for weak solutions by using density approach.

4.1 Recovering the Energy Outside the Region $\Omega ^*$

Let $\Omega ^*\subset {\mathcal {M}}$ be an open and bounded set with smooth boundary $\partial \Omega ^*$, such that $\Omega \subset \subset \Omega ^*$. Let us consider $\varphi \in C^\infty ({\mathcal {M}})$ satisfying:

• (i) $\varphi \equiv 1$ in ${\mathcal {M}}\backslash \Omega ^*$;

• (ii) $\varphi \equiv 0$ in $\overline{\Omega }$;

• (iii) $0 \le \varphi \le 1$ in ${\mathcal {M}}$.

Multiplying equation (3.1) by $\varphi u$ and integrating over $[0,T]\times {\mathcal {M}}$, we obtain

$$\begin{aligned} \int _0^T\int _{\mathcal {M}}[u_{tt}-\Delta u +a(x)g(u_t)+u]\varphi u \ d{\mathcal {M}}dt=0. \end{aligned}$$

(4.3)

We observe that

$$\begin{aligned} \frac{d}{dt}\left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right]= & {} \int _{\mathcal {M}}u_{tt} \varphi u \ d{\mathcal {M}}+ \int _{\mathcal {M}}u_t \varphi u_t \ d{\mathcal {M}}\\= & {} \int _{\mathcal {M}}u_{tt} \varphi u \ d{\mathcal {M}}+\int _{\mathcal {M}}u_t^2 \varphi \ d{\mathcal {M}}. \end{aligned}$$

Integrating over [0, T] we conclude that

$$\begin{aligned} \int _0^T\int _{\mathcal {M}}u_{tt}\varphi u \ d{\mathcal {M}}dt = \left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T - \int _{\mathcal {M}}u_t^2 \varphi \ d{\mathcal {M}}dt. \end{aligned}$$

(4.4)

In addition,

$$\begin{aligned} \int _0^T\int _{\mathcal {M}}-\Delta u(\varphi u)\ d{\mathcal {M}}dt= & {} \int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla (\varphi u)\rangle \ d{\mathcal {M}}dt \nonumber \\= & {} \int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u +\varphi |\nabla u|^2 \ d{\mathcal {M}}dt. \end{aligned}$$

(4.5)

Combining (4.3), (4.4) and (4.5), we deduce

$$\begin{aligned} 0= & {} \left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T - \int _{\mathcal {M}}u_t^2 \varphi \ d{\mathcal {M}}dt \nonumber \\&+ \int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u +\varphi |\nabla u|^2 \ d{\mathcal {M}}dt \nonumber \\&+ \int _0^T\int _{\mathcal {M}}\varphi u^2 \ d{\mathcal {M}}dt + \int _0^T\int _{\mathcal {M}}a(x) g(u_t) \varphi u \ d{\mathcal {M}}dt. \end{aligned}$$

(4.6)

Taking into account the properties of function $\varphi$, we can write

$$\begin{aligned}&\int _0^T\int _{\Omega ^* \setminus \Omega }\varphi [u^2+ |\nabla u|^2] \ d{\mathcal {M}}dt+ \int _0^T\int _{{\mathcal {M}}\setminus \Omega ^*}u^2 +|\nabla u|^2 \ d{\mathcal {M}}dt \nonumber \\&\quad = -\left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T +\int _{\Omega ^* \setminus \Omega } \varphi u_t^2 \ d{\mathcal {M}}dt + \int _{{\mathcal {M}}\setminus \Omega ^*} u_t^2 \ d{\mathcal {M}}dt \nonumber \\&\quad - \int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u \ d{\mathcal {M}}dt - \int _0^T\int _{{\mathcal {M}}\setminus \Omega } a(x) g(u_t) \varphi u \ d{\mathcal {M}}dt. \end{aligned}$$

(4.7)

Observing that $a(x)\ge a_0>0$ in ${\mathcal {M}}\setminus \overline{\Omega }$, it holds that

$$\begin{aligned}&\int _0^T\int _{{\mathcal {M}}\setminus \Omega ^*}u^2 +|\nabla u|^2 \ d{\mathcal {M}}dt \le -\left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T\nonumber \\&\quad +\,a_0^{-1}\int _0^T\int _{\Omega ^* \setminus \Omega }a(x) \varphi u_t^2 \ d{\mathcal {M}}dt +a_0^{-1}\int _0^T\int _{{\mathcal {M}}\setminus \Omega ^*}a(x) u_t^2 \ d{\mathcal {M}}dt\nonumber \\&\quad - \int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u \ d{\mathcal {M}}dt - \int _0^T\int _{{\mathcal {M}}\setminus \Omega } a(x) g(u_t) \varphi u \ d{\mathcal {M}}dt. \end{aligned}$$

(4.8)

Adding $\int _0^T\int _{{\mathcal {M}}\setminus \Omega ^*} u_t^2 \ d{\mathcal {M}}dt$ in the previous inequality and using the fact that $0\le \varphi \le 1$, we obtain

$$\begin{aligned}&\int _0^T\int _{{\mathcal {M}}\setminus \Omega ^*}u^2 +|\nabla u|^2 +u_t^2 \ d{\mathcal {M}}dt \le -\left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T\nonumber \\&\quad + \, a_0^{-1}\int _0^T\int _{\mathcal {M}}a(x) u_t^2 \ d{\mathcal {M}}dt - \int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u \ d{\mathcal {M}}dt\nonumber \\&\quad -\int _0^T\int _{{\mathcal {M}}\setminus \Omega } a(x) g(u_t) \varphi u \ d{\mathcal {M}}dt +\int _0^T\int _{{\mathcal {M}}\setminus \Omega ^*} u_t^2 \ d{\mathcal {M}}dt \end{aligned}$$

(4.9)

Next, we shall estimate the term $\int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u \ d{\mathcal {M}}dt$.

Observe that $\langle \nabla u,\nabla \varphi \rangle = 0$ in ${\mathcal {M}}\backslash \Omega ^*$.

We shall make use of a generalized Green’s identity, that makes sense in our case, because $u\in H^1(\Omega ^*)$, then $u^2\in W^{1,1}(\Omega ^*)$, thus $u^2\big |_{\partial \Omega ^*}\in L^1(\partial \Omega ^*)$ (see [6]) and therefore by using the dense and continuous immersion ${\mathcal {D}}(\overline{\Omega ^*})\hookrightarrow W^{1,1}(\Omega ^*)$ we can show the desired. Note that in our case $\varphi \in C^\infty (\overline{\Omega ^*})$ and $\varphi \equiv 1$ in $\partial \Omega ^*$.

So we can write

$$\begin{aligned} \int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u \ d{\mathcal {M}}= & {} \int _{\Omega ^*} \langle \nabla u,\nabla \varphi \rangle u \ d{\mathcal {M}}\nonumber \\= & {} \frac{1}{2}\int _{\Omega ^*}\langle \nabla (u^2),\nabla \varphi \rangle \ d{\mathcal {M}}\nonumber \\= & {} -\frac{1}{2}\int _{\Omega ^*}\Delta \varphi \ u^2 \ d{\mathcal {M}}+\int _{\partial \Omega ^*}\partial _{\nu }\varphi \ u^2 \ d\Gamma \nonumber \\= & {} -\frac{1}{2}\int _{\Omega ^*}\Delta \varphi \ u^2 \ d{\mathcal {M}}\end{aligned}$$

(4.10)

Therefore,

$$\begin{aligned} \left| \int _0^T\int _{\mathcal {M}}\langle \nabla u,\nabla \varphi \rangle u \ d{\mathcal {M}}dt \right| \le \frac{c}{2}\int _0^T\int _{\Omega ^*}u^2 \ d{\mathcal {M}}dt, \end{aligned}$$

(4.11)

where $c:= max_{x \in \overline{\Omega ^*}} \ |\Delta \varphi (x)|$.

We conclude so, from (4.9) and (4.11) that

$$\begin{aligned}&\int _0^T\int _{{\mathcal {M}}\setminus \ \Omega ^*}u^2 +|\nabla u|^2 +u_t^2 \ d{\mathcal {M}}dt \le -\left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T\nonumber \\&\quad + \,3a_0^{-1}\int _0^T\int _{\mathcal {M}}a(x) u_t^2 \ d{\mathcal {M}}dt + \frac{c}{2}\int _0^T\int _{\Omega ^*\backslash \Omega }u^2 \ d{\mathcal {M}}dt \nonumber \\&\quad - \int _0^T\int _{{\mathcal {M}}\setminus \ \Omega } a(x) g(u_t) \varphi u \ d{\mathcal {M}}dt. \end{aligned}$$

(4.12)

From (4.12), it remains to estimate $\int _0^T\int _{\Omega ^*}u^2 +|\nabla u|^2 +u_t^2 \ d{\mathcal {M}}dt$ in terms of “good terms”. Indeed, we observe that $\Omega ^*$ is a submanifold with smooth boundary $\partial \Omega ^*$.

4.2 Recovering the Energy Inside $\Omega ^*$

Let q be a vector field in $\overline{\Omega ^*}$ of class $C^1$. Multiplying equation (3.1) by $\langle \nabla u,q\rangle$ and integrating over $[0,T]\times \Omega ^*$ we infer

$$\begin{aligned} 0=\int _0^T\int _{\Omega ^*}(u_{tt}-\Delta u +u +a(x)g(u_t))\langle \nabla u,q\rangle d{\mathcal {M}}dt. \end{aligned}$$

(4.13)

$\bullet$ Estimate for $I_1:=\int _0^T\int _{\Omega ^*}u_{tt}\langle \nabla u,q\rangle d{\mathcal {M}}dt.$

We note that

$$\begin{aligned} \frac{d}{dt}\int _{\Omega ^*}u_{t}\langle \nabla u,q\rangle d{\mathcal {M}}=\int _{\Omega ^*}u_{tt}\langle \nabla u,q\rangle d{\mathcal {M}}+ \int _{\Omega ^*}u_{t}\langle \nabla u_t,q\rangle d{\mathcal {M}}. \end{aligned}$$

Thus

$$\begin{aligned} I_1=\left[ \int _{\Omega ^*}u_{t}\langle \nabla u,q\rangle d{\mathcal {M}}\right] _0^T-\int _0^T\int _{\Omega ^*}u_{t}\langle \nabla u_t,q\rangle d{\mathcal {M}}dt. \end{aligned}$$

Recall that

$$\begin{aligned} u_t\langle q, \nabla u_t\rangle = \frac{1}{2}\langle q, \nabla u_t^2\rangle . \end{aligned}$$

(4.14)

Then, taking (4.14) into account and making use of Theorem 2.2, we can write

$$\begin{aligned} I_1= & {} \left[ \int _{\Omega ^*}u_{t}\langle \nabla u,q\rangle d{\mathcal {M}}\right] _0^T -\int _0^T\int _{\Omega ^*} \frac{1}{2}\langle q, \nabla u_t^2\rangle \ d{\mathcal {M}}dt \nonumber \\= & {} \left[ \int _{\Omega ^*}u_{t}\langle \nabla u,q\rangle d{\mathcal {M}}\right] _0^T+\frac{1}{2}\int _0^T\int _{\Omega ^*}div(q)u_t^2 \ d{\mathcal {M}}dt \nonumber \\&- \frac{1}{2}\int _0^T\int _{\partial \Omega ^*}\langle q, \nu \rangle u_t^2 \ d\Gamma dt. \end{aligned}$$

(4.15)

$\bullet$ Estimate for $I_2:=\int _0^T\int _{\Omega ^*}-\Delta u\langle q, \nabla u\rangle \ d{\mathcal {M}}dt.$

Employing Green’s theorem before mentioned and from Lemma 2.1, we deduce

$$\begin{aligned} I_2= & {} \int _0^T\int _{\Omega ^*}\langle \nabla u, \nabla (\langle q, \nabla u\rangle )\rangle \ d{\mathcal {M}}dt - \int _0^T\int _{\partial \Omega ^*}\partial _{\nu }u\langle q, \nabla u\rangle \ d\Gamma dt \nonumber \\= & {} \int _0^T\int _{\Omega ^*}\nabla q(\nabla u,\nabla u) \ d{\mathcal {M}}dt\nonumber \\&+\int _0^T\int _{\Omega ^*}\frac{1}{2}\left[ div(|\nabla u|^2q)-div(q)|\nabla u|^2\right] \ d{\mathcal {M}}dt \nonumber \\&-\int _0^T\int _{\partial \Omega ^*}\partial _{\nu }u\langle q, \nabla u\rangle \ d\Gamma dt \nonumber \\= & {} \int _0^T\int _{\Omega ^*}\nabla q(\nabla u,\nabla u) \ d{\mathcal {M}}dt + \int _0^T\int _{\Omega ^*}\frac{1}{2}\langle q, \nabla (|\nabla u|^2)\rangle \ d{\mathcal {M}}dt \nonumber \\&-\int _0^T\int _{\partial \Omega ^*}\partial _{\nu }u\langle q, \nabla u\rangle \ d\Gamma dt \nonumber \\= & {} \int _0^T\int _{\Omega ^*}\nabla q(\nabla u,\nabla u) \ d{\mathcal {M}}dt - \frac{1}{2}\int _0^T\int _{\Omega ^*}div(q)|\nabla u|^2 \ d{\mathcal {M}}dt \nonumber \\&+\frac{1}{2}\int _0^T\int _{\partial \Omega ^*}\langle q,\nu \rangle |\nabla u|^2 \ d\Gamma dt - \int _0^T\int _{\partial \Omega ^*}\partial _{\nu }u\langle q, \nabla u\rangle \ d\Gamma dt. \end{aligned}$$

(4.16)

$\bullet$ Estimate for $I_3:=\int _0^T\int _{\Omega ^*} u\langle q, \nabla u\rangle \ d{\mathcal {M}}dt.$

$$\begin{aligned} I_3= & {} \frac{1}{2} \int _0^T\int _{\Omega ^*} \langle q, \nabla u^2\rangle \ d{\mathcal {M}}dt \nonumber \\= & {} -\frac{1}{2}\int _0^T\int _{\Omega ^*} div(q)u^2 \ d{\mathcal {M}}dt + \frac{1}{2}\int _0^T\int _{\partial \Omega ^*} \langle q,\nu \rangle u^2 \ d\Gamma dt \end{aligned}$$

(4.17)

Combining (4.13), (4.15), (4.16) and (4.17), we can state the following lemma:

Lemma 4.1

Let q be a vector field of class $C^1$ in $\overline{\Omega ^*}$. For all regular solution u to problem (3.1), we have the following identity:

$$\begin{aligned}&\left[ \int _{\Omega ^*}u_{t}\langle \nabla u,q\rangle d{\mathcal {M}}\right] _0^T+\frac{1}{2}\int _0^T\int _{\Omega ^*}div(q)[u_t^2-|\nabla u|^2-u^2] \ d{\mathcal {M}}dt \\&\quad +\int _0^T\int _{\Omega ^*}\nabla q(\nabla u,\nabla u) \ d{\mathcal {M}}dt +\int _0^T\int _{\Omega ^*}a(x)g(u_t)\langle q, \nabla u \rangle \ d{\mathcal {M}}dt \\&\quad = \int _0^T\int _{\partial \Omega ^*}\partial _\nu u\langle q, \nabla u \rangle \ d\Gamma dt + \frac{1}{2}\int _0^T\int _{\partial \Omega ^*}\langle q, \nu \rangle [u_t^2-|\nabla u|^2-u^2] \ d\Gamma dt. \end{aligned}$$

Exploiting Lemma 4.1 com $q=\nabla f$, where $f:\overline{\Omega ^*}\longrightarrow {\mathbb {R}}$ is a $C^\infty$ map to be determined later on, we obtain

$$\begin{aligned}&\left[ \int _{\Omega ^*}u_{t}\langle \nabla u,\nabla f\rangle d{\mathcal {M}}\right] _0^T+\frac{1}{2}\int _0^T\int _{\Omega ^*}\Delta f[u_t^2-|\nabla u|^2-u^2] \ d{\mathcal {M}}dt \nonumber \\&\qquad +\int _0^T\int _{\Omega ^*}\nabla ^2 f (\nabla u,\nabla u) \ d{\mathcal {M}}dt +\int _0^T\int _{\Omega ^*}a(x)g(u_t)\langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}dt\nonumber \\&\quad = \frac{1}{2}\int _0^T\int _{\partial \Omega ^*}\langle \nabla f, \nu \rangle [ u_t^2-|\nabla u|^2-u^2] \ d\Gamma dt\nonumber \\&\qquad +\int _0^T\int _{\partial \Omega ^*}\partial _\nu u\langle \nabla f, \nabla u \rangle \ d\Gamma dt. \end{aligned}$$

(4.18)

Lemma 4.2

Let u be a regular solution to problem (3.1) and $\alpha \in C^1(\overline{\Omega ^*})$. Thus

$$\begin{aligned} \left[ \int _{\Omega ^*}u_t\alpha u \ d{\mathcal {M}}\right] _0^T= & {} \int _0^T\int _{\Omega ^*}\alpha [u_t^2-|\nabla u|^2 - u^2] \ d{\mathcal {M}}dt - \int _0^T\int _{\Omega ^*}\langle \nabla u, \nabla \alpha \rangle u \ d{\mathcal {M}}dt\\&-\int _0^T\int _{\Omega ^*}a(x)g(u_t)\alpha u \ d{\mathcal {M}}dt + \int _0^T\int _{\partial \Omega ^*}\partial _\nu u \alpha u \ d\Gamma dt. \end{aligned}$$

Proof

Multiplying equation (3.1) by $\alpha u$ and performing an integration by parts we obtain the desired result. $\square$

Adding (4.18) and the identity given in Lemma 4.2 we deduce

$$\begin{aligned}&\int _0^T\int _{\Omega ^*}\left( \frac{\Delta f}{2}-\alpha \right) u_t^2 \ d{\mathcal {M}}dt +\int _0^T\int _{\Omega ^*}\nabla ^2 f\left( \nabla u,\nabla u\right) \ d{\mathcal {M}}dt \nonumber \\&\quad + \int _0^T\int _{\Omega ^*}\left( \alpha - \frac{\Delta f}{2}\right) |\nabla u|^2 \ d{\mathcal {M}}dt = \int _0^T\int _{\Omega ^*}\left( \frac{\Delta f}{2}-\alpha \right) u^2 \ d{\mathcal {M}}dt\nonumber \\&\quad -\left[ \int _{\Omega ^*}u_t\langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}\right] _0^T -\left[ \int _{\Omega ^*}u_t\alpha u \ d{\mathcal {M}}\right] _0^T \nonumber \\&\quad -\int _0^T\int _{\Omega ^*}a(x)g(u_t)\alpha u \ d{\mathcal {M}}dt -\int _0^T\int _{\Omega ^*}a(x)g(u_t)\langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}dt\nonumber \\&\quad + \int _0^T\int _{\partial \Omega ^*}\partial _\nu u \langle \nabla f, \nabla u \rangle \ d\Gamma dt \nonumber \\&\quad +\frac{1}{2}\int _0^T\int _{\partial \Omega ^*}\langle \nabla f, \nu \rangle \left[ u_t^2 - |\nabla u |^2 - u^2\right] \ d\Gamma dt \nonumber \\&\quad -\int _0^T\int _{\Omega ^*}\langle \nabla u, \nabla \alpha \rangle u \ d{\mathcal {M}}dt + \int _0^T\int _{\partial \Omega ^*}\partial _\nu u \alpha u \ d\Gamma dt. \end{aligned}$$

(4.19)

$\bullet$ $\mathbf {Cut}$-$\mathbf {off}$:

Let $\varepsilon >0$ be sufficiently small such that the tubular neighbourhood

$$\begin{aligned} \omega _{\varepsilon }=\left\{ x \in {\mathcal {M}}; \ d(x,\partial \Omega ^*)\le \frac{\varepsilon }{2} \right\} \end{aligned}$$

is contained in ${\mathcal {M}}\backslash \overline{\Omega }$. Let us consider $\psi \in {\mathcal {D}}({\mathcal {M}})$ such that (see Fig. 5)

• (i) $\psi =1 \ in \ \omega _{\frac{\varepsilon }{2}}:=\omega _{\varepsilon }\backslash \Omega ^*;$

• (ii) $\psi =0 \ in \ \overline{\Omega ^*}\backslash int(\omega '_{\frac{\varepsilon }{2}}),$ where $\omega '_{\frac{\varepsilon }{2}}:= \omega _{\varepsilon }\backslash \omega _{\frac{\varepsilon }{2}};$

• (iii) $0\le \psi \le 1 \ in \ {\mathcal {M}}.$

Taking $q = \psi \nabla f$ in Lemma 4.1, we infer,

$$\begin{aligned}&\left[ \int _{\Omega ^*}u_{t}\psi \langle \nabla u,\nabla f\rangle d{\mathcal {M}}\right] _0^T+\frac{1}{2}\int _0^T\int _{\Omega ^*}div(\psi \nabla f)\left[ u_t^2-|\nabla u|^2-u^2\right] \ d{\mathcal {M}}dt \\&\quad +\int _0^T\int _{\Omega ^*}\nabla (\psi \nabla f)(\nabla u,\nabla u) \ d{\mathcal {M}}dt +\int _0^T\int _{\Omega ^*}a(x)g(u_t)\langle \psi \nabla f, \nabla u \rangle \ d{\mathcal {M}}dt\\&\quad = \int _0^T\int _{\partial \Omega ^*}\partial _\nu u\langle \psi \nabla f, \nabla u \rangle \ d\Gamma dt + \frac{1}{2}\int _0^T\int _{\partial \Omega ^*}\langle \psi \nabla f, \nu \rangle \left[ u_t^2-|\nabla u|^2-u^2\right] \ d\Gamma dt, \end{aligned}$$

that is,

$$\begin{aligned}&\left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_{t}\psi \langle \nabla u,\nabla f\rangle d{\mathcal {M}}\right] _0^T+ \frac{1}{2}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\psi \Delta f[u_t^2-|\nabla u|^2-u^2] \ d{\mathcal {M}}dt\nonumber \\&\qquad +\frac{1}{2}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\langle \nabla f,\nabla \psi \rangle \left[ u_t^2-|\nabla u|^2-u^2\right] \ d{\mathcal {M}}dt\nonumber \\&\qquad + \int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\psi \nabla ^2 f(\nabla u, \nabla u) \ d{\mathcal {M}}dt +\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\langle \nabla u,\nabla \psi \rangle \langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}dt \nonumber \\&\qquad +\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)g(u_t)\psi \langle \nabla f ,\nabla u \rangle \ d{\mathcal {M}}dt\nonumber \\&\quad = \frac{1}{2}\int _0^T\int _{\partial \Omega ^*}\langle \nabla f, \nu \rangle \left[ u_t^2-|\nabla u|^2-u^2\right] \ d\Gamma dt\nonumber \\&\qquad +\int _0^T\int _{\partial \Omega ^*}\partial _\nu u\langle \nabla f, \nabla u \rangle \ d\Gamma dt. \end{aligned}$$

(4.20)

Fig. 5

Properties of function $\psi$

Substituting (4.20) in (4.19) it results that

$$\begin{aligned}&\int _0^T\int _{\Omega ^*}\left( \frac{\Delta f}{2}-\alpha \right) u_t^2 \ d{\mathcal {M}}dt +\int _0^T\int _{\Omega ^*}\nabla ^2 f(\nabla u,\nabla u) \ d{\mathcal {M}}dt + \int _0^T\int _{\Omega ^*}\left( \alpha - \frac{\Delta f}{2}\right) |\nabla u|^2 \ d{\mathcal {M}}dt\nonumber \\&\quad = \int _0^T\int _{\Omega ^*}\left( \frac{\Delta f}{2}-\alpha \right) u^2 \ d{\mathcal {M}}dt -\left[ \int _{\Omega ^*}u_t\langle \nabla f, \nabla u\rangle \ d{\mathcal {M}}\right] _0^T -\left[ \int _{\Omega ^*}u_t\alpha u \ d{\mathcal {M}}\right] _0^T \nonumber \\&\quad - \int _0^T\int _{\Omega ^*}a(x)g(u_t)\alpha u \ d{\mathcal {M}}dt- \int _0^T\int _{\Omega ^*}a(x)g(u_t)\langle \nabla f, \nabla u\rangle \ d{\mathcal {M}}dt \nonumber \\&\quad - \int _0^T\int _{\Omega ^*}\langle \nabla u, \nabla \alpha \rangle u \ d{\mathcal {M}}dt + \frac{1}{2}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\psi \Delta f\left[ u_t^2-|\nabla u|^2-u^2\right] \ d{\mathcal {M}}dt\nonumber \\&\quad + \frac{1}{2}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\langle \nabla f,\nabla \psi \rangle \left[ u_t^2-|\nabla u|^2-u^2\right] \ d{\mathcal {M}}dt +\left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_{t}\psi \langle \nabla u, \nabla f\rangle d{\mathcal {M}}\right] _0^T\nonumber \\&\quad + \int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\psi \nabla ^2 f(\nabla u, \nabla u) \ d{\mathcal {M}}dt+ \int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\langle \nabla u, \nabla \psi \rangle \langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}dt\nonumber \\&\quad + \int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)g(u_t)\psi \langle \nabla f ,\nabla u \rangle \ d{\mathcal {M}}dt+\int _0^T\int _{\partial \Omega ^*}\partial _\nu u \ \alpha \ u \ d\Gamma dt. \end{aligned}$$

(4.21)

Remark 4

This is the precise moment when the properties of the function f will be essential. Note that we need to find a subset $V \subset \overline{\Omega ^*}$ with regular smooth boundary $\overline{\partial _1 V}$ which intercepts $\partial \overline{\Omega ^*}$ transversally, in such way that $meas(V)\ge meas( \overline{\Omega ^*})- \varepsilon$ and $meas(V\cap \partial \overline{\Omega ^*})\ge meas(\partial \overline{\Omega ^*})- \varepsilon$, for all $\varepsilon > 0$. In addition, we need to find regular functions $\alpha , f: \overline{\Omega ^*}\longrightarrow \mathbb {R}$, $\alpha \ge 0$ and $\nabla \alpha \mid _{V}\equiv 0$ such that

$$\begin{aligned} C\int _0^T\int _V u_t^2 +|\nabla u|^2 \ d{\mathcal {M}}dt\le & {} \int _0^T\int _V(\frac{\Delta f}{2}-\alpha )u_t^2 \ d{\mathcal {M}}dt\nonumber \\&+\int _0^T\int _V \nabla ^2 f(\nabla u,\nabla u)+(\alpha -\frac{\Delta f}{2})|\nabla u|^2 \ d{\mathcal {M}}dt,\nonumber \\ \end{aligned}$$

(4.22)

for some positive constant C.

The construction of a smooth function f satisfying the Remark 4 can be found in [13]. The general idea is construct this function locally. Afterwards we glue them. The compactness of $\overline{\Omega ^*}$ is a crucial ingredient. We can put radially symmetric open sets satisfying some conditions inside V (and outside the damping region), remembering that we say that an open set $V\subset \overline{\Omega ^*}$ is radially symmetric with respect to $p\in V$ if the expression of the metric in polar coordinates $(r,\theta )=(r,\theta _1,\theta _2, \ldots , \theta _{n-1})$ centered in p is given by $ds^2=dr^2+Q^2(r)d\theta ^2$.

Thus we will omit the construction and quote the present theorem that guarantees the existence of such a function whose proof is given in detail in [13].

Theorem 4.2

Let $({\mathcal {M}}^n,\mathbf g)$ be a n-dimensional Riemannian manifold and let $\overline{\Omega ^*} \subset {\mathcal {M}}$ the compact subset mentioned before. Fix $\epsilon >0$. Then there exist an open subset $V\subset \overline{\Omega ^*}$ and smooth functions $\alpha , f: \overline{\Omega ^*} \rightarrow \mathbb {R}$ such that $meas(V)\ge meas ({\overline{\Omega ^*}})-\epsilon$, $meas(V\cap \partial \overline{\Omega ^*})\ge meas(\partial \overline{\Omega ^*})-\epsilon$, $\alpha \ge 0$, $\nabla \alpha \mid _V\equiv 0$ and

$$\begin{aligned}&C \int _0^T \int _{V}\left[ u_t^2 + |\nabla u|^2 \right] d{M}dt \le \int _{0}^{T}\int _{V}\left( \frac{\Delta f}{2}-\alpha \right) u_{t}^{2} \, d{{\mathcal {M}}} dt\\&\quad +\int _0^T \int _{V} \left[ \nabla ^2 f(\nabla u,\nabla u)+\left( \alpha -\frac{\Delta f}{2}\right) |\nabla u|^2\right] \,d{{\mathcal {M}}} dt, \end{aligned}$$

for some positive constant C.

Moreover if  $\overline{\Omega ^*}$ contains radially symmetric subsets, then we can choose V in such a way that a precise part of these radially symmetric subsets is contained in V, namely, the radially symmetric regions that can get rid of dissipative effects are those that satisfy the following condition

$$\begin{aligned} Q^\prime (r)\in \left[ \frac{2}{n}\left( \alpha +C\right) ,\frac{\alpha -C}{\frac{n}{2}-1}\right] \end{aligned}$$

(4.23)

(if $n=2$, $Q^\prime (r)$ does not need to satisfy any upper bound), where Q is function at the metric expression $ds^2=dr^2+Q^2(r)d\theta ^2.$

Once established that the Remark 4 is valid, we are able to complete the stabilization energy.

Taking into account that $0\le \psi \le 1$, we conclude, from (4.21), that

$$\begin{aligned}&C\int _0^T\int _{\Omega ^*} u_t^2 + |\nabla u|^2 \ d{\mathcal {M}}dt\le C^* \int _0^T\int _{\Omega ^*\backslash V} u_t^2 + |\nabla u|^2 \ d{\mathcal {M}}dt \nonumber \\&\quad +\left| \left[ \int _{\Omega ^*}u_t\langle \nabla f, \nabla u \rangle d{\mathcal {M}}\right] _0^T\right| + \ C_2\left| \left[ \int _{\Omega ^*}u_t u \ d{\mathcal {M}}\right] _0^T\right| \nonumber \\&\quad + C_1 \int _0^T\int _{\Omega ^*}u^2 \ d{\mathcal {M}}dt +C_2\int _0^T\int _{\Omega ^*}a(x)|g(u_t)||u| \ d{\mathcal {M}}dt\nonumber \\&\quad +\int _0^T\int _{\Omega ^*}a(x)|g(u_t)||\nabla f||\nabla u| \ d{\mathcal {M}}dt +\int _0^T\int _{\Omega ^*\backslash V}|\langle \nabla u, \nabla \alpha \rangle ||u| \ d{\mathcal {M}}dt \nonumber \\&\quad +\left| \left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_t\langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}\right] _0^T\right| +C_3\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}u_t^2 + |\nabla u|^2+u^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +C_4\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}|\nabla u|^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\,C_5\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}} |\nabla u|^2\ d{\mathcal {M}}dt +\frac{C_4}{2}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}u_t^2 + |\nabla u|^2+u^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)|g(u_t)||\nabla u||\nabla f| \ d{\mathcal {M}}dt +C_6\left| \int _0^T\int _{\partial \Omega ^*}\partial _\nu u \ u \ d\Gamma dt\right| ,\qquad \end{aligned}$$

(4.24)

where $C_1:=\displaystyle \max _{x \in \overline{\Omega ^*}}\left| \left( \alpha -\frac{\Delta f}{2}\right) (x)\right|$, $C_2:=\displaystyle \max _{x \in \overline{\Omega ^*}}\ \alpha (x)$, $C_3:=\displaystyle \max _{x \in \overline{\Omega ^*}}|\Delta f(x)|$,

$$\begin{aligned} C_4:=\displaystyle \max _{x \in \overline{\Omega ^*}}(|\nabla f(x)||\nabla \psi (x)|), \ \ C_5 \ \text{ is } \text{ such } \text{ that } \ |\nabla ^2 f(\nabla u,\nabla u)(x)|\le C_5|\nabla u(x)|^2, \end{aligned}$$

$\forall \ x \in \overline{\Omega ^*}, \ \ C_6:=\displaystyle \max _{x \in {\overline{\partial \Omega ^*}}}\ \alpha (x).$

Adding $C\int _0^T\int _{\Omega ^*}u^2 \ d{\mathcal {M}}dt$ in (4.24), we conclude that

$$\begin{aligned}&C\int _0^T\int _{\Omega ^*} u_t^2 + |\nabla u|^2 +u^2 \ d{\mathcal {M}}dt\le \displaystyle C^* \int _0^T\int _{\Omega ^*\backslash V} u_t^2 + |\nabla u|^2 \ d{\mathcal {M}}dt \nonumber \\&\quad +\int _0^T\int _{\Omega ^*\backslash V}|\langle \nabla u, \nabla \alpha \rangle ||u| \ d{\mathcal {M}}dt +\widetilde{C}a_0^{-1}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)u_t^2\ d{\mathcal {M}}dt \nonumber \\&\quad +\, \widetilde{C}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}|\nabla u|^2 \ d{\mathcal {M}}dt + \left| \left[ \int _{\Omega ^*}u_t\langle \nabla f,\nabla u\rangle \ d{\mathcal {M}}\right] _0^T\right| \nonumber \\&\quad + \,\widetilde{C}\,\left| \left[ \int _{\Omega ^*}u_t u \ d{\mathcal {M}}\right] _0^T\right| +\widetilde{C}\int _0^T\int _{\Omega ^*}u^2 \ d{\mathcal {M}}dt +\widetilde{C}\int _0^T\int _{\Omega ^*}a(x)|g(u_t)||u| \ d{\mathcal {M}}dt\nonumber \\&\quad +\, \widetilde{C}\int _0^T\int _{\Omega ^*}a(x)|g(u_t)||\nabla u| \ d{\mathcal {M}}dt +\frac{\widetilde{C}}{2}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}} u^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\left| \left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_t\langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}\right] _0^T\right| \nonumber \\&\quad +\,\widetilde{C}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)|g(u_t)||\nabla u| \ d{\mathcal {M}}dt + \widetilde{C}\left| \int _0^T\int _{\partial \Omega ^*}\partial _\nu u \ u \ d\Gamma dt\right| . \end{aligned}$$

(4.25)

In the sequel, let us analyze some terms in (4.25). We shall use Hölder inequality as well as the inequality $ab\le \frac{a^2}{4\beta }+\beta b^2$, where $\beta$ is an arbitrary positive number.

$\bullet$ Analysis of $J_1:= \int _0^T\int _{\Omega ^*}a(x)|g(u_t)||u| \ d{\mathcal {M}}dt.$

$$\begin{aligned} J_1\le & {} \int _0^T\left( \int _{\Omega ^*}a(x)|g(u_t)|^2 \ d{\mathcal {M}}\right) ^{\frac{1}{2}}\left( \int _{\Omega ^*}a(x)|u|^2 \ d{\mathcal {M}}\right) ^{\frac{1}{2}} dt \nonumber \\\le & {} \int _0^T\left[ \frac{1}{4\beta }\int _{\Omega ^*}a(x)|g(u_t)|^2 \ d{\mathcal {M}}+\beta \int _{\Omega ^*}a(x)|u|^2 \ d{\mathcal {M}}\right] dt \nonumber \\\le & {} \frac{1}{4\beta }\int _0^T\int _{\Omega ^*}a(x)|g(u_t)|^2 \ d{\mathcal {M}}dt + \Vert a\Vert _{\infty }2\beta \int _0^T \int _{\Omega ^*}\frac{1}{2}|u|^2 \ d{\mathcal {M}}dt\nonumber \\\le & {} \frac{1}{4\beta }\int _0^T\int _{{\mathcal {M}}}a(x)|g(u_t)|^2 \ d{\mathcal {M}}dt + \widehat{C}\beta \int _0^T E(t) \ dt, \end{aligned}$$

(4.26)

where E(t) is the energy defined in (3.2).

$\bullet$ Analysis of $J_2:=\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)|u_t|^2 \ d{\mathcal {M}}dt.$

$$\begin{aligned}&J_2\le \int _0^T\int _{{\mathcal {M}}}a(x)|u_t|^2 \ d{\mathcal {M}}dt \end{aligned}$$

(4.27)

$\bullet$ Analysis of $J_3:= \int _0^T\int _{\Omega ^*}a(x)|g(u_t)||\nabla u| \ d{\mathcal {M}}dt + \int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)|g(u_t)||\nabla u| \ d{\mathcal {M}}dt.$

$$\begin{aligned} J_3\le & {} 2\int _0^T\int _{\Omega ^*}a(x)|g(u_t)||\nabla u| \ d{\mathcal {M}}dt \nonumber \\\le & {} \frac{1}{2\beta }\int _0^T\int _{{\mathcal {M}}}a(x)|g(u_t)|^2 \ d{\mathcal {M}}dt + \widehat{C}\beta \int _0^T E(t) \ dt \end{aligned}$$

(4.28)

$\bullet$ Analysis of $J_4:= \left| \int _0^T\int _{\partial \Omega ^*}\partial _\nu u \ u \ d\Gamma dt\right| .$

Taking $\alpha =\psi$ in Lemma 4.2 we obtain

$$\begin{aligned} \left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_t\psi u \ d{\mathcal {M}}\right] _0^T= & {} \int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\psi \left[ u_t^2-|\nabla u|^2 - u^2\right] \ d{\mathcal {M}}dt\nonumber \\&-\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\langle \nabla u, \nabla \psi \rangle u \ d{\mathcal {M}}dt \nonumber \\&-\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)g(u_t)\psi u \ d{\mathcal {M}}dt + \int _0^T\int _{\partial \Omega ^*}\partial _\nu u \ u \ d\Gamma dt.\nonumber \\ \end{aligned}$$

(4.29)

Thus,

$$\begin{aligned} J_4\le & {} \left| \left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_t u \ d{\mathcal {M}}\right] _0^T\right| +\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}u_t^2 + |\nabla u|^2 \ d{\mathcal {M}}dt\nonumber \\&+\widetilde{C}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}\frac{|\nabla u|^2}{2}+ \frac{u^2}{2} \ d{\mathcal {M}}dt +\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}a(x)|g(u_t)||u| \ d{\mathcal {M}}dt \nonumber \\&+\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}u^2 \ d{\mathcal {M}}dt. \end{aligned}$$

(4.30)

Employing the same procedure we have followed in (4.26) yields:

$$\begin{aligned} J_4\le & {} \left| \left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_t u \ d{\mathcal {M}}\right] _0^T\right| +a_0^{-1}\int _0^T\int _{\Omega ^*}a(x)u_t^2 \ d{\mathcal {M}}dt\nonumber \\&+ \widehat{C}\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}|\nabla u|^2 \ d{\mathcal {M}}dt+ \widehat{C}\int _0^T\int _{\Omega ^*}u^2 \ d{\mathcal {M}}dt \nonumber \\&+\frac{1}{4\beta }\int _0^T\int _{{\mathcal {M}}}a(x)|g(u_t)|^2 \ d{\mathcal {M}}dt + \widehat{C}\beta \int _0^T E(t) \ dt \end{aligned}$$

(4.31)

Therefore, from (4.25), (4.26), (4.27), (4.28) and (4.31), setting

$$\begin{aligned} \chi:= & {} \left| \left[ \int _{\Omega ^*}u_t\langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}\right] _0^T\right| + \left| \left[ \int _{\Omega ^*}u_t u \ d{\mathcal {M}}\right] _0^T \right| \nonumber \\&+ \left| \left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_t\langle \nabla f, \nabla u \rangle \ d{\mathcal {M}}\right] _0^T\right| + \left| \left[ \int _{\omega '_{\frac{\varepsilon }{2}}}u_t u \ d{\mathcal {M}}\right] _0^T\right| , \end{aligned}$$

(4.32)

we conclude that

$$\begin{aligned}&C\int _0^T\int _{\Omega ^*}u_t^2+|\nabla u|^2+u^2\ d{\mathcal {M}}dt\le C'|\chi | + C'\beta \int _0^T E(t)dt\nonumber \\&\quad +\ C'\int _0^T\int _{\Omega ^*\backslash V}|\nabla u|^2+u^2 \ d{\mathcal {M}}dt +\ C^* \int _0^T\int _{\Omega ^*\backslash V} u_t^2 + |\nabla u|^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\ C'\int _0^T\int _{{\mathcal {M}}} a(x)[|g(u_t)|^2+u_t^2] \ d{\mathcal {M}}dt + C'\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}|\nabla u|^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\,C' \int _0^T\int _{\Omega ^*}u^2 \ d{\mathcal {M}}dt, \end{aligned}$$

(4.33)

where $C'=C'(\Vert a\Vert _{\infty }, f, \psi , \alpha , a_0^{-1})$.

In the sequel, we shall estimate the term $\int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}|\nabla u|^2 \ d{\mathcal {M}}dt$. For this purpose, we shall construct a “cut-off ” $\eta _\delta$ defined in an specific neighborhood of $\omega '_{\frac{\varepsilon }{2}}$ [see [13], Sect. 7, p. 955].

Initially let $\Omega ^{**}\subset {\mathcal {M}}$ be an open bounded set with smooth boundary $\partial \Omega ^{**}$ such that $\Omega \subset \subset \Omega ^*\subset \subset \Omega ^{**}$ and $\tilde{\eta }:\mathbb {R}\longrightarrow \mathbb {R}$ satisfying

$$\begin{aligned} \tilde{\eta }(x)=\left\{ \begin{array}{ccl} 1 &{} if &{} x\le 0\\ (x-1)^2 &{} if &{} x\in [1/2,1]\\ 0 &{} if &{} x>1\end{array}\right. \end{aligned}$$

and it is defined on (0, 1 / 2) in such way that $\tilde{\eta }$ is a non-increasing function of class $C^1$. For $\delta >0$, we set, $\tilde{\eta }_\delta (x):=\tilde{\eta }(\dfrac{x}{\delta })$. We observe that exists a constant M which does not depend on $\delta$, such that

$$\begin{aligned} \dfrac{|\tilde{\eta }'_\delta (x)|^2}{\tilde{\eta }_\delta (x)}\le \dfrac{M}{\delta ^2} \quad \text{ para } \text{ todo } x<\delta \end{aligned}$$

Now, let $\delta >0$ sufficiently small such that

$$\begin{aligned} \tilde{\omega }_\delta :=\big \{x\in \Omega ^{**}\,;\,d\big (x,\partial \omega '_{\frac{\varepsilon }{2}}\big )<\delta \big \} \end{aligned}$$

is totally contained in $\Omega ^{**}\backslash \Omega$, that is, $\tilde{\omega }_\delta$ is a tubular neighborhood of $\omega '_{\frac{\varepsilon }{2}}$ totally contained in $\Omega ^{**}\backslash \Omega$.

We define $\eta _\delta :\overline{\Omega ^{**}}\longrightarrow \mathbb {R}$ where (see Fig. 6)

$$\begin{aligned} \eta _\delta (x)=\left\{ \begin{array}{ccl} 1 &{} \text {if} &{} x\in \omega '_{\frac{\varepsilon }{2}}\\ \tilde{\eta }_\delta (d(x, \omega '_{\frac{\varepsilon }{2}})) &{} \text {if} &{} x\in \tilde{\omega }_\delta \\ 0 &{} &{} \text{ otherwise, } \end{array}\right. \end{aligned}$$

Fig. 6

Properties of function $\eta _\delta$

We have that $\eta _\delta$ is a $C^1$ function defined in $\overline{\Omega ^{**}}$, because $\partial \omega '_{\frac{\varepsilon }{2}}$ and $\partial (\tilde{\omega }_\delta \cup \omega '_{\frac{\varepsilon }{2}})$ are regular. We note also, that

$$\begin{aligned} \nabla \eta _\delta (x)=\nabla (\tilde{\eta }'_\delta (d(x,\omega '_{\frac{\varepsilon }{2}}))) =\tilde{\eta }_\delta '(d(x,\omega '_{\frac{\varepsilon }{2}}))\nabla d(x,\omega '_{\frac{\varepsilon }{2}}), \end{aligned}$$

(4.34)

soon

$$\begin{aligned} \dfrac{|\nabla \eta _\delta (x)|}{\eta _\delta (x)}^2 =\dfrac{|\tilde{\eta }'_\delta (d(x, \omega '_{\frac{\varepsilon }{2}})|^2}{\tilde{\eta }_\delta (d(x, \omega '_{\frac{\varepsilon }{2}}))} \le \dfrac{M}{\delta ^2} \end{aligned}$$

(4.35)

for all $x\in \tilde{\omega }_\delta$. Particularly, $\dfrac{|\nabla \eta _\delta (x)|}{\eta _\delta (x)}^2 \in L^\infty (\tilde{\omega }_\delta \cup \omega '_{\frac{\varepsilon }{2}})$.

Using Lemma 4.2 with $\eta _\delta = \alpha$ and $\Omega ^*=\Omega ^{**}$, we infer,

$$\begin{aligned} \left[ \int _{\Omega ^{**}}u_t\eta _\delta u \ d{\mathcal {M}}\right] _0^T= & {} \int _0^T\int _{\Omega ^{**}}\eta _\delta [u_t^2-|\nabla u|^2 - u^2] \ d{\mathcal {M}}dt - \int _0^T\int _{\Omega ^{**}}\langle \nabla u, \nabla \eta _\delta \rangle u \ d{\mathcal {M}}dt\\&-\int _0^T\int _{\Omega ^{**}}a(x)g(u_t)\eta _\delta u \ d{\mathcal {M}}dt + \int _0^T\int _{\partial \Omega ^{**}}\partial _\nu u \eta _\delta u \ d\Gamma dt . \end{aligned}$$

Thus, defining, $V_{\delta }:=\tilde{\omega }_{\delta }\cup \omega '_{\frac{\varepsilon }{2}}$ it holds that

$$\begin{aligned} \left[ \int _{V_{\delta }}u_t\eta _\delta u \ d{\mathcal {M}}\right] _0^T= & {} \int _0^T\int _{V_{\delta }}\eta _\delta [u_t^2-|\nabla u|^2 - u^2] \ d{\mathcal {M}}dt - \int _0^T\int _{V_{\delta }}\langle \nabla u, \nabla \eta _\delta \rangle u \ d{\mathcal {M}}dt\nonumber \\&-\int _0^T\int _{V_{\delta }}a(x)g(u_t)\eta _\delta u \ d{\mathcal {M}}dt. \end{aligned}$$

(4.36)

$\bullet$ Estimate for $K_1:= \int _0^T\int _{V_{\delta }}\eta _\delta |u_t|^2 \ d{\mathcal {M}}dt$

$$\begin{aligned} |K_1|\le a_0^{-1}\int _0^T\int _{V_{\delta }}a(x)|u_t|^2 \ d{\mathcal {M}}dt\le a_0^{-1}\int _0^T\int _{{\mathcal {M}}}a(x)|u_t|^2 \ d{\mathcal {M}}dt. \end{aligned}$$

(4.37)

$\bullet$ Estimate for $K_2:= \int _0^T\int _{V_{\delta }}a(x)g(u_t)\eta _\delta u \ d{\mathcal {M}}dt$

$$\begin{aligned} |K_2|\le \frac{C}{4\beta }\int _0^T\int _{{\mathcal {M}}}a(x)|g(u_t)|^2 \ d{\mathcal {M}}dt+ 2\beta \int _0^T E(t)dt. \end{aligned}$$

(4.38)

$\bullet$ Estimate for $K_3:= \int _0^T\int _{V_{\delta }}u\langle \nabla u, \nabla \eta _\delta \rangle \ d{\mathcal {M}}dt$

$$\begin{aligned} |K_3|\le & {} \int _0^T\int _{V_{\delta }}|u||\nabla u| |\nabla \eta _\delta | \ d{\mathcal {M}}dt\nonumber \\\le & {} \int _0^T\int _{V_{\delta }}\frac{1}{2}\eta _\delta |\nabla u|^2 \ d{\mathcal {M}}dt + \int _0^T\int _{V_{\delta }}\frac{1}{2}\frac{|\nabla \eta _\delta |^2}{\eta _\delta }|u|^2| \ d{\mathcal {M}}dt \nonumber \\\le & {} \frac{1}{2}\int _0^T\int _{V_{\delta }}\eta _\delta |\nabla u|^2 \ d{\mathcal {M}}dt + \frac{M}{2\delta ^2}\int _0^T\int _{V_{\delta }}|u|^2 \ d{\mathcal {M}}dt. \end{aligned}$$

(4.39)

Denoting $\chi _1:= -\left[ \int _{V_\delta }u_t\eta _\delta u \ d{\mathcal {M}}\right] _0^T$ it results that, according to (4.36), (4.37), (4.38) and (4.39), that

$$\begin{aligned}&\frac{1}{2}\int _0^T\int _{V_{\delta }}\eta _\delta |\nabla u|^2 \ d{\mathcal {M}}dt\le |\chi _1| + \tilde{C}\int _0^T\int _{\mathcal {M}}a(x)u_t^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\,\frac{\tilde{C}}{4\beta }\int _0^T\int _{\mathcal {M}}a(x) |g(u_t)|^2 \ d{\mathcal {M}}+ \tilde{C}\int _0^T\int _{\Omega ^{**}}|u|^2 \ d{\mathcal {M}}dt+\ 2\beta \int _0^T E(t)dt.\nonumber \\ \end{aligned}$$

(4.40)

We observe that

$$\begin{aligned} \int _0^T\int _{\omega '_{\frac{\varepsilon }{2}}}|\nabla u|\ d{\mathcal {M}}dt\le \int _0^T\int _{V_\delta }\eta _\delta |\nabla u|^2 \ d{\mathcal {M}}dt, \end{aligned}$$

(4.41)

remembering that $V_{\delta }:=\tilde{\omega }_{\delta }\cup \omega '_{\frac{\varepsilon }{2}}.$

Consequently, substituting (4.40) in (4.33), taking (4.41) into account, grouping some terms and noting that

$$\begin{aligned} \int _0^T\int _{\Omega ^*\backslash V}u^2 \ d{\mathcal {M}}dt\le \int _0^T\int _{\Omega ^{**}}u^2 \ d{\mathcal {M}}dt \end{aligned}$$

we conclude that

$$\begin{aligned}&C\int _0^T\int _{\Omega ^*}u_t^2 + |\nabla u|^2 + u^2 \ d{\mathcal {M}}dt\le C'|\chi |+\widehat{C}|\chi _1|\nonumber \\&\quad +\,\widehat{C}\int _0^T\int _{\mathcal {M}}a(x)[g(u_t)|^2+u_t^2] \ d{\mathcal {M}}dt +\widehat{C}\int _0^T\int _{\Omega ^{**}}u^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\, C^*_1\int _0^T\int _{\Omega ^*\backslash V}|\nabla u|^2+u_t^2 \ d{\mathcal {M}}dt +\, \widehat{C} \beta \int _0^T E(t)dt. \end{aligned}$$

(4.42)

where $\widehat{C}= \widehat{C}(\Vert a\Vert _\infty , a_0^{-1}, C', \frac{M}{\delta ^2}, \psi , f, \alpha ).$

We will now estimate the term $\int _0^T\int _{\Omega ^*\backslash V}|\nabla u|^2 \ d{\mathcal {M}}dt$.

Let ${\mathcal {M}}_*\subset {\mathcal {M}}$ a open subset, so that $\overline{\Omega ^*}\backslash V\subset {\mathcal {M}}_* \subset \subset \Omega ^{**}$. Note that by hypothesis $a(x)\ge a_0>0$ in ${\mathcal {M}}_*$.

Consider $\theta >0$ sufficiently small such that

$$\begin{aligned} \widetilde{\omega }_\theta :=\{x\in \Omega ^{**}; d(x,\partial V)< \theta \}\subset {\mathcal {M}}_*, \end{aligned}$$

$\widetilde{\omega }_\theta$ is a tubular neighborhood of $\partial V$ that has only one connected component.

Proceeding similarly to what was done for estimating the term $\int _0^T\int _{\omega '_{\frac{\epsilon }{2}}}|\nabla u|^2 \ d{\mathcal {M}}dt$, we conclude that

$$\begin{aligned}&\frac{1}{2}\int _0^T\int _{\Omega ^*\backslash V}|\nabla u|^2 \ d{\mathcal {M}}dt\le |\chi _2| + C_2\int _0^T\int _{\mathcal {M}}a(x)u_t^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\,\frac{C_2}{4\beta }\int _0^T\int _{\mathcal {M}}a(x) |g(u_t)|^2 \ d{\mathcal {M}}+ C_2\int _0^T\int _{\Omega ^{**}}|u|^2 \ d{\mathcal {M}}dt+\ 2\beta \int _0^T E(t)dt.\nonumber \\ \end{aligned}$$

(4.43)

where $\omega _\theta :=\widetilde{\omega }_\theta \cup \Omega ^*\backslash V \subset {\mathcal {M}}_*$ and $\chi _2:= -\left[ \int _{\omega _\theta }u_t\eta _\theta u \ d{\mathcal {M}}\right] _0^T.$

Therefore, from (4.43) and (4.42), we can write

$$\begin{aligned}&C\int _0^T\int _{\Omega ^*}u_t^2 + |\nabla u|^2 + u^2 \ d{\mathcal {M}}dt\le C'|\chi |+ \widehat{C}|\chi _1|+ C_3|\chi _2|\nonumber \\&\quad +\ C_3\int _0^T\int _{\mathcal {M}}a(x)[g(u_t)|^2+u_t^2]\ d{\mathcal {M}}dt +C_3\int _0^T\int _{\Omega ^{**}}u^2 \ d{\mathcal {M}}dt\nonumber \\&\quad +\ C_3 \beta \int _0^T E(t)dt. \end{aligned}$$

(4.44)

The estimate (4.44) is almost what we want. It is necessary to estimate the term $\int _0^T\int _{\Omega ^{**}}u^2 \ d{\mathcal {M}}dt$ as a function of “good terms”, this is the next step, already with the total energy of the system recovered.

4.3 Recovering the Total Energy

Multiplying (4.12) by C and adding with (4.44), we obtain the following inequality:

$$\begin{aligned}&C\int _0^T\int _{\mathcal {M}}u_t^2+ |\nabla u|^2 + u^2 \ d{\mathcal {M}}dt\le C\left| \left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T\right| \nonumber \\&\quad +\, C'|\chi |+\widehat{C} |\chi _1|+ C_3|\chi _2|+ C_4\int _0^T\int _{\mathcal {M}}a(x)(|g(u_t)|^2 + u_t^2) \ d{\mathcal {M}}dt \nonumber \\&\quad +\ C\int _0^T\int _{{\mathcal {M}}\backslash \Omega }a(x)|g(u_t)||u| \varphi \ d{\mathcal {M}}dt + C_4\int _0^T\int _{\Omega ^{**}}u^2 \ d{\mathcal {M}}dt \nonumber \\&\quad +\ C_3 \beta \int _0^T E(t)dt. \end{aligned}$$

(4.45)

$\bullet$ Estimate for the term $\int _0^T\int _{{\mathcal {M}}\backslash \Omega }a(x)|g(u_t)||u| \varphi \ d{\mathcal {M}}dt.$

$$\begin{aligned} \int _0^T\int _{{\mathcal {M}}\backslash \Omega }a(x)|g(u_t)||u| \varphi \ d{\mathcal {M}}dt\le & {} \int _0^T\left( \int _{{\mathcal {M}}} a(x)|g(u_t)|^2 \ d{\mathcal {M}}\right) ^{\frac{1}{2}} \left( \int _{{\mathcal {M}}} a(x)|u|^2 \ d{\mathcal {M}}\right) ^{\frac{1}{2}} dt\nonumber \\\le & {} \frac{1}{4\beta }\int _0^T\int _{{\mathcal {M}}}a(x)|g(u_t)|^2 \ d{\mathcal {M}}dt + C_1\beta \int _0^T E(t) dt\nonumber \\ \end{aligned}$$

(4.46)

Thus by (4.45) it follows that

$$\begin{aligned}&2C\int _0^T E(t)dt\le C \left| \left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T\right| + C'|\chi |+\widehat{C} |\chi _1|+ C_3|\chi _2|\nonumber \\&\quad +\,\tilde{C}\int _0^T\int _{\mathcal {M}}a(x)(|g(u_t)|^2 + u_t^2) \ d{\mathcal {M}}dt + \tilde{C}\int _0^T\int _{\Omega ^{**}}u^2 \ d{\mathcal {M}}dt\nonumber \\&\quad + \, \tilde{C}\beta \int _0^T E(t) dt . \end{aligned}$$

(4.47)

where $\tilde{C}=\tilde{C}(\Vert a\Vert _\infty , a_0^{-1}, C, C',C_3, C_4, \frac{M}{\delta ^2}, \psi , f, \alpha , \varphi ).$

Next, we shall estimate $|\chi |$, $|\chi _1|$, $|\chi _2|$ and $\left| \left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T\right|$.

Using the identity (4.1), we have that

$$\begin{aligned} |\chi _1|= & {} \left| \left[ \int _{V_\delta }u_t \eta _\delta u \ d{\mathcal {M}}\right] _0^T\right| \\\le & {} C_1\left[ \int _{{\mathcal {M}}}|u_t(T,x)|| u(T,x)| \ d{\mathcal {M}}+ \int _{{\mathcal {M}}}|u_t(0,x)|| u(0,x)| \ d{\mathcal {M}}\right] \\\le & {} C_1(E(T)+E(0))\\= & {} C_1\left( 2E(T) + \int _0^T\int _{\mathcal {M}}a(x)g(u_t)u_t \ d{\mathcal {M}}dt\right) \\\le & {} 2C_1 E(T) + C_2\int _0^T\int _{\mathcal {M}}a(x)\left[ |g(u_t)|^2+|u_t|^2\right] \ d{\mathcal {M}}dt. \end{aligned}$$

Making use of analogous arguments we conclude that

$$\begin{aligned}&C'|\chi |+\widehat{C}|\chi _1|+ C_3|\chi _2| + C\left| \left[ \int _{\mathcal {M}}u_t \varphi u \ d{\mathcal {M}}\right] _0^T\right| \le C_0E(T) \nonumber \\&\quad +\, C_0\int _0^T\int _{\mathcal {M}}a(x)\left[ |g(u_t)|^2+|u_t|^2\right] \ d{\mathcal {M}}dt. \end{aligned}$$

(4.48)

where $C_0$ does not depend on T.

It remains in (4.47), to estimate the term $\int _0^T\int _{\Omega ^{**}}u^2 \ d{\mathcal {M}}dt$. For this purpose, we need the following lemma:

Lemma 4.3

Let u be a regular solution to problem (3.1). Thus, for all $T>T_0$, where $T_0$ is a positive constant large enough, there exists a positive constant $C(T_0,E(0))$ such that the following inequality holds

$$\begin{aligned} \int _0^T\int _{\overline{\Omega ^{**}}} u^2 \ d{\mathcal {M}}dt \le C(T_0,E(0))\left\{ \int _0^T\int _{\mathcal {M}}a(x)\left[ |g(u_t)|^2 + |u_t|^2\right] \ d{\mathcal {M}}dt\right\} . \end{aligned}$$

(4.49)

Proof

We argue by contradiction. To ease notations, we will set $u':=u_t$. Assume that (4.49) is not verified. Then there exists a sequence of initial data $\{u_k(0), u'_k(0)\}$, such that the corresponding regular solutions $\{u_k\}$ of (3.1) with $E_k(0)$ assumed uniformly bounded in k, verifies

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _0^T\int _{\overline{\Omega ^{**}}}\ {u}_k^2 \ d{\mathcal {M}}dt}{\int _0^T\int _{{\mathcal {M}}} a(x)(g^2(u'_k)+{u'}_k^2)\ d{\mathcal {M}}dt}=+ \infty , \end{aligned}$$

(4.50)

this is,

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _0^T\int _{{\mathcal {M}}} a(x)(g^2(u'_k)+{u'}_k^2)\ d{\mathcal {M}}dt}{\int _0^T\int _{\overline{\Omega ^{**}}}\ {u}_k^2 \ d{\mathcal {M}}dt}= 0 \end{aligned}$$

(4.51)

Since the energy is a non-increasing function, it holds that $E_k(t)\le E_k(0)\le L$, consequently,

$$\begin{aligned} \int _{\mathcal {M}}{u'}_k^2(x,t)+|\nabla u_k(x,t)|^2+ u_k^2(x,t)\ d{\mathcal {M}}\le 2L, \end{aligned}$$

that is,

$$\begin{aligned} \int _0^T\int _{\mathcal {M}}{u'}_k^2(x,t)+|\nabla u_k(x,t)|^2+ u_k^2(x,t)\ d{\mathcal {M}}dt\le 2LT, \end{aligned}$$

thus,

$$\begin{aligned} \Vert u_k\Vert ^2_{H^1(\Sigma _T)}\le 2LT, \end{aligned}$$

(4.52)

where $\Sigma _T = (0,T)\times {\mathcal {M}}.$

In addition,

$$\begin{aligned} \int _{\mathcal {M}}|\nabla u_k(x,t)|^2+ u_k^2(x,t)\ d{\mathcal {M}}\le 2L \ \ \text{ and } \ \ \int _{\mathcal {M}}{u'}_k^2(x,t) d{\mathcal {M}}\le 2L, \end{aligned}$$

consequently

$$\begin{aligned}&\Vert u_k\Vert ^2_{L^\infty (0,T;H^1({\mathcal {M}}))}\le 2L; \end{aligned}$$

(4.53)

$$\begin{aligned}&\Vert {u'}_k\Vert ^2_{L^\infty (0,T;L^2({\mathcal {M}}))}\le 2L. \end{aligned}$$

(4.54)

Thus, there exists a subsequence of $\{u_k\}$, still denote by the same index, which verifies the following convergence

$$\begin{aligned}&u_k\rightharpoonup u \text{ in } H^1(\Sigma _T); \end{aligned}$$

(4.55)

$$\begin{aligned}&u_k\mathop {\rightharpoonup }\limits ^{\star } u \text{ in } L^\infty (0,T;H^1({\mathcal {M}}));\end{aligned}$$

(4.56)

$$\begin{aligned}&u_k'\mathop {\rightharpoonup }\limits ^{\star } u' \text{ in } L^\infty (0,T;L^2(\mathcal {M})). \end{aligned}$$

(4.57)

From the fact that $H^1(\Omega ^{**})\mathop {\hookrightarrow }\limits ^{c}L^2(\Omega ^{**})$ and both spaces are reflexive, it holds from (4.53), (4.54) and employing Aubin–Lions Theorem, that

$$\begin{aligned} u_k\longrightarrow u \text{ in } L^2(0,T;L^2(\Omega ^{**})). \end{aligned}$$

(4.58)

We note that u does not depend on $\Omega ^{**}$ in view of (4.55).

At this point we shall divide the proof into two cases, namely: $u\ne 0$ e $u=0$.

(i) $u\ne 0$ em $(0,T)\times {\mathcal {M}}$.

We observe that taking (4.58) into account, we have

$$\begin{aligned} \Vert u_k\Vert ^2_{L^2(0,T;L^2(\Omega ^{**}))}\le C. \end{aligned}$$

Thus, considering (4.51) we obtain

$$\begin{aligned} \lim _{k\rightarrow +\infty }\int _0^T\int _{\mathcal {M}}a(x)(g^2(u_k')+{u'}_k^2) \ d{\mathcal {M}}dt=0 \end{aligned}$$

(4.59)

and from (4.59) we conclude that

$$\begin{aligned}&\lim _{k\rightarrow +\infty }\int _0^T\int _{\mathcal {M}}|a(x) g(u_k')|^2 \ d{\mathcal {M}}dt \\&\quad \le \Vert a\Vert _{\infty }\lim _{k\rightarrow +\infty }\int _0^T\int _{\mathcal {M}}a(x)(g^2(u_k')+{u'}_k^2) \ d{\mathcal {M}}dt = 0, \end{aligned}$$

that is

$$\begin{aligned} ag(u'_k)\longrightarrow 0\ \text{ in } \ L^2(0,T;L^2({\mathcal {M}})). \end{aligned}$$

(4.60)

Our aim is taking limit in the equation

$$\begin{aligned} \left\{ \begin{array}{ll} {u^{''}_k-\Delta u_k +u_k +a(x)g(u^{'}_k)=0} &{} {in\,\,\,{\mathcal {M}}\times (0,\infty )} \\ {u_k(0)=u_{0,k}}\,\ ,\,\,u^{'}_k(0)=u_{1,k} &{} {in\,\,\,{\mathcal {M}}} \\ \end{array}\right. \end{aligned}$$

(4.61)

We observe that according to (4.59) we infer

$$\begin{aligned} \int _0^T\int _{({\mathcal {M}}\backslash \Omega )\cup {\mathcal {M}}_*}{u'}_k^2 \ d{\mathcal {M}}dt\le & {} a_0^{-1}\int _0^T\int _{({\mathcal {M}}\backslash \Omega )\cup {\mathcal {M}}_*}a(x) {u'}_k^2 \ d{\mathcal {M}}dt \\\le & {} a_0^{-1}\int _0^T\int _{{\mathcal {M}}}a(x) {u'}_k^2 \ d{\mathcal {M}}dt\rightarrow 0, \end{aligned}$$

when $k \rightarrow \infty$.

Thus, from (4.57) it follows, for a. e. $t\in [0,T]$, $u'(t)$ satisfies

$$\begin{aligned} u'(t)=\left\{ \begin{array}{ll} {u'(t)} &{} {in\,\,\,\overline{\Omega }} \\ {0} &{} {in\,\,\,({\mathcal {M}}\backslash \Omega )\cup {\mathcal {M}}_*} \\ \end{array}\right. \end{aligned}$$

(4.62)

Passing to the limit, we deduce

$$\begin{aligned} \left\{ \begin{array}{ll} {u^{''}-\Delta u +u =0} &{} {in\,\,\,L^\infty (0,T;H^{-1}({\mathcal {M}}))} \\ {u'=0} &{} {in\,\,\,({\mathcal {M}}\backslash \Omega )\cup {\mathcal {M}}_*} \\ \end{array}\right. \end{aligned}$$

(4.63)

with $u\in L^\infty (0,T;H^1({\mathcal {M}}))$ e $u'\in L^\infty (0,T;L^2({\mathcal {M}}))$.

Considering $\Omega ^*\supset \supset \Omega$ satisfying the conditions mentioned before and setting $v=u'$, from (4.63) we obtain the following problem

$$\begin{aligned} \left\{ \begin{array}{ll} {v^{''}-\Delta v +v =0} &{} {in\,\,\,\Omega ^*\times (0,T)} \\ {v=0} &{} {in\,\,\,(\overline{\Omega ^*}\backslash \Omega )\cup {\mathcal {M}}_*}\times (0,T) \\ \end{array}\right. \end{aligned}$$

(4.64)

in the distributional sense, with $v\in L^\infty (0,T;L^2({\mathcal {M}}))$ e $v'\in L^\infty (0,T;H^{-1}({\mathcal {M}}))$. In addition, employing Lions-Magenes ([30], Chap. 3, Theor. 9.3, p. 288), we have

$$\begin{aligned} v\in C([0,T];L^2({\mathcal {M}}))\cap C^1([0,T];H^{-1}({\mathcal {M}})). \end{aligned}$$

From Appendix, we deduce that $v=u'=0$ em ${\mathcal {M}}$.

It follows that (4.63) can be written as

$$\begin{aligned} \left\{ \begin{array}{ll} {-\Delta u +u =0} &{} {in\,\,\,{\mathcal {M}}\times (0,\infty )} \\ {u'=0} &{} {in\,\,\,{\mathcal {M}}} \\ \end{array}\right. \end{aligned}$$

(4.65)

Since $u\in L^\infty (0,T;H^1({\mathcal {M}}))$ we conclude that $\Delta u \in L^\infty (0,T;H^1({\mathcal {M}}))$, then $u(t)\in H^1({\mathcal {M}})$ and $\Delta u(t)\in H^1({\mathcal {M}})\hookrightarrow L^2({\mathcal {M}})$. Multiplying (4.65) by u(t) and integrating over ${\mathcal {M}}$, we deduce

$$\begin{aligned} \int _{\mathcal {M}}-\Delta u(t) u(t) \ d{\mathcal {M}}+\int _{\mathcal {M}}|u(t)|^2 \ d{\mathcal {M}}= 0, \end{aligned}$$

that is,

$$\begin{aligned} \int _{\mathcal {M}}|\nabla u(t)|^2 \ d{\mathcal {M}}+\int _{\mathcal {M}}|u(t)|^2 \ d{\mathcal {M}}= 0, \end{aligned}$$

Consequently $u(t)=0$ in $H^1({\mathcal {M}})$ for a. e. $t\in [0,T]$, that is $u=0$, which is a contradiction.

(ii) $u=0$ em $(0,T)\times {\mathcal {M}}$.

We define

$$\begin{aligned}&c_k:=\left[ \int _0^T\int _{\overline{\Omega ^{**}}}|u_k|^2\ d\mathcal {M}dt\right] ^{\frac{1}{2}} \end{aligned}$$

(4.66)

$$\begin{aligned}&\bar{u}_k:=\dfrac{1}{c_k}u_k \end{aligned}$$

(4.67)

Thus, we deduce

$$\begin{aligned} \left\| \bar{u}_k\right\| ^2_{L^2(0,T;L^2(\Omega ^{**}))}=1 \end{aligned}$$

(4.68)

If one considers

$$\begin{aligned} \overline{E}_k(t):=\dfrac{1}{2}\displaystyle \int _{\mathcal {M}}\left| \bar{u}_k'\right| ^2 +\left| \bar{u}_k\right| ^2+\left| \nabla \bar{u}_k\right| ^2d\mathcal {M}, \end{aligned}$$

(4.69)

then

$$\begin{aligned} \overline{E}_k(t)=\dfrac{E_k(t)}{c_k^2} \end{aligned}$$

(4.70)

According to (4.47) and (4.48) it holds, for T sufficiently large that

$$\begin{aligned} E(T)\le \widehat{C}\left[ \int _0^T\int _{\mathcal {M}}a(x)(g^2(u')+{u'}^2) \ d{\mathcal {M}}dt +\int _0^T\int _{\overline{\Omega ^{**}}}u^2 \ d{\mathcal {M}}dt\right] , \end{aligned}$$

(4.71)

and according to the identity of energy that follows

$$\begin{aligned} E(0)= & {} E(T)+\int _0^T\int _{\mathcal {M}}a(x)g(u')u' \ d{\mathcal {M}}dt\\\le & {} E(T) + \int _0^T\int _{\mathcal {M}}a(x)(g^2(u')+{u'}^2) \ d{\mathcal {M}}dt. \end{aligned}$$

Thus,

$$\begin{aligned} E(t)\le E(0) \le \tilde{C}\left[ \int _0^T\int _{\mathcal {M}}\big (a(x)g^2(u')+a(x){u'}^2\big )\ d\mathcal {M}dt +\int _0^T\int _{\overline{\Omega ^{**}}}|u|^2d\mathcal {M}dt\right] \end{aligned}$$

for all $t\in (0,T)$, with T sufficiently large.

From the last inequality and taking (4.70) into account, we obtain,

$$\begin{aligned} \overline{E}_k(t)=\dfrac{E_k(t)}{c_k^2}\le \left[ \dfrac{\int _0^T\int _{\mathcal {M}}\big (a(x)g^2(u_k') +a(x)u_k'^2\big )d\mathcal {M}dt}{\int _0^T\int _{\overline{\Omega ^{**}}}|u_{k}|^2d\mathcal {M}dt}+1\right] \end{aligned}$$

(4.72)

Then, from (4.51) and (4.72), we can guarantee the existence of a constant $L_1>0$ such that

$$\begin{aligned} \overline{E}_k(t)\le L_1\,,\,\forall t\in [0,T],\,\forall k\in \mathbb {N} \end{aligned}$$

Analogously we have done to the case (i), we obtain a subsequence such that

$$\begin{aligned}&\bar{u}_k\rightharpoonup \bar{u} \text{ in } H^1(\Sigma _T); \end{aligned}$$

(4.73)

$$\begin{aligned}&\bar{u}_k\mathop {\rightharpoonup }\limits ^{\star } \bar{u} \text{ in } L^\infty (0,T;H^1({\mathcal {M}})); \end{aligned}$$

(4.74)

$$\begin{aligned}&\bar{u}_k'\mathop {\rightharpoonup }\limits ^{\star } \bar{u'} \text{ in } L^\infty (0,T;L^2(\mathcal {M})), \end{aligned}$$

(4.75)

and, therefore, making use of compactness arguments, it holds that

$$\begin{aligned} \bar{u}_k\longrightarrow \bar{u} \text{ in } L^2(0,T;L^2(\Omega ^{**})) \end{aligned}$$

(4.76)

Furthermore, we observe that from (4.51)

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _0^T\int _{{\mathcal {M}}} a(x)g^2(u'_k)\ d{\mathcal {M}}dt}{c_k^2}= 0 \end{aligned}$$

(4.77)

and

$$\begin{aligned} \int _0^T\int _{({\mathcal {M}}\backslash \Omega )\cup {\mathcal {M}}_*}\bar{u}_k'^2 \ d{\mathcal {M}}dt\le & {} a_0^{-1}\int _0^T\int _{({\mathcal {M}}\backslash \Omega )\cup {\mathcal {M}}_*}a(x)\bar{u}_k'^2 \ d{\mathcal {M}}dt\nonumber \\\le & {} a_0^{-1}\int _0^T\int _{{\mathcal {M}}}a(x)\bar{u}_k'^2 \ d{\mathcal {M}}dt\longrightarrow 0 \end{aligned}$$

(4.78)

We have that $\bar{u}_k$ satisfies

$$\begin{aligned} \bar{u}''_k-\Delta \bar{u}_k + \bar{u}_k +a(x)\frac{g(u'_k)}{c_k}= 0 \ \ \text{ in }\ \ {\mathcal {M}}\times (0,T) \end{aligned}$$

(4.79)

Thus, passing to the limit in (4.79) when $k\rightarrow \infty$ we obtain, from (4.73), (4.74), (4.75), (4.76), (4.77) and (4.78), that

$$\begin{aligned} \left\{ \begin{array}{rl} \bar{u}''-\Delta \bar{u}+\bar{u}=0&{} \text{ in } \mathcal {M}\times (0,T)\\ \\ \bar{u}'=0&{} \text{ in } ({\mathcal {M}}\backslash \Omega )\cup {\mathcal {M}}_* \times (0,T). \end{array}\right. \end{aligned}$$

(4.80)

Using the same arguments employed in the first case (i),we conclude that $\bar{u}=0$ in ${\mathcal {M}}\times (0,T)$, which is a contradiction having in mind (4.68) and (4.76).

This is finishes the proof of the lemma. $\square$

Thus, from (4.47), (4.48) and Lemma 4.3 we obtain the following

$$\begin{aligned} 2C\int _0^T E(t)dt\le & {} C_0E(T) + \tilde{C_1}\int _0^T\int _{\mathcal {M}}a(x)(|g(u_t)|^2 + u_t^2) \ d{\mathcal {M}}dt \nonumber \\&+ \tilde{C}\beta \int _0^T E(t) dt . \end{aligned}$$

(4.81)

for all $T\ge T_0$, $T_0$ sufficiently large such that $T_0(2C-\tilde{C}\beta )-C_0>0$. Note that $C_0$ not depend on T.

Considering $\beta$ sufficiently small so that $\tilde{C_2}:=2C-\tilde{C}\beta >0$ we obtain

$$\begin{aligned} \tilde{C_2}\int _0^T E(t)dt\le C_0E(T) + \tilde{C_1}\int _0^T\int _{\mathcal {M}}a(x)(|g(u_t)|^2 + u_t^2) \ d{\mathcal {M}}dt, \end{aligned}$$

(4.82)

for all $T\ge T_0$.

Observe that, for all $T\ge T_0$ and $t\in [0,T_0]$ we have

$$\begin{aligned} E(T)\le E(T_0)\le E(t)\Longrightarrow T_0E(T)\le \int _0^{T_0}E(t)dt\le \int _0^T E(t)dt \end{aligned}$$

Then

$$\begin{aligned} \tilde{C_2}T_0E(T)\le & {} \tilde{C_2}\int _0^TE(t)dt\nonumber \\\le & {} C_0E(T) + \tilde{C_1}\int _0^T\int _{\mathcal {M}}a(x)(|g(u_t)|^2 + u_t^2) d{\mathcal {M}}dt \end{aligned}$$

(4.83)

Therefore

$$\begin{aligned} E(T) \le \frac{\tilde{C_1}}{\tilde{C_2}T_0-C_0}\int _0^T\int _{\mathcal {M}}a(x)(|g(u_t)|^2 + u_t^2) \ d{\mathcal {M}}dt, \end{aligned}$$

which leads us to the following result:

Proposition 4.1

For $T\ge T_0$, $T_0$ sufficiently large, the solution u to problem (3.1) satisfies

$$\begin{aligned} E(T)\le C\int _0^T\int _{\mathcal {M}}\big [a(x)|u_t|^2+a(x)|g(u_t)|^2\big ]d\mathcal {M}dt \end{aligned}$$

(4.84)

where the constant $C=C\big (T_0,E(0),\Vert a\Vert _{L^\infty (\mathcal {M})},\alpha ,f,\psi ,a_0,M/\delta ^2\big )$.

Proof of Theorem 4.1

Let us suppose initially that u is a regular solution to problem (3.1). By hypotheses, we have that $|g(s)|\ge k|s|$, $\forall s \in {\mathbb {R}}$.

From the identity of the energy it holds that

$$\begin{aligned} E(0)= & {} E(T) + \int _0^T\int _{\mathcal {M}}a(x)g(u_t)u_t \ d{\mathcal {M}}dt\\\ge & {} E(T) + k\int _0^T\int _{\mathcal {M}}a(x)u_t^2 \ d{\mathcal {M}}dt, \end{aligned}$$

that is,

$$\begin{aligned} E(T)-E(0)\le -k\int _0^T\int _{\mathcal {M}}a(x)u_t^2 \ d{\mathcal {M}}dt \end{aligned}$$

(4.85)

In addition, from Proposition (4.1) and from the fact that $|g(s)|\le K|s|$, $\forall s \in {\mathbb {R}}$ we obtain

$$\begin{aligned} E(T)\le & {} C \int _0^T\int _{\mathcal {M}}a(x)(g^2(u_t)+u_t^2) \ d{\mathcal {M}}dt\\\le & {} C_1 \int _0^T\int _{\mathcal {M}}a(x)u_t^2 \ d{\mathcal {M}}dt, \ \ \ \forall \ T\ge T_0. \end{aligned}$$

Consequently,

$$\begin{aligned} -E(T)\ge -C_1\int _0^T\int _{\mathcal {M}}a(x)u_t^2 \ d{\mathcal {M}}dt, \ \ \ \forall \ T\ge T_0. \end{aligned}$$

(4.86)

Multiplying (4.85) by $C_1$ and considering (4.86) we obtain

$$\begin{aligned} C_1\left[ E(T)-E(0)\right]\le & {} -kC_1\int _0^T\int _{\mathcal {M}}a(x)u_t^2 \ d{\mathcal {M}}dt\\\le & {} -k E(T),\ \ \ \ \ \ \ \ \forall \ T \ge T_0, \end{aligned}$$

that is,

$$\begin{aligned} E(T)\le \frac{C_1}{C_1+k}E(0)=\frac{1}{1+C}E(0), \ \ \ \forall \ T \ge T_0, \end{aligned}$$

where $C=\frac{k}{C_1}$.

Repeating the process for 2T, we arrive at

$$\begin{aligned} E(2T)\le \frac{1}{1+C}E(T)\le \frac{1}{(1+C)^2}E(0),\ \ \ \forall \ T \ge T_0. \end{aligned}$$

In general, we conclude that

$$\begin{aligned} E(nT)\le \frac{1}{(1+C)^n}E(0),\ \ \ \forall \ T \ge T_0. \end{aligned}$$

(4.87)

Now, let $t\ge T_0$, then $t=nT_0+r$, $0\le r<T_0$. Since the energy is a non-increasing function, we obtain

$$\begin{aligned} E(t)\le E(t-r)=E(nT_0)\le \frac{1}{(1+C)^n}E(0)= \frac{1}{(1+C)^{\frac{t-r}{T_0}}}E(0) \end{aligned}$$

Considering $C_0 = \textit{e}^{\frac{r}{T_0}ln(1+C)}$ and $\lambda _0= \frac{ln(1+C)}{T_0}>0$ it follows the desired for regular solutions, or still,

$$\begin{aligned} E(t)\le C_0 \textit{e}^{-\lambda _0 t}E(0); \ \ \forall t\ge T_0. \end{aligned}$$

(4.88)

By density arguments the exponential decay is recovered for weak solutions. $\square$

5 Manifolds that Admit Open Subsets Without Damping

5.1 The Manifold $(\mathbb {R}^n, g_{\varphi })$

Consider the manifold ${\mathcal {M}}= \mathbb {R}^n$, $n\ge 2$, endowed with the radial metric described in polar coordinates $(r,\theta )\in \big [0,+\infty \big [\times \mathbb S^{n-1}$ by the formula:

$$\begin{aligned} g_\varphi =\mathrm dr^2+\varphi (r)^2\mathrm d\theta ^2, \end{aligned}$$

(5.1)

where $\mathrm d\theta ^2$ is the standard round metric (of radius 1) of $\mathbb S^{n-1}$ and $\varphi :\big ]0,+\infty \big [\rightarrow \mathbb {R}^+$ is a smooth function satisfying:

• $\varphi ^{(2k)}(0)=0$ for all $k\ge 0$;

• $\varphi '(0)=1$.

The metric $g_\varphi$ is complete. Discreteness, existence of strongly convex functions, as well as many other spectral and geometric properties of the manifold $\mathbb {R}^n$ endowed with the metric (5.1) can be given in terms of (asymptotic) properties of the function $\varphi$. For instance:

Lemma 5.1

The following statements hold true.

• (1) $( \mathbb {R}^n,g_\varphi )$ has finite volume if and only if $\displaystyle \int _0^{+\infty }\varphi (r)^{n-1}\,\mathrm dr<+\infty$.

• (2) If $\lim \limits _{r\rightarrow +\infty }\left| \dfrac{\varphi '(r)}{\varphi (r)}\right| =+\infty$, then $(\mathbb {R}^n,g_\varphi )$ is discrete.

• (3) If $\varphi '(r)\ge c>0$ for all r, then $(\mathbb {R}^n,g_\varphi )$ admits a proper, strongly convex function which is bounded from below.

• (4) $( \mathbb {R}^n,g_\varphi )$ is stochastically complete ¹ if and only if

  $$\begin{aligned} \int _a^{+\infty } \frac{\int _0^r \varphi ^{n-1}(s)\, \mathrm ds }{\varphi ^{n-1}(r)}\,\mathrm dr = +\infty . \end{aligned}$$

• (5) $(\mathbb {R}^n,g_\varphi )$ is parabolic ² if and only if $\displaystyle \int _a^{+\infty } \varphi (r)^{-n+1}\,\mathrm dr = +\infty$ for some $a\ge 0$.

Proof

The volume of $( \mathbb {R}^n,g_\varphi )$ is easily computed as

$$\begin{aligned} c_{n-1}\int _0^{+\infty }\varphi (r)^{n-1}\,\mathrm dr, \end{aligned}$$

where $c_{n-1}$ is the volume of the unit sphere of dimension $n-1$. This settles (1).

For part (3), set $f(r)=\int _0^r \varphi (t)\,\mathrm dt$. The Hessian of f is computed as:

$$\begin{aligned} \mathrm {Hess}(f)= & {} f''(r)\, \mathrm dr\otimes \mathrm dr + f'(r) \mathrm {Hess}(r)\\= & {} \varphi '(r)\,\mathrm dr\otimes \mathrm dr +\varphi (\varphi '/\varphi ) (g_\varphi - \mathrm dr\otimes \mathrm dr) = \varphi '(r) g_\varphi . \end{aligned}$$

Thus, f is strongly convex when $\varphi '\ge c>0$. In this situation, such a function is proper, and positive.

For part (2), (4) and (5), see for instance [32]. $\square$

Let us assume that ${\mathbb {R}}^n$ is endowed by the metric $g_\varphi$ described in the example (5.1), so that $\varphi '(r)\ge c>0,$ $\forall r$. According to Lemma 5.1, item (3), it holds that $({\mathbb {R}}^n,g_\varphi )$ admits a regular function $f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ proper, strongly convex and bounded from below.

In addition, let us assume that the set $\Omega \subset {\mathbb {R}}^n$, is an open bounded set with smooth boundary, and $\varphi$ satisfies

$$\begin{aligned} \varphi '(r)\in \left[ \frac{2}{n}(\alpha +C), \frac{\alpha -C}{\frac{n}{2}-1}\right] , \end{aligned}$$

where $C\in (0,\frac{1}{2}]$ and $\alpha \in \left[ \frac{n}{2}-1+C, \frac{n}{2}-C\right]$, $n>2$. If $n=2$, it is not necessary to have an upper bound to $\varphi '$.

According to the proof of item (3), $f(r,\theta )= \int _0^t\varphi (t) dt$. Consequently, we deduce

$$\begin{aligned}&\nabla ^2 f\left( \frac{\partial }{\partial r},\frac{\partial }{\partial r}\right) =\frac{\partial ^2 f}{\partial r^2}= \varphi '(r),\nonumber \\&\nabla ^2 f\left( \frac{\partial }{\partial r},\frac{\partial }{\partial \theta }\right) =0,\nonumber \\&\nabla ^2 f\left( \frac{\partial }{\partial \theta _i},\frac{\partial }{\partial \theta _j}\right) =0, \end{aligned}$$

(5.2)

if $i \ne j$ and

$$\begin{aligned} \nabla ^2 f\left( \frac{\partial }{\partial \theta _i},\frac{\partial }{\partial \theta _i}\right) =\frac{\partial f}{\partial r}\, \varphi (r)\, \varphi '(r)= \varphi ^2(r)\, \varphi '(r), \end{aligned}$$

(5.3)

thus $\nabla ^2 f$ is proportional to the metric $g_\varphi$.

Thus, from the Theorem 4.2 remains true in the whole $\Omega$, which allow us to avoid putting dissipative effects totally in $\Omega$.

Remark 5

From Lemma 5.1, item (1), the manifold $({\mathbb {R}}^n, g_\varphi )$ has finite volume if and only if $\int _0^\infty \varphi (r)^{n-1} dr \, <\infty$. Then, assuming this can occur, we can consider a nonlinear dissipation g satisfying

$$\begin{aligned} ks^2\le g(s)s\le Ks^2 \end{aligned}$$

for $|s|>1$. In other words, it is not necessary to consider the previous property for $\forall s$, since, in this case, we can prove analogously what we have done before: the existence of solutions as well as very general decay rate estimates by employing the method developed in Lasiecka-Tataru (ver[27]) (see also Cavalcanti et. al. [11]).

5.2 Manifolds with Curvature Conditions

Let $\Omega ^* \subset {\mathcal {M}}$ be as defined before. We denote $K =\overline{\Omega ^*}$.

The main goal of the present subsection is to determine an important class of Riemannian manifolds, namely, Riemannian manifolds $({\mathcal {M}},\mathbf g )$ with sectional curvature verifying $k_1\le sec_g\le k_2<0$, such that it is possible to guarantee the existence of open and disjoint subsets, $V_1$, ..., $V_k$, with $\bigcup _{i=1}^k V_i\subset V$, where $V\subset K$ can be free of dissipative effects and, in addition, $meas(V)>meas(K)-\epsilon$, $meas(V\cap \partial K)>meas(\partial K)-\epsilon$, for all $\epsilon > 0$, more precisely, Remark 4 occurs for such class of Riemannian manifolds with $V\supset \bigcup _{i=1}^k V_i.$

We shall determine precisely an open subset $V_i\subset K$ and function $\alpha _i,f_i:V_i\rightarrow {\mathbb {R}}$, where the inequality (4.22) holds, and consequently such an open can stay free of dissipative effects.

Initially we start with the following result:

Theorem 5.1

Let $\alpha ,\ C \in {\mathbb {R}}$ such that $\alpha > (n-1)C$, $n\in {\mathbb {N}}$ and $n\ge 2$. Let us consider $W \subset {\mathcal {M}}^n$ and $f:{\mathcal {M}}\rightarrow {\mathbb {R}}$ verifying

$$\begin{aligned} \frac{2}{n}(\alpha + C)|v|_p^2\le \nabla ^2f(v,v)\le \left[ \frac{4}{n^2}(\alpha +C)+\frac{2}{n}(\alpha -C)\right] |v|_p^2, \end{aligned}$$

$\forall \ p \in W$ e $\forall \ v \in T_p{\mathcal {M}}$. Then, the following inequality holds

$$\begin{aligned} C\int _0^T\int _W u_t^2 +|\nabla u|^2 \ d{\mathcal {M}}dt\le & {} \int _0^T\int _W(\frac{\Delta f}{2}-\alpha )u_t^2 \ d{\mathcal {M}}dt\nonumber \\+ & {} \int _0^T\int _W \nabla ^2 f(\nabla u,\nabla u)+(\alpha -\frac{\Delta f}{2})|\nabla u|^2 \ d{\mathcal {M}}dt,\nonumber \\ \end{aligned}$$

(5.4)

for all regular solution u to problem (3.1)

Proof

Initially we observe that

$$\begin{aligned} \Delta f(p)=\sum _{i=1}^n\nabla ^2 f(e_i,e_i), \end{aligned}$$

where $\{e_1, \ldots , e_n\}$ is an orthonormal basis in $T_p{\mathcal {M}}$. Thus, we obtain,

$$\begin{aligned} 2(\alpha + C)\le \Delta f(p)\le \frac{4}{n}(\alpha +C)+2(\alpha -C), \ \forall p\in W. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{1}{2}\Delta f(p)|\nabla u|^2\le \left[ \frac{2}{n}(\alpha +C)+(\alpha -C)\right] |\nabla u|^2, \end{aligned}$$

for all $p\in W$.

Consequently

$$\begin{aligned}&\nabla ^2 f(p) (\nabla u, \nabla u)+\left( \alpha -\frac{1}{2}\Delta f(p)\right) |\nabla u|^2 \ge \frac{2}{n}(\alpha + C)|\nabla u|^2\nonumber \\&+\ \alpha |\nabla u|^2-\left[ \frac{2}{n}(\alpha +C)+(\alpha -C)\right] |\nabla u|^2 \nonumber \\&= \ C|\nabla u|^2 \end{aligned}$$

(5.5)

for all $p\in W$.

On the other hand,

$$\begin{aligned} \left( \frac{1}{2}\Delta f(p)-\alpha \right) u_t^2 \ge (\alpha + C)-\alpha =C u_t^2 \end{aligned}$$

(5.6)

for all $p \in W$. Integrating the inequalities (5.5) and (5.6) over $W \times (0,T)$ we deduce the desired. $\square$

5.2.1 Manifolds Admitting Smooth Functions with Bounds on the Hessian

Given a smooth function $f:({\mathcal {M}},\mathbf g ) \rightarrow {\mathbb {R}}$ on a Riemannian manifold, let us denote by $\mathrm {Hess}(f)$ its Hessian, which is the tensor field of symmetric (0, 2)-tensors on ${\mathcal {M}}$ defined by:

$$\begin{aligned} \mathrm {Hess}(f)(v,w)= \mathbf g (\nabla _v(\nabla f),w), \end{aligned}$$

for all $p\in {\mathcal {M}}$ and all $v,w\in T_p{\mathcal {M}}$. Given a function $h:{\mathcal {M}}\rightarrow {\mathbb {R}}$, we write $\mathrm {Hess}(f)\ge h$ (resp., $\mathrm {Hess}(f)\le h$) if $\mathrm {Hess}(f)_p(v,v)\ge h(p) \mathbf g _p(v,v)$ (resp., $\mathrm {Hess}(f)_p(v,v)\le h(p) \mathbf g _p(v,v)$) for all $p\in {\mathcal {M}}$ and all $v\in T_p{\mathcal {M}}$.

We would like to describe a class of manifolds admitting smooth functions f that satisfy an inequality of the form:

$$\begin{aligned} A\le \mathrm {Hess}(f)\le B \end{aligned}$$

(5.7)

for some prescribed constants $B>A>0$. Our motivation is to satisfy (5.4), which amounts to determining positive constants $\alpha$, C and a smooth function f such that the following inequalities:

$$\begin{aligned} \mathrm {Hess}(f)\ge C +\tfrac{1}{2}\Delta f-\alpha \quad \text{ and }\quad \tfrac{1}{2}\Delta f-\alpha \ge C \end{aligned}$$

(5.8)

are satisfied in some open subset $W\subset {\mathcal {M}}$. Observe that, taking traces in (5.7), one obtains immediately the following inequalities for the Laplacian $\Delta f$:

$$\begin{aligned} \tfrac{1}{2} n A\le \frac{1}{2}\Delta f\le \tfrac{1}{2}nB, \end{aligned}$$

(5.9)

where $n=\mathrm {dim}({\mathcal {M}})$.

Using (5.7) and (5.9), we see that (5.8) are satisfied, provided that A and B satisfy:

$$\begin{aligned} A-\tfrac{1}{2}nB\ge C-\alpha ,\quad \text{ and }\quad \tfrac{1}{2}nA\ge C+\alpha . \end{aligned}$$

Equalities in the above expressions are obtained by setting:

$$\begin{aligned} A=\tfrac{2}{n}(C+\alpha ),\quad B=\tfrac{2}{n}\big [\tfrac{2-n}{n}C+\tfrac{2+n}{n}\alpha \big ]=\tfrac{4}{n^2}(C+\alpha )-\frac{2}{n}(C-\alpha ). \end{aligned}$$

The condition $B>A$ gives:

$$\begin{aligned} C+\alpha <\tfrac{2-n}{n}C+\tfrac{2+n}{n}\alpha , \end{aligned}$$

i.e.,

$$\begin{aligned} \alpha >(n-1)C. \end{aligned}$$

(5.10)

5.2.2 Warped Products

The class of warped products offers many examples of Riemannian manifolds (M, g) that admit smooth functions f satisfying the pinching condition (5.7). It is not hard to show that a complete n-dimensional Riemannian manifold admitting a smooth function with Hessian bounded from below by a positive constant and having a (necessarily unique and minimal) critical point, must be diffeomorphic to $\mathbb {R}^n$. However, if one requires the existence of smooth strongly convex functions, without the assumption of admitting a minimum, then one has no topological obstruction. Note that for our purposes, it is not necessary to assume the existence of critical points for functions satisfying the pinching conditions (5.7) on the Hessian, as we will explain below.

A basic examples of manifolds admitting a globally defined function satisfying the pinching condition (5.7) is given as follows: Let $(\widetilde{M},\widetilde{g})$, and consider the product manifold $M=\mathbb {R} \times \widetilde{M}$. Let M be endowed with the warped product metric:

$$\begin{aligned} g=\mathrm dt^2+w(t)^2\,\widetilde{g}, \end{aligned}$$

(5.11)

where t is the coordinate in $\mathbb {R}$. One could consider, more generally, metrics on $\mathbb {R} \times \widetilde{M}$ of the form $g=\alpha (t)^2\,\mathrm dt^2+w(t)^2\,\widetilde{g}$. However, by a standard change of coordinates, one could rewrite such metric as $\mathrm d\rho ^2+w\big (t(\rho )\big )^2\,\widetilde{g}$, where $r=\int \alpha (t)$.

Given the metric (5.11), consider the radial function $f:M\rightarrow \mathbb {R}$ defined by $f(t,x)=\int _0^tw(s)\,\mathrm ds$. An easy computation shows that $\mathrm {Hess}(f)$ at any point p is a multiple of the metric g:

$$\begin{aligned} \mathrm {Hess}(f)=w'(t)\,g. \end{aligned}$$

In particular, if $w:\mathbb {R}\rightarrow \mathbb {R}$ is a smooth diffeomorphism satisfying:

$$\begin{aligned} A\le w'(t)\le B,\qquad \forall \,t\in \mathbb {R}, \end{aligned}$$

then the corresponding function f satisfies (5.7).

It is also interesting to observe that, by a characterization of Cheeger and Colding in [14], Riemannian manifolds admitting functions whose Hessian is a multiple of the metric are basically warped products of the type described here.

5.2.3 The Hessian Comparison Theorem

We will now change our point of view, and try to determine regions of a Riemannian manifold that are domains of smooth functions that satisfy the pinching condition on the Hessian (5.7) using curvature conditions.

Let us recall that, given a Riemannian manifold $({\mathcal {M}},\mathbf g )$ and a point $p\in {\mathcal {M}}$, the injectivity radius at p, denoted by $\mathrm {inj}(p)$ is the supremum of the set

$$\begin{aligned} \big \{r>0:\exp _p\big \vert _{B_p(0,r)}\ \text{ is } \text{ a } \text{ diffeomorphism }\big \}. \end{aligned}$$

Here, $B_p(0,r)$ is the open ball of radius r and center 0 in $T_p{\mathcal {M}}$. Equivalently, $\mathrm {inj}(p)$ is the distance between p and its cut locus. A point $p\in {\mathcal {M}}$ is said to be a pole if $\mathrm {inj}(p)=+\infty$. Given $p\in {\mathcal {M}}$ and $v,w\in T_p{\mathcal {M}}$ linearly independent, we will denote by $\mathrm {sec}_\mathbf g (v\wedge w)$ the sectional curvature of the plane spanned by v and w.

Let us recall the following result on the Hessian of the distance function.

Theorem 5.2

Let $({\mathcal {M}}^n_i,\mathbf g _i)$, $i=1,2$, be complete Riemannian manifolds, and let $\gamma _i:[0,L]\rightarrow {\mathcal {M}}_i$ be geodesics parameterized by arc-length, such that $\gamma _i$ does not intersect the cut locus of $\gamma _i(0)$. Denote by $r_i=\mathrm {dist}\big (\cdot ,\gamma _i(0)\big )$, $i=1,2$. Assume that for all $t\in [0,L]$, one has:

$$\begin{aligned} \mathrm {sec}_{g_1}\big (v_1\wedge \dot{\gamma }_1(t)\big )\ge \mathrm {sec}_{g_2}\big (v_2\wedge \dot{\gamma }_2(t)\big ) \end{aligned}$$

for all $v_i\in \dot{\gamma }_i(t)^\perp$. Then:

$$\begin{aligned} \mathrm {Hess}(r_1)(v_1,v_1)\le \mathrm {Hess}(r_2)(v_2,v_2), \end{aligned}$$

for all $v_i\in \dot{\gamma }_i(t)^\perp$ and all $t\in [0,L]$.

Remark 6

In the above statement, $r_i\big (\gamma _i(t)\big )=t$ for all $t\in [0,L]$ (because $\gamma _i$ is parameterized by arc-length, hence $\mathrm {Hess}(r_i)\big (\dot{\gamma }_i(t),\dot{\gamma }_i(t)\big )=0$ for all t. In order to have a function with control on the Hessian in all directions, one should consider the squared distance function. An easy computation shows that, for all smooth function $f:{\mathcal {M}}\rightarrow {\mathbb {R}}$:

$$\begin{aligned} \mathrm {Hess}(\tfrac{1}{2}f^2)(v,v)=\mathbf g (\nabla f,v)^2+f\cdot \mathrm {Hess}(f)(v,v). \end{aligned}$$

In particular, since $\nabla r_i\big (\gamma _i(t)\big )=\dot{\gamma }_i(t)$, then:

$$\begin{aligned} \mathrm {Hess}(\tfrac{1}{2}r_i^2)(\dot{\gamma }_i,\dot{\gamma }_i)=1. \end{aligned}$$

More generally, if $F:{\mathbb {R}}\rightarrow {\mathbb {R}}$ and $r:{\mathcal {M}}\rightarrow {\mathbb {R}}$ are smooth functions, then:

$$\begin{aligned} \mathrm {Hess}(F\circ r)(X,X)=F''(r)\mathbf g (\nabla r,X)^2+F'(r)\,\mathrm {Hess}(r)(X,X). \end{aligned}$$

(5.12)

Thus, in computing the Hessian of a distance function, one has to distinguish radial directions, i.e., parallel to the tangent field $\dot{\gamma }(t)$, and spherical directions, i.e., orthogonal to $\dot{\gamma }(t)$.

5.2.4 Space Forms

Let $g_{\mathbb S^n}$ be the standard round metric on the unit sphere $\mathbb S^n$. For $k\in {\mathbb {R}}$, consider the manifold ${\mathcal {M}}^n= {\mathbb {R}}^+\times \mathbb S^{n-1}$ endowed with the metric

$$\begin{aligned} g_k=\mathrm dr^2+S_k(r)\,g_{\mathbb S^{n-1}}, \end{aligned}$$

where:

$$\begin{aligned} S_k(r)={\left\{ \begin{array}{ll}\frac{1}{\sqrt{k}}\,\sin \big (\sqrt{k}\,r\big ),&{}\text {if } k>0,\\ \;r,&{}\text {if } k=0,\\ \frac{1}{\sqrt{\vert k\vert }}\sinh \big (\sqrt{\vert k\vert }\,r\big ),&{}\text {if } k<0. \end{array}\right. } \end{aligned}$$

The (completion of the) Riemannian manifold $({\mathcal {M}}^n,g_k)$ is the³ n-dimensional space form of curvature k. An explicit calculation of the Hessian of the distance function for spaceforms yields the following immediate corollary of the Hessian comparison theorem.

Proposition 5.1

(Special case of the Hessian comparison theorem) Given a Riemannian manifold $({\mathcal {M}}^n,g)$ and a point $p\in {\mathcal {M}}$, set $r=\mathrm {dist}(\cdot , p)$. If $\mathrm {sec}_g\ge k$, then, in the spherical directions, $\mathrm {Hess}(r)\le \frac{1}{n-1}H_k(r)$, where:

$$\begin{aligned} H_k(r)={\left\{ \begin{array}{ll}(n-1)\sqrt{k}\cdot \cot \big (\sqrt{k}\cdot r\big ),&{}\text {if } k>0;\\ \dfrac{n-1}{r},&{}\text {if } k=0;\\ (n-1)\sqrt{-k}\cdot \coth \big (\sqrt{-k}\cdot r\big ),&{}\text {if } k<0. \end{array}\right. } \end{aligned}$$

Similarly, $\mathrm {sec}_g\le k$, then $\mathrm {Hess}(r)\ge \frac{1}{n-1}H_k(f)$ in the spherical directions.

Remark 7

One can consider more generally a metric on ${\mathcal {M}}^n$ of the form:

$$\begin{aligned} \mathbf g =\mathrm dr^2+\phi (r)^2\,g_{\mathbb S^{n-1}}, \end{aligned}$$

(5.13)

where $\phi :\big ]0,+\infty \big [\rightarrow {\mathbb {R}}^+$ is a smooth function with $\lim \limits _{r\rightarrow 0}\phi (r)=0$ and $\lim \limits _{r\rightarrow 0}\phi '(r)=1$. The following formulas for the sectional curvature of these metrics hold:

$$\begin{aligned} \mathrm {sec}_\mathrm {rad}=\frac{\phi ''}{\phi },\quad \mathrm {sec}_{\mathrm {sph}}=\frac{1-(\phi ')^2}{\phi ^2}, \end{aligned}$$

(5.14)

where $\mathrm {sec}_\mathrm {rad}$ is the sectional curvature of planes containing the radial vector, and $\mathrm {sec}_{\mathrm {sph}}$ is the curvature of planes perpendicular to the radial vector.

5.2.5 Conclusion

Let us consider a simply connected complete Riemannian manifold $({\mathcal {M}}^n,\mathbf g )$ satisfying:

$$\begin{aligned} k_1\le \mathrm {sec}_g\le k_2<0, \end{aligned}$$

for some (negative) constants $k_1,k_2$. It is well known that every point p of such a manifold, which is necessarily diffeomorphic to ${\mathbb {R}}^n$, is a pole. Let us fix positive constants $0<A<B$; for computational convenience, let us choose⁴ $B>n-1$ and $A<n-1$. We will exhibit a smooth function $f:{\mathcal {M}}\rightarrow \big [0,+\infty \big [$ and a non empty open subset $W\subset {\mathcal {M}}$ such that (5.7) holds in W.

Let us fix $p_0\in {\mathcal {M}}$, set $r=\mathrm {dist}(\cdot ,p_0)$, choose $\gamma \in \big ]A,B\big [$, and let $f=\frac{\gamma }{2}r^2$. Using (5.12), the Hessian of f is given by:

$$\begin{aligned} \mathrm {Hess}(f)(X,X)=\gamma \cdot g(\nabla r,X)^2+\gamma r\,\mathrm {Hess}(r)(X,X). \end{aligned}$$

If X is a radial vector, then $\mathrm {Hess}(r)(X,X)=0$, and since $g(\nabla r,\nabla r)=1$, then $\mathrm {Hess}(f)(X,X)=\gamma g(X,X)$, i.e., inequality (5.7) is always satisfied for radial vectors. As to spherical directions X, since $g(\nabla r,X)=0$, then

$$\begin{aligned} \mathrm {Hess}(f)(X,X)=\gamma r\,\mathrm {Hess}(r)(X,X), \end{aligned}$$

and by the Hessian comparison theorem we then get:

$$\begin{aligned} \gamma r\sqrt{-k_2}\,\mathrm {coth}\big (r\sqrt{-k_2}\big )\le \mathrm {Hess}(f)(X,X)\le \gamma r\sqrt{-k_1}\,\mathrm {coth}\big (r\sqrt{-k_1}\big ). \end{aligned}$$

Thus, inequality (5.7) is satisfied in the region:

$$\begin{aligned} r_1\le r\le r_2, \end{aligned}$$

where $r_1$ and $r_2$ are defined by:

$$\begin{aligned} r_1\sqrt{-k_2}\,\mathrm {coth}\big (r_1\sqrt{-k_2}\big )=\frac{A}{\gamma },\quad r_2\sqrt{-k_1}\,\mathrm {coth}\big (r_2\sqrt{-k_1}\big )=\frac{B}{\gamma }. \end{aligned}$$

By elementary arguments, one can determine a large class of examples of smooth functions satisfying pinching conditions on the Hessian, using wisely formulas (5.12) for the Hessian of composite functions and formulas (5.14) for the radial curvature of warped metrics of the type (5.13) in the Hessian comparison theorem.

Appendix

We start this section by considering a uniqueness result due to Bur and Gérard [10] (see p. 80, Sect. 6) or Lasiecka et. al. [26]. Let us consider the wave equation posed in a compact Riemannian manifold (M, g) with boundary. If $(\omega , T_0)$ controls M, then the following observability inequality holds for ultra weak solutions to problem

$$\begin{aligned} \left\{ \begin{aligned}&v_{tt} - \Delta _g v=0 ~\hbox { in }M \times (0,T),\\&v=0 ~\hbox { on }\partial M \times (0,T),\\&v(0)=v_0\in L^2(M); v_t(0)=v_1 \in H^{-1}(M), \end{aligned} \right. \end{aligned}$$

(6.1)

holds:

$$\begin{aligned} ||v_0||_{L^2(M)}^2 + ||v_1||_{H^{-1}(M)} \le C \int _0^T \int _\omega |v(x,t)|^2\,d{\mathcal {M}}dt, \end{aligned}$$

(6.2)

for a certain constant C and for all $T\ge T_0$ and for all $\{v_0,v_1\}\in L^2(M) \times H^{-1}(M)$.

Following we will show that if the Riemannian manifold M admits a function with positive defined Hessian in some subset $U \subset \text{ int }M$, then any geodesic on U hits $\partial U$.

We consider a Riemannian manifold M Riemannian metric $G=\left< \cdot , \cdot \right>$ and Riemannian connection $\tilde{\nabla }$. We denote its Laplace–Beltrami operator by $\Delta$. Fix a coordinate system $(x_1, \ldots , x_n)$ on M, denote $G_{ij}=\left<\partial /\partial x_i,\partial /\partial x_j\right>$, let $G^{ij}$ be the inverse matrix of $G_{ij}$. The Laplace–Beltrami operator in this coordinate system is given by

$$\begin{aligned} \Delta u=\frac{1}{\sqrt{\det G_{ij}}}\sum _{i,j=1}^n \frac{\partial }{\partial x_i}\left( \sqrt{\det G_{ij}}G^{ij} \frac{\partial u}{\partial x_j} \right) , \end{aligned}$$

where $\nabla$ is the usual gradient correspondent to the Euclidean metric on the domain $(x_1, \ldots , x_n)$.

The Hessian of a smooth function $\phi :M\rightarrow \mathbb R$ is a symmetric 2-form on M defined as

$$\begin{aligned} \nabla ^2 \phi (X,Y)=XY(\phi )-\tilde{\nabla }_XY (\phi ), \end{aligned}$$

(6.3)

where X and Y are vector fields on M. Here $X(\phi )$ denotes the directional derivative of $\phi$ in the direction of the vector field X. It is well known that the value of $\nabla ^2(X,Y)(p)$ depends only on the values of X and Y on p. It means that the right-hand side of (6.3) does not depend on the smooth extension we take for X(p) and Y(p).

A curve $\gamma :(-\varepsilon ,\varepsilon ) \rightarrow M$ is a geodesic if $\tilde{\nabla }_{\gamma ^\prime (t)}\gamma ^\prime (t)\equiv 0$.

Lemma 6.1

Let M be a complete Riemannian manifold, eventually with boundary, and let $\phi :\text {int}M\rightarrow \mathbb R$ be a smooth function. Suppose that $\phi$ is bounded and $\nabla ^2 \phi (v,v)\ge c\Vert v \Vert ^2$ on an open subset $U\subset \text {int} M$, where $c>0$ is a constant. Then any geodesic on U hits $\partial U$.

Proof

Let $\gamma$ be a geodesic on U. Then

$$\begin{aligned} \frac{d^2}{dt^2}\phi (\gamma (t))= & {} \gamma ^\prime (t) \gamma ^\prime (t) \phi = \nabla ^2 \phi (\gamma ^\prime (t),\gamma ^\prime (t))+ ( \nabla _{\gamma ^\prime (t)}\gamma ^\prime (t))(\phi )\\= & {} \tilde{\nabla }^2 \phi (\gamma ^\prime (t),\gamma ^\prime (t))\ge c \Vert \gamma ^\prime (t)\Vert ^2 \end{aligned}$$

where the last equality holds because $\gamma$ is a geodesic. Observe that the last term is a positive real constant because $\Vert \gamma ^\prime (t) \Vert$ does not depend on t.

Then $\phi (\gamma (t))$ is a smooth real valued function which second derivative is bounded below by a strictly positive constant. Then it is not difficult to prove that if $\gamma$ is defined on a interval $(a,\infty )$, then $\lim _{t\rightarrow \infty } \phi (\gamma (t))=\infty$. Analogously $\lim _{t\rightarrow -\infty } \phi (\gamma (t))=\infty$ whenever $\gamma$ is defined on a interval $(-\infty ,a)$. But neither of the cases are possible if $\gamma$ remain forever in U because $\phi$ is bounded there. Therefore $\gamma$ must hit $\partial U$. $\square$

According to the construction of the function f give in [13] and mentioned in Remark 4 we have that $\nabla ^2f(v,v)\ge C\Vert v\Vert _g^2$ in V, where $V=\bigcup _{i=1}^{k} V_i$. Thus, we ensure that the geodesics find the effective dissipation region, namely $(M\backslash \overline{\Omega }) \cup M_*,$ That is, there are no geodesic “trapped” within the free sets of dissipative effects $V_i$, for all $i=1, \ldots , k$. It is worth remembering that $\Omega \subset \subset \Omega ^*$.

Now we can show that, in fact, the function v given in (4.64) is null in all $\Omega ^*$. For this we have to show that $v=0$ in $V_i$, for all $i=1, \ldots , k,$ since $(\overline{\Omega ^*} \backslash V) \subset {\mathcal {M}}_*$ and $v=0$ in ${\mathcal {M}}_*.$

By (4.64), we have

$$\begin{aligned} \left\{ \begin{array}{ll} {v^{''}-\Delta v +v =0} &{} {in\,\,\,\Omega ^*\times (0,T)} \\ {v=0} &{} {in\,\,\,\partial \Omega ^*}\times (0,T) \\ {v(0)=v_0 \in L^2(\Omega ^*);} &{} {v'(0)=v_1 \in H^{-1}(\Omega ^*),} \end{array}\right. \end{aligned}$$

(6.4)

with $v=0$ in $(\Omega ^*\backslash \Omega )\cup {\mathcal {M}}_*$.

Now, consider the problems

$$\begin{aligned} \left\{ \begin{array}{ll} {\varphi ^{''}-\Delta \varphi =0} &{} {in\,\,\,\Omega ^*\times (0,T)} \\ {\varphi =0} &{} {in\,\,\,\partial \Omega ^*}\times (0,T) \\ {\varphi (0)=v_0 \in L^2(\Omega ^*);} &{} {\varphi '(0)=v_1 \in H^{-1}(\Omega ^*),} \end{array}\right. \end{aligned}$$

(6.5)

and,

$$\begin{aligned} \left\{ \begin{array}{ll} {z^{''}-\Delta z =-v} &{} {in\,\,\,\Omega ^*\times (0,T)} \\ {z=0} &{} {in\,\,\,\partial \Omega ^*}\times (0,T) \\ {z(0)=z'(0)=0.} \end{array}\right. \end{aligned}$$

(6.6)

Defining $w= \varphi + z$, we have that w is solution of

$$\begin{aligned} \left\{ \begin{array}{ll} {w^{''}-\Delta w =-v} &{} {in\,\,\,\Omega ^*\times (0,T)} \\ {w=0} &{} {in\,\,\,\partial \Omega ^*}\times (0,T) \\ {w(0)=v_0 \in L^2(\Omega ^*);} &{} {w'(0)=v_1 \in H^{-1}(\Omega ^*),} \end{array}\right. \end{aligned}$$

(6.7)

and, if $y=v-w$ then y is solution of

$$\begin{aligned} \left\{ \begin{array}{ll} {y^{''}-\Delta y =0} &{} {in\,\,\,\Omega ^*\times (0,T)} \\ {y=0} &{} {in\,\,\,\partial \Omega ^*}\times (0,T) \\ {y(0)=y'(0)=0.} \end{array}\right. \end{aligned}$$

(6.8)

By uniqueness of solution, we conclude that $y=0$, that is, $v=w=\varphi +z.$ Note that $z''-\Delta v= -v =0$ is $(\overline{\Omega ^*}\backslash \Omega )\cup {\mathcal {M}}_*$ with $z(0)=0=z'(0)$, where it follows that $z=0$ in $(\overline{\Omega ^*}\backslash \Omega )\cup {\mathcal {M}}_*$, and consequently, $\varphi =0$ in $(\overline{\Omega ^*}\backslash \Omega )\cup {\mathcal {M}}_*$.

As $\varphi =0$ in ${\mathcal {M}}_*\supset \supset \overline{\Omega ^*}\backslash V$ we have that exists an open neighborhood $\omega _i$ of $\partial V_i$, with $\omega _i\subset V_i$, such that $\varphi =0$ in $\omega _i.$

Restricting the problem (6.5) to $V_i$, follows by (6.2) that

$$\begin{aligned} ||v_0||_{L^2(V_i)}^2 + ||v_1||_{H^{-1}(V_i)} \le C \int _0^T \int _{\omega _i} |\varphi (x,t)|^2\,d{\mathcal {M}}dt, \end{aligned}$$

(6.9)

where it follows that $v_0=v_1=0$ em $V_i,$ for all $i=1, \ldots , k,$ and therefore, by the problem (6.8), we conclude that $v=0$ in $V_i,$ for all $i=1, \ldots , k,$ as we wanted to show. $\square$