1 Introduction

In this paper we study the following problem

$$\begin{aligned}&\displaystyle {u_{tt}-M\left( \int _{{\mathbb {B}}^N}|\nabla _{{\mathbb {B}}^N}u|^2\,dV\right) \Delta _{{\mathbb {B}}^N} u+\delta u_t=0}\,\, \text{ in }\,{\mathbb {B}}^N\times (0,\infty ), \end{aligned}$$
(1)
$$\begin{aligned}&u(x,0)=u_0(x), \, u_t(x,0)=u_1(x),\,\, x\in {\mathbb {B}}^N, \end{aligned}$$
(2)

where \(\nabla _{{\mathbb {B}}^N}\) and \(\Delta _{{\mathbb {B}}^N}\) are, respectively, the gradient and the Laplace–Beltrami operator in the disc model of the hyperbolic space \({\mathbb {B}}^N\); \(M:[0,\infty )\rightarrow {\mathbb {R}}\) is a known function; \(u_0\) and \(u_1\) are the initial data and \(\delta \) is a positive constant.

The space \({\mathbb {B}}^N\) is the unit disc \(\{x\in {\mathbb {R}}^N:\,|x|<1\}\) of \({\mathbb {R}}^N\) endowed with the Riemannian metric g given by \(g_{ij}=p^2\delta _{ij}\), where \( p(x)= \frac{2}{1-|x|^2}\) and \(\delta _{ij}=1\), if \(i=j\) and \(\delta _{ij}=0\), if \(i\ne j\). The hyperbolic gradient \(\nabla _{{\mathbb {B}}^N}\) and the hyperbolic Laplacian \(\Delta _{{\mathbb {B}}^N}\) are given by

$$\begin{aligned} \nabla _{{\mathbb {B}}^N} u = \frac{\nabla u}{p}\quad \text{ and }\quad \Delta _{{\mathbb {B}}^N} u = p^{-N}div(p^{N-2}\nabla u)=p^{-2}\Delta + \frac{(N-2)}{p} x\cdot \nabla , \end{aligned}$$
(3)

where \(\cdot \) is the standard scalar product in \({\mathbb {R}}^N\); and \(\nabla \) and \(\Delta \) are the usual gradient and Laplacian of \({\mathbb {R}}^N\). Details can be found in the references [15, 16, 31, 32].

There are large literature concerned with existence, uniqueness, and stability of Kirchhoff model. We can cite the works of Bae and Nakao [1], Cavalcanti, Domingos Cavalcanti and Soriano [10], Cavalcanti et al. [11, 12], Ghisi [17, 18], Louredo and Miranda [21], Miranda and Jutuca [22], Miranda, Louredo and Medeiros [23], Perla Menzala [19], Ono [27,28,29], Nishihara [26], Yamada [33], and references therein.

We would like to emphasize the work of Miranda and Jutuca [22] where the authors proved the existence, uniqueness and decay for the problem with boundary damping. Precisely, they studied the problem

$$\begin{aligned}&\displaystyle {u_{tt}-M\left( t,\int _{\Omega }|\nabla u|^2\,dx\right) \Delta u=0} \,\, \text{ in }\,\Omega \times (0,\infty ), \end{aligned}$$
(4)
$$\begin{aligned}&u=0\,\, \text{ on }\,\Gamma _0\times (0,\infty ), \end{aligned}$$
(5)
$$\begin{aligned}&\frac{\partial u}{\partial \nu }+\delta u_t=0\,\, \text{ on }\,\Gamma _1\times (0,\infty ), \end{aligned}$$
(6)
$$\begin{aligned}&u(x,0)=u_0(x), \, u_t(x,0)=u_1(x),\,\, x\in \Omega , \end{aligned}$$
(7)

where the domain \(\Omega \) is an open and bounded subset of \({\mathbb {R}}^N\) and its boundary is given by \(\Gamma =\Gamma _0\cup \Gamma _1\). To prove the existence of solution, the authors used fixed point theorem combined with the use of Faedo–Galerkin method. They proved the exponential decay for the strong energy associated to the problem.

In [10], Cavalcanti, Domingos Cavalcanti and Soriano extended the results of Miranda and Jutuca [22] to the nonlinear case. Precisely, they studied the problem

$$\begin{aligned}&\displaystyle {u_{tt}-M\left( t,\int _{\Omega }|\nabla u|^2\,dx\right) \Delta u=0} \,\, \text{ in }\,\Omega \times (0,\infty ), \end{aligned}$$
(8)
$$\begin{aligned}&u=0\,\, \text{ on }\,\Gamma _0\times (0,\infty ), \end{aligned}$$
(9)
$$\begin{aligned}&\frac{\partial u}{\partial \nu }+g(u_t)=0\,\, \text{ on }\,\Gamma _1\times (0,\infty ), \end{aligned}$$
(10)
$$\begin{aligned}&u(x,0)=u_0(x), \, u_t(x,0)=u_1(x),\,\, x\in \Omega , \end{aligned}$$
(11)

where the domain \(\Omega \) is an open, bounded star-shaped subset of \({\mathbb {R}}^N\) and its boundary is given by \(\Gamma =\Gamma _0\cup \Gamma _1\). The authors proved the existence and uniqueness of regular solutions without the classical assumption involving the smallness on the initial data.

When the domain is whole space \({\mathbb {R}}^N\), there are some additional difficulties and the exponential stabilization is not expected. In fact, it is well known that an ingredient to prove the exponential stability (without restriction on the initial data) is to use the Poincaré inequality, which does not hold in whole \({\mathbb {R}}^N\). In this direction, we can cite the work of Ono [30], where the author proved that the energy decays with polynomial rate.

Recently Dias Silva, Pitot, and Vicente [14] studied the Kirchhoff equation defined on whole \({\mathbb {R}}^N\) space. Inspired on work of Bjorland and Schonbek [2], they defined suitable Hilbert spaces V, H and an operator \(A=-\Delta \) by the triple \(\{V, H, a(u, v)\}\), where a(u, v) is a bilinear, continuous and coercive form defined in V. This strategy allows the authors to prove that the energy decays exponentially.

On the other hand, studies involving the wave equation defined in whole hyperbolic space can be found in [34], where Wang, Ning, and Yang considered the following problem

$$\begin{aligned}&\displaystyle {u_{tt}-\Delta _{g} u+a(x) u_t=0} \,\, \text{ in }\,{\mathbb {H}}^N\times (0,\infty ), \end{aligned}$$
(12)
$$\begin{aligned}&u(x,0)=u_0(x), \, u_t(x,0)=u_1(x),\,\, x\in {\mathbb {H}}^N, \end{aligned}$$
(13)

where \(\Delta _{g}\) is the Laplace–Beltrami operator in \({\mathbb {H}}^N\). Using multiplier methods and compactness-uniqueness arguments, they proved the exponential stabilization. An important tool to prove the stability of (12)–(13) is the following Poincaré inequality

$$\begin{aligned} \int _{{\mathbb {H}}^N}u^2\,dx_g\le C\int _{{\mathbb {H}}^N}|\nabla _g u|_g^2\,dx_g, \end{aligned}$$
(14)

for \(u\in H^1({\mathbb {H}}^N)\), where \(\nabla _g\) is the gradient operator associated to the Riemannian metric g. As described before, in whole \({\mathbb {R}}^N\) the inequality above does not hold. Therefore, it is not possible to prove the exponential stability without restriction on the initial data.

Another way to prove the exponential stability of wave equation defined in \({\mathbb {B}}^N\) was proved by Carrião, Miyagaki, and Vicente [9]. Indeed, the authors considered the semilinear problem with localized damping

$$\begin{aligned}&\displaystyle {u_{tt}-\Delta _{{\mathbb {B}}^N} u+f(u)+a(x)u_t=0}\,\,\text{ in }\,{\mathbb {B}}^N\times (0,\infty ), \end{aligned}$$
(15)
$$\begin{aligned}&u(x,0)=u_0(x),\, u_t(x,0)=u_1(x)\,\, \text{ for }\,\,x\in {\mathbb {B}}^N, \end{aligned}$$
(16)

where a, f, \(u_0\) and \(u_1\) are known functions and \(\Delta _{{\mathbb {B}}^N}\) is the Laplace–Beltrami operator in the disc model of the Hyperbolic \({\mathbb {B}}^N\). Making the appropriate change \(v:= p^{\frac{N-2}{2}}u\), we have that u satisfies (15)–(16) if, and only if, v satisfies the following singular problem

$$\begin{aligned}&\displaystyle {p^2 v_{tt}-\Delta v+\beta _0p^2v+p^{\frac{N+2}{2}}f(p^{-\frac{N}{2}+1}v)+a(x)p^2v_t=0}\,\,\text{ in }\,B_1\times (0,\infty ), \end{aligned}$$
(17)
$$\begin{aligned}&v(x,0)=v_0(x),\, v_t(x,0)=v_1(x)\,\, \text{ for }\,\, x\in B_1, \end{aligned}$$
(18)

where \(\beta _0=\frac{N(N-2)}{4}\) and \(B_1\) is the unit disc \(\{x\in {\mathbb {R}}^N:\,|x|<1\}\) of \({\mathbb {R}}^N\) endowed with the Euclidean metric. Therefore, the authors worked with (17)–(18) which is a singular problem in \(B_1\). To overcome the difficulty of deal with the singularities, they used the Hardy inequality, in a version due to the Brezis and Marcus [4] and Brezis, Marcus, and Shafrir [5]. To best of our knowledge, the technique used by Carrião, Miyagaki, and Vicente [9] is new in the context of hyperbolic equations. Before that, only elliptic equations were studied with this tool. See Carrião et al. [6,7,8].

As described before, the main tool used in [9] is Hardy’s inequality. Precisely, using this inequality, it is possible to prove that

$$\begin{aligned} \int _{B_1} p^2 w^2 \,dx\le C \int _{B_1} |\nabla w|^2\,dx, \end{aligned}$$
(19)

for all \(w \in H^1_{0}(B_1)\), which is a kind of Poincaré’s inequality with a weight p. This inequality is shown in Lemma 1 and it will be used many times in the paper.

We also would like to cite the work of Bortot et al. [3], where they studied the Klein Gordon equation, subject to a nonlinear and localized damping, in a complete and non-compact Riemannian manifold without boundary. Precisely, they studied the problem

$$\begin{aligned}&\displaystyle {u_{tt}-\Delta u+f(u)+a(x)g(u_t)=0}\,\,\text{ in }\,{\mathcal {M}}\times (0,\infty ), \end{aligned}$$
(20)
$$\begin{aligned}&u(x,0)=u_0(x),\, u_t(x,0)=u_1(x)\,\, \text{ for }\,\, x\in {\mathcal {M}}, \end{aligned}$$
(21)

where \({\mathcal {M}}\) (endowed with a Riemannian metric) is a complete and non-compact N dimensional Riemannian manifold without boundary and \(\Delta \) denotes the Laplace–Beltrami operator. The function a, responsible by the damping localization, acts in \({\mathcal {M}}\setminus {\overline{\Omega }}\), where \(\Omega \) is an arbitrary open and bounded set in \({\mathcal {M}}\).

In the present paper, we use the strategy of [9] to change the original problem into a singular problem. Therefore, defining \(v:= p^{\frac{N-2}{2}}u\) and observing (3), we have that u satisfies (1)–(2) if, and only if, v satisfies the following singular problem

$$\begin{aligned}&p^2 v_{tt}-M\left( \lambda (t)\right) \Big (\Delta v-\beta _0p^2v\Big ) +\delta p^2v_t=0\,\,\text{ in }\,B_1\times (0,\infty ), \end{aligned}$$
(22)
$$\begin{aligned}&v=0\,\,\text{ on }\,\partial B_1\times (0,\infty ), \end{aligned}$$
(23)
$$\begin{aligned}&v(x,0)=v_0(x),\, v_t(x,0)=v_1(x)\,\, \text{ for }\,\, x\in B_1, \end{aligned}$$
(24)

where \(\beta _0=\frac{N(N-2)}{4}\) and \(B_1\) is the unit disc \(\{x\in {\mathbb {R}}^N:\,|x|<1\}\) of \({\mathbb {R}}^N\) endowed with the Euclidean metric and

$$\begin{aligned} \lambda (t)=\int _{B_1}\Big (|\nabla v|^2-(N-2)\langle \nabla v,xpv\rangle +\left( \frac{N-2}{2}\right) ^2p^2v^2|x|^2\Big )\,dx. \end{aligned}$$
(25)

We work, equivalently, with (22)–(24) which is a singular problem in \(B_1\). Therefore, since \(p(s)\rightarrow \infty \), as \(|s|\rightarrow 1\), the difficulty of deal with the wave equation in whole space \({\mathbb {B}}^N\) is replaced by the difficulty of deal with a singular problem in \(B_1\).

The main goal of the present paper is bring the technique used by Carrião, Miyagaki, and Vicente [9] and Carrião et al. [6,7,8] to the context of Kirchhoff equation. This strategy allows us to prove that the energy associated to the problem decay exponentially.

The main technical difficulty of the present paper is to control the singularities. It is well known that when Kirchhoff equation is in place, it is not possible to use semigroups techniques to prove the existence of solution. In the \({\mathbb {R}}^N\) case many authors have been used fixed point methods and Faedo–Galerkin method. But, due to the presence of singularities, some usual calculus does not hold here and it is necessary some additional arguments. Indeed, when the Kirchhoff model is considered in a domain \(\Omega \) of \({\mathbb {R}}^N\) it is usual to use Faedo–Galerkin method with special bases (given by eigenvector of the Laplace operator). This allows to estimate the norm of the sequence of approximate solution in the space \(L^{\infty }(0,T;H_0^1(\Omega )\cap H^2(\Omega ))\). But, the singularities does not allow us to take the same way. This difficult can be seen in the proof of the existence of solution. To overcome this problem, it is necessary to make four estimates. It will be clarified in Sect. 4.

Our paper is organized as follows. In Sect. 2 we present the notation, assumptions, and preliminaries. We also enunciate the theorem which gives the exponential stability. Moreover, we enunciate a result which gives the existence and uniqueness of solution. The stability is proved in Sect. 3. In Sect. 4 we prove the existence and uniqueness of solution.

2 Preliminaries and main result

As described in the introduction, in this section we establish some notations and the main result. We also enunciate the tool which is the main novelty in the context of class of Kirchhoff equations, a class of Hardy inequality. This inequality is used many times into the paper. The reader can see that Lemma 1 is called many times through the paper.

Thus, we start defining some usual spaces. Let \(L^2(B_1)\) be endowed with the norm and inner product

$$\begin{aligned} \Vert u\Vert _2=\left( \int _{B_1}u^2\,dx\right) ^{\frac{1}{2}} \quad \text{ and } \quad (u,v)=\int _{B_1}uv\,dx. \end{aligned}$$

In the space \(H_0^1(B_1)\) we consider the norm and inner product defined by

$$\begin{aligned} \Vert u\Vert _{H_0^1(B_1)}=\Vert \nabla u\Vert _2 \quad \text{ and }\quad (u,v)=\int _{B_1}\nabla u\cdot \nabla v\,dx. \end{aligned}$$

Now, we enunciate the Lemma 1, which gives us a Hardy inequality class and, after this, the classical Nakao’s Lemma.

Lemma 1

There exists a positive constant \(C_H\) such that the following inequality holds

$$\begin{aligned} \int _{B_1} p^2 w^2 \,dx\le C_H^2 \int _{B_1} |\nabla w|^2\,dx, \end{aligned}$$
(26)

for all \(w \in H^1_{0}(B_1)\).

Proof

See Carrião, Miyagaki, and Vicente [9]. See also Carrião et al. [6,7,8].

It is well known that Nakao’s lemma is an important tool to prove the stability to problems involving the Kirchhoff equation. Below, we enunciate the lemma whose proof can be found in Nakao [24, 25]. \(\square \)

Lemma 2

(Nakao’s Lemma) Let \(\phi :[0,\infty )\rightarrow [0,\infty )\) be a bounded function satisfying

$$\begin{aligned} \sup _{t\le s\le t+1} \phi (s)\le C_0 (\phi (t)-\phi (t+1)), \end{aligned}$$
(27)

for \(t\ge 0\), where \(C_0\) is a positive constant. Then, it holds

$$\begin{aligned} \phi (t)\le \theta _1 e^{-\theta _2 t} \end{aligned}$$
(28)

for all \(t\ge 0\), where \(\theta _1\) and \(\theta _2\) are positive real number which depends on known constants.

Now, it is important to observe that, from the inequality

$$\begin{aligned} (N-2)\langle \nabla v,xpv\rangle \le |\nabla v|^2+\left( \frac{N-2}{2}\right) ^2p^2v^2|x|^2 \end{aligned}$$

we infer that

$$\begin{aligned} \lambda (t)\ge 0, \quad \text{ for } \text{ all } t\ge 0. \end{aligned}$$

Thus, we have the control in the sign of \(\lambda (t)\), but we do not control the sign of each term of \(\lambda (t)\), and it is one difficulty that needs being overcome.

Throughout this paper, we denote the specific constants by \(C_1,C_2,\ldots \) and the generic ones only by C.

Below, we enunciate two assumptions. The first involves the function M and the second one is called into the literature of assumption of small initial data, and it is well used in the context of Kirchhoff models.

Assumption 1

\(M:[0,\infty )\rightarrow (0,\infty )\) is an increasing and continuously differentiable function. There exist positive constants \(m_0\), \(C_1\) and a real number \(q\ge 1\) such that

$$\begin{aligned} 0<m_0\le M(s),\quad \text{ for } \text{ all } s\in [0,\infty ) \end{aligned}$$
(29)

and

$$\begin{aligned} |M(s)| \le C_1|s|^q,\quad \text{ for } \text{ all } s\in [1,\infty ). \end{aligned}$$
(30)

Thus, it holds

$$\begin{aligned} |M(s)| \le C_1(1+|s|^q),\quad \text{ for } \text{ all } s\in [0,\infty ). \end{aligned}$$
(31)

We define the energy associated to the problem (22)–(24) by

$$\begin{aligned} E(t)=\int _{B_1}p^2 v_t^2\,dV+{\overline{M}}\left( \lambda (t)\right) , \end{aligned}$$
(32)

where

$$\begin{aligned} {\overline{M}}(s)=\int _0^sM(\xi )\,d\xi . \end{aligned}$$
(33)

We also define the following auxiliary functional

$$\begin{aligned} \Psi (t)=E(t)+\frac{\delta }{2} \int _{B_1}p^2v_tv\,dx+\frac{\delta ^2}{4} \int _{B_1}p^2v^2\,dx. \end{aligned}$$
(34)

Assumption 2

We suppose that the initial data \((v_0,v_1)\in H_0^1(B_1)\cap H^2(B_1)\times H_0^1(B_1)\) satisfy

$$\begin{aligned} \max \left\{ C_2, C_3\Psi ^{q}(0) \right\} <\delta , \end{aligned}$$
(35)

where

$$\begin{aligned} C_2 =\frac{4C_1C_6}{\sqrt{m_0}}, \quad C_3=\frac{\sqrt{2}C_2}{2m_0^q}, \end{aligned}$$
(36)

here

$$\begin{aligned} C_6=\frac{(N-2)(N+5)C_H}{2}+2(N-2). \end{aligned}$$

To prove the stability of the problem, we need working with \(\Psi (t)\) instead of E(t). First, we observe that

$$\begin{aligned} \left| \frac{\delta }{2} \int _{B_1}p^2v_tv\,dx \right| \le \frac{1}{2} \int _{B_1}p^2v_t^2\,dx +\frac{\delta ^2}{8}\int _{B_1}p^2v^2\,dx. \end{aligned}$$
(37)

Thus,

$$\begin{aligned} \frac{\delta }{2} \int _{B_1}p^2v_tv\,dx \ge -\frac{1}{2} \int _{B_1}p^2v_t^2\,dx -\frac{\delta ^2}{8}\int _{B_1}p^2v^2\,dx. \end{aligned}$$
(38)

We also have

$$\begin{aligned} {\overline{M}}\left( \lambda (t)\right) \ge m_0\lambda (t)\ge 0, \end{aligned}$$
(39)

for all \(t\ge 0\).

Thus, we infer

$$\begin{aligned} \Psi (t)\ge \frac{1}{2}\int _{B_1}p^2v_t^2\,dx +{\overline{M}}\left( \lambda (t)\right) +\frac{\delta ^2}{8}\int _{B_1}p^2v^2\,dx. \end{aligned}$$
(40)

Therefore,

$$\begin{aligned} \Psi (t)\ge \frac{1}{2} \left[ \int _{B_1}p^2v_t^2\,dx +{\overline{M}}\left( \lambda (t)\right) \right] =\frac{1}{2}E(t). \end{aligned}$$
(41)

From (41), we see that to show that the energy associated to (22)–(24) decay exponentially, it is enough to prove that there exist positive constants \(\alpha _1\) and \(\alpha _2\) such that

$$\begin{aligned} \Psi (t)\le \alpha _1e^{-\alpha _2 t}, \end{aligned}$$
(42)

for all \(t\ge 0\). Therefore, we can enunciate the following result which gives us the exponential decay for the energy associated to (22)–(24).

Theorem 1

Assume that Assumptions 1 and 2 hold. Let v be a solution of (22)–(24) in the class

$$\begin{aligned} \begin{array}{c} v\in L^{\infty }(0,T;H_{0}^{1}(B_1)\cap H^{2}(B_{1})),\quad v_{t}\in L^{\infty }(0,T;H_{0}^{1}(B_1)),\\ pv_{tt} \in L^{\infty }(0,T;L^{2}(B_{1})). \end{array} \end{aligned}$$
(43)

Then there exist positive constants \(\alpha _1\) and \(\alpha _2\) such that

$$\begin{aligned} \Psi (t)\le \alpha _{1} e^{-\alpha _{2} t},\quad \text{ for } \text{ all } t\ge 0. \end{aligned}$$
(44)

Next task is to establish a result which ensures the existence and uniqueness of solution to (22)–(24). For this purpose, we need an additional assumption. First, we define

$$\begin{aligned}&{\widetilde{\Psi }}(0) =\Psi (0)+\left[ \frac{M_0}{2}\left( \int _{B_1} |\Delta v_0|^2\,dx \right) ^{\frac{1}{2}} +\beta _{0}M_{0}\left( \int _{B_1} p^{2}|v_{0}|^2\,dx \right) ^{\frac{1}{2}}\right. \\&\left. +\delta \left( \int _{B_1} p^{2} |v_{1}|^2\,dx \right) ^{\frac{1}{2}}\right] ^{2} +\int _{B_1}|\nabla v_1|^2\,dx +\beta _0\int _{B_1}\frac{p^{2}|v_{1}|^{2}}{M(\lambda (0))}\,dx, \end{aligned}$$

where

$$\begin{aligned} M_{0}=\max _{0\le s\le \left( \frac{\delta ^{2}m_{0}}{8 C_{1}^{2}C_{6}^{2}}\right) ^{\frac{1}{2q}}}|M(s)|. \end{aligned}$$

Thus, we consider the following assumption

Assumption 3

We suppose that the initial data \((v_0,v_1)\in H_0^1(B_1)\cap H^2(B_1)\times H_0^1(B_1)\) satisfy

$$\begin{aligned} C_4{\widetilde{\Psi }}^{\frac{1}{2}}(0) +\delta C_5{\widetilde{\Psi }}(0)<\delta , \end{aligned}$$
(45)

where

$$\begin{aligned} C_4=\frac{M_1\sqrt{L_2}[8+8C_{H}(N-2)+C_{H}^{2}(N-2)^2]}{m_0} \quad \text{ and }\quad C_5=\frac{C_{4}^{2}}{m_0}, \end{aligned}$$

here

$$\begin{aligned} M_1=\max _{0\le s\le \left( \frac{\delta ^2m_0}{8 C_1^2C_6^2}\right) ^{\frac{1}{2q}}}|M'(s)| \end{aligned}$$

and

$$\begin{aligned} L_2=2\left( \frac{1}{m_0}+\frac{3(N-2)^2}{4} \right) \frac{\Psi (0)}{\min \left\{ \frac{1}{2},\frac{\delta ^2}{8}\right\} }. \end{aligned}$$
(46)

We observe that it is possible to find \(\delta ,M,v_0\), and \(v_1\) such that the Assumptions 2 and 3 hold. Indeed, given M, we can calculate \(C_2\) and \(C_3\). After this, we choose \(\delta >C_2\). Now, as \(\Psi (0)\) and \({\widetilde{\Psi }}(0)\) depend of \(v_0\) and \(v_1\), it is possible to take a couple of initial data \((v_0,v_1)\) sufficiently small such that \(C_3\Psi ^{q}(0)\) and \(C_4{\widetilde{\Psi }}^{\frac{1}{2}}(0)+\delta C_5{\widetilde{\Psi }}(0)\) become so small that (35) holds.

Theorem 2

(Existence and uniqueness of solution) If Assumptions 1, 2, and 3 are in place, then there exists a unique solution of (22)–(24) in the class (43).

3 Exponential decay

In this section, we prove the exponential decay for the energy associated to the problem (22)–(24). We start with a lemma.

Lemma 3

Let v the solution of (22)–(24) in the class (43). It holds

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Psi (t) +\frac{\delta }{4}\int _{B_1}p^{2}v_{t}^{2}\,dx +\frac{\beta _0m_0\delta }{4}\int _{B_1}p^{2}v^{2}\,dx \le 0, \end{aligned}$$
(47)

for all \(t\ge 0\).

Proof

Multiplying (22) by \(v_t\) and integrating over \(B_1\), we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _{B_1}p^{2}v_{t}^{2}\,dx +\frac{1}{2}M(\lambda (t))\frac{d}{dt} \left[ \int _{B_1}\Big (|\nabla v|^{2}+\beta _0 p^{2}v^{2}\Big )\,dx \right] \end{aligned}$$
$$\begin{aligned} +\delta \int _{B_1}p^{2}{v}_{t}^{2}\,dx=0. \end{aligned}$$
(48)

\(\square \)

We observe that

$$\begin{aligned}&\frac{1}{2}M(\lambda (t))\frac{d}{dt} \left[ \int _{B_1}\Big (|\nabla v|^2+\beta _{0}p^{2}v^{2}\Big )\,dx \right] =\frac{1}{2}M(\lambda (t))\frac{d}{dt}\lambda (t)\\&\quad +\frac{1}{2}M(\lambda (t))\frac{d}{dt} \left\{ \int _{B_1}\left[ \left( \beta _0-\beta _1|x|^2 \right) p^{2}v^{2}+(N-2)\langle \nabla v,xpv\rangle \right] \,dx\right\} \\&=\frac{1}{2}\frac{d}{dt}{\overline{M}}(\lambda (t))\\&\quad +\frac{1}{2}M(\lambda (t))\frac{d}{dt} \left\{ \int _{B_1}\left[ \left( \beta _0-\beta _1|x|^2\right) p^{2}v^{2}+(N-2)\langle \nabla v,xpv\rangle \right] \,dx\right\} \\&=\frac{1}{2}\frac{d}{dt}{\overline{M}}(\lambda (t)) +M(\lambda (t))\left\{ \int _{B_1}\left[ \left( \beta _0-\beta _1|x|^2\right) p^{2}vv_{t}\right. \right. \end{aligned}$$
$$\begin{aligned} +(N-2)(\langle \nabla v,xpv_t\rangle +\langle \nabla v_t,xpv\rangle ) \Big ] \,dx \Big \}. \end{aligned}$$
(49)

Combining (48) with (49), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \int _{B_1}p^2|v_t|^2\,dx +{\overline{M}}(\lambda (t)) \right] +\delta \int _{B_1}p^2|v_t|^2\,dx\nonumber \\&=-M(\lambda (t))\Big \{\int _{B_1} \Big [\left( \beta _0-\beta _1|x|^2 \right) p^2vv_t\nonumber \\&\qquad +(N-2)(\langle \nabla v,xpv_t\rangle +\langle \nabla v_t,xpv\rangle ) \Big ]\,dx\Big \}. \end{aligned}$$
(50)

Now, we are going to estimate the right hand side of (50). Using (31), Hölder’s inequality, and Lemma 1, we have

$$\begin{aligned}&M(\lambda (t))\int _{B_1} \left( \beta _0-\beta _1|x|^2 \right) p^2vv_t\,dx\nonumber \\&\le C_1\left( 1+|\lambda (t)|^q \right) \left( \beta _0+\beta _1 \right) \left( \int _{B_1}p^2v^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}. \end{aligned}$$
(51)

From (51) and using Lemma 1, we infer

$$\begin{aligned}&M(\lambda (t)) \int _{B_1}\left( \beta _0 -\beta _1|x|^2\right) p^2vv_t\,dx\nonumber \\&\le C_1\left( 1+|\lambda (t)|^q \right) \left( \beta _0+\beta _1 \right) C_H\left( \int _{B_1}|\nabla v|^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}. \end{aligned}$$
(52)

Now, from (31) and Hölder’s inequality, we also obtain

$$\begin{aligned}&M(\lambda (t)) \int _{B_1} \langle \nabla v,xpv_t\rangle \,dx\nonumber \\&\le C_1\left( 1+|\lambda (t)|^q \right) \left( \int _{B_1}|\nabla v|^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}. \end{aligned}$$
(53)

Using (31), Lemma 1, Gauss’ theorem, and Hölder’s inequality, we have

$$\begin{aligned}&M(\lambda (t)) \int _{B_1} \langle \nabla v_t,xpv\rangle \,dx=-M(\lambda (t))\sum _{i=1}^n \int _{B_1}v_t\frac{\partial (x_ipv)}{\partial x_i} \,dx\nonumber \\&=-M(\lambda (t))\int _{B_1} v_t\left( p\langle x,\nabla v\rangle +p^2v|x|^2+pv \right) \,dx\nonumber \\&\le C_1\left( 1+|\lambda (t)|^q\right) \int _{B_1}\left( p|v_t||\nabla v|+ p^2|v_t||v|+ \frac{p^2}{p}|v||v_t| \right) \,dx\nonumber \\&\le C_1\left( 1+|\lambda (t)|^q \right) \left( 1+3C_H \right) \left( \int _{B_1}|\nabla v|^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}. \end{aligned}$$
(54)

From (50), (52)–(54), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \int _{B_1}p^2|v_t|^2\,dx +{\overline{M}}(\lambda (t)) \right] +\delta \int _{B_1}p^2|v_t|^2\,dx\nonumber \\&\le C_1\left( 1+|\lambda (t)|^q \right) C_6\left( \int _{B_1}|\nabla v|^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}, \end{aligned}$$
(55)

where

$$\begin{aligned} C_6=\frac{(N-2)(N+5)C_H}{2}+2(N-2). \end{aligned}$$

On the other hand, multiplying (22) by v, we have

$$\begin{aligned}&\int _{B_1}p^2v_{tt}v\,dx +M(\lambda (t))\int _{B_1}|\nabla v|^2\,dx +\beta _0 M(\lambda (t))\int _{B_1} p^2v^2\,dx\nonumber \\&+\delta \int _{B_1}p^2vv_t\,dx=0. \end{aligned}$$
(56)

We observe that

$$\begin{aligned} \int _{B_1}p^2v_{tt}v\,dx =\frac{d}{dt}\int _{B_1}p^2v_tv\,dx -\int _{B_1}p^2|v_t|^2\,dx. \end{aligned}$$
(57)

From (56) and (57), we infer

$$\begin{aligned}&\frac{d}{dt} \left[ \int _{B_1}p^2v_tv\,dx +\frac{\delta }{2} \int _{B_1}p^2v^2\,dx \right] -\int _{B_1}p^2|v_t|^2\,dx +M(\lambda (t))\int _{B_1}|\nabla v|^2\,dx\nonumber \\&+\beta _0M(\lambda (t))\int _{B_1} p^2v^2\,dx=0. \end{aligned}$$
(58)

Multiplying (58) by \(\frac{\delta }{4}\), adding the resultant equation with (55), and observing the definition of \(\Psi \) (see (34)), we have

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \Psi (t) +\frac{3\delta }{4} \int _{B_1}p^2|v_t|^2\,dx +\frac{\delta }{4}M(\lambda (t))\int _{B_1}|\nabla v|^2\,dx +\frac{\delta \beta _0}{4}M(\lambda (t))\int _{B_1} p^2v^2\,dx\nonumber \\&\le C_1C_6\left( 1+|\lambda (t)|^q \right) \left( \int _{B_1}|\nabla v|^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}. \end{aligned}$$
(59)

From the elementary inequality \(2ab\le a^2+b^2\), we have

$$\begin{aligned}&C_1C_6\left( \int _{B_1}|\nabla v|^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}\nonumber \\&\le \frac{C_1^2C_6^2}{\delta }\int _{B_1}|\nabla v|^2\,dx +\frac{\delta }{4}\int _{B_1}p^2|v_t|^2\,dx \end{aligned}$$
(60)

and

$$\begin{aligned}&C_1C_6|\lambda (t)|^q \left( \int _{B_1}|\nabla v|^2\,dx\right) ^{\frac{1}{2}} \left( \int _{B_1}p^2|v_t|^2\,dx\right) ^{\frac{1}{2}}\nonumber \\&\le \frac{m_0\delta }{8}\int _{B_1}|\nabla v|^2\,dx +\frac{2C_1^2C_6^2|\lambda (t)|^{2q}}{m_0\delta }\int _{B_1}p^2|v_t|^2\,dx. \end{aligned}$$
(61)

From (59)–(61), we conclude that

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \Psi (t) +\frac{1}{\delta } \left( \frac{\delta ^2}{2} -\frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0} \right) \int _{B_1}p^2|v_t|^2\,dx\\&+\left( \frac{\delta m_0}{8} -\frac{C_1^2C_6^2}{\delta } \right) \int _{B_1} |\nabla v|^2\,dx +\frac{\delta m_0 \beta _0}{4}\int _{B_1} p^2v^2\,dx \le 0. \end{aligned}$$

Since \(\delta >C_2\), we infer

$$\begin{aligned} \frac{\delta m_0}{8} -\frac{C_1^2C_6^2}{\delta } >\frac{m_0\delta }{16}. \end{aligned}$$

Therefore,

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \Psi (t)+\frac{1}{\delta } \left( \frac{\delta ^2}{2} -\frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0} \right) \int _{B_1}p^2|v_t|^2\,dx\nonumber \\&+\frac{\delta m_0}{16}\int _{B_1} |\nabla v|^2\,dx +\frac{\delta m_0 \beta _0}{4}\int _{B_1} p^2v^2\,dx \le 0. \end{aligned}$$
(62)

As \(\lambda (t)\ge 0\), for all \(t\in [0,T]\), and \(0<m_0\le M(s)\), for all \(s\ge 0\), we have

$$\begin{aligned} \frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0} \le \frac{2 C_1^2C_6^2[{\overline{M}}(\lambda (t))]^{2q}}{m_0^{2q+1}}, \end{aligned}$$
(63)

for all \(t\in [0,T]\). We observe that

$$\begin{aligned} -\frac{\delta }{2}\int _{B_1}p^2v_tv\,dx \le \frac{1}{2}\int _{B_1}p^2|v_t|^2\,dx +\frac{\delta ^2}{8}\int _{B_1}p^2v^2\,dx. \end{aligned}$$
(64)

Thus,

$$\begin{aligned} {\overline{M}}(\lambda (t)) \le {\overline{M}}(\lambda (t))+ \frac{\delta }{2}\int _{B_1}p^2v_tv\,dx +\frac{1}{2}\int _{B_1}p^2|v_t|^2\,dx +\frac{\delta ^2}{8}\int _{B_1}p^2v^2\,dx\nonumber \\ \le \Psi (t). \end{aligned}$$
(65)

Therefore, from (63)–(65), we have

$$\begin{aligned} \frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0} \le \frac{2 C_1^2C_6^2\Psi ^{2q}(t)}{m_0^{2q+1}}, \end{aligned}$$
(66)

for all \(t\in [0, T]\). Using (66) and the Assumption 2, we infer

$$\begin{aligned} \frac{2 C_1^2C_6^2|\lambda (0)|^{2q}}{m_0} \le \frac{2 C_1^2C_6^2\Psi ^{2q}(0)}{m_0^{2q+1}} <\frac{ \delta ^2}{4}. \end{aligned}$$
(67)

Next task is to prove that

$$\begin{aligned} \frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0} <\frac{\delta ^2}{4}, \end{aligned}$$
(68)

for all \(t\in [0, T]\). We suppose that (68) does not hold. From (67) and of the continuity of the function \(t\mapsto \frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0}\) there exists \(t^*\in (0,T]\) such that

$$\begin{aligned} \frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0} <\frac{\delta ^2}{4}, \end{aligned}$$
(69)

for all \(t\in [0,t^*)\), and

$$\begin{aligned} \frac{2 C_1^2C_6^2|\lambda (t^*)|^{2q}}{m_0} =\frac{\delta ^2}{4}. \end{aligned}$$
(70)

Integrating (62) from 0 to \(t^*\), we have

$$\begin{aligned} \frac{1}{2}\left[ \Psi _m(t^*)-\Psi (0) \right] +\frac{1}{\delta } \int _0^{t^*}\left( \frac{\delta ^2}{2}-\frac{2 C_1^2C_6^2|\lambda (t)|^{2q}}{m_0}\right) \int _{B_1}p^2|v_t|^2\,dx\,dt \le 0. \end{aligned}$$
(71)

Combining (69) with (71), we obtain

$$\begin{aligned} \Psi (t^*)\le \Psi (0). \end{aligned}$$
(72)

The estimate (63), (65), (72), and the Assumption 2 give us that

$$\begin{aligned} \frac{2 C_1^2C_6^2|\lambda (t^* )|^{2q}}{m_0} \le \frac{2 C_1^2C_6^2\Psi ^{2q}(0)}{m_0^{2q+1}} <\frac{\delta ^2}{4}, \end{aligned}$$
(73)

which is a contradiction with (70). Thus, (68) holds.

Therefore, (62) and (68) allow us to conclude that (47) holds. \(\Box \)

Proof of Theorem

1. Observing Nakao’s Lemma, it is enough to prove that here exists a positive constant C such that

$$\begin{aligned} \Psi (t)\le C\left( \Psi (t)-\Psi (t+1)\right) \end{aligned}$$
(74)

for all \(t\ge 0\).

Thus, let \(t\ge 0\) be a fixed real number. To simplify the notation, we define \(F^2(t)=\Psi (t)-\Psi (t+1)\). Integrating (47) from t to \(t+1\), we have

$$\begin{aligned} \int _t^{t+1}\int _{B_1}p^2v_t^2\,dx\,dt \le \frac{2}{\delta } F^2(t), \end{aligned}$$
(75)

for all \(t\ge 0\). Using the mean value theorem for integrals, there exist \(t_1\in \Big [t,t+\frac{1}{4}\Big ]\) and \(t_2\in \Big [t+\frac{3}{4},t+1\Big ]\) such that

$$\begin{aligned} \int _t^{t+\frac{1}{4}}\int _{B_1}p^2v_t^2\,dx\,dt =\frac{1}{4}\int _{B_1}p^2(x)v_t^2(x,t_1)\,dx \end{aligned}$$
(76)

and

$$\begin{aligned} \int _{t+\frac{3}{4}}^{t+1}\int _{B_1}p^2v_t^2\,dx\,dt =\frac{1}{4}\int _{B_1}p^2(x)v_t^2(x,t_2)\,dx. \end{aligned}$$
(77)

Thus, (75)–(77) give us

$$\begin{aligned} \int _{B_1}p^2(x)v_t^2(x,t_1)\,dx +\int _{B_1}p^2(x)v_t^2(x,t_2)\,dx \le \frac{8}{\delta }F^2(t). \end{aligned}$$
(78)

On the other hand, multiplying (22) by v and integrating over \(B_1\times (t_1,t_2)\), we have

$$\begin{aligned}&\int _{t_1}^{t_2}M(\lambda (t)) \int _{B_1}\left[ |\nabla v|^2+\beta _0 p^2v^2\right] \,dx\,dt =\int _{B_1}p^2(x)v_t(x,t_1)v(x,t_1)\,dx\nonumber \\&-\int _{B_1}p^2(x)v_t(x,t_2)v(x,t_2)\,dx +\int _{t_1}^{t_2}\int _{B_1}p^2v_t^2\,dx\,dt -\delta \int _{t_1}^{t_2}\int _{B_1}p^2v_tv\,dx\,dt.\nonumber \\ \end{aligned}$$
(79)

Using the assumption that M is an increasing function, we obtain

$$\begin{aligned}&{\overline{M}}(\lambda (t)) =\int _0^{\lambda (t)}M(\xi )\,d\xi \le M(\lambda (t))\lambda (t)\nonumber \\&=M(\lambda (t))\left[ \int _{B_1}\Big (|\nabla v|^2-(N-2)\langle \nabla v,xpv\rangle +\beta _1p^2v^2|x|^2\Big )\,dx \right] \nonumber \\&\le C M(\lambda (t))\int _{B_1}\Big (|\nabla v|^2+p^2v^2|x|^2\Big )\,dx\nonumber \\&\le C M(\lambda (t)) \int _{B_1}\Big (|\nabla v|^2+\frac{N(N-2)}{4}p^2v^2\Big )\,dx. \end{aligned}$$
(80)

From (79) and (80), we have

$$\begin{aligned} {\overline{M}}(\lambda (t))\le & {} C\left( \int _{B_1}p^2(x)|v_t(x,t_1)||v(x,t_1)|\,dx +\int _{B_1}p^2(x)|v_t(x,t_2)||v(x,t_2)|\,dx \right. \nonumber \\&\left. +\int _{t_1}^{t_2}\int _{B_1}p^2v_t^2\,dx\,dt +\delta \int _{t_1}^{t_2}\int _{B_1}p^2|v_t||v|\,dx\,dt \right) . \end{aligned}$$
(81)

Now, we are going to estimate the right-hand side of (81). Observing (40) and that \(\Psi \) is decreasing (see Lemma 3), we obtain

$$\begin{aligned} \int _{B_1}p^2(x)v^2(x,t_i)\,dx \le \frac{8}{\delta ^2}\Psi (t_i) \le \frac{8}{\delta ^2}\Psi (t), \end{aligned}$$
(82)

for \(i=1,2\).

Using (78), (82), and the elementary inequality \(2ab\le \varepsilon a^2+\frac{b^2}{\varepsilon }\), for each \(\varepsilon >0\), we have

$$\begin{aligned} \int _{B_1}p^2(x)v_t(x,t_i)v(x,t_i)\,dx \le \frac{4}{\varepsilon \delta }F^2(t) +\frac{4\varepsilon }{\delta ^2}\Psi (t), \end{aligned}$$
(83)

for \(i=1,2\). Using one more time the inequality \(2ab\le \varepsilon a^2+\frac{b^2}{\varepsilon }\) and (75), we infer

$$\begin{aligned} \delta \int _{t_1}^{t_2}\int _{B_1}p^2v_tv\,dx\,dt\le & {} \frac{\delta }{2\varepsilon }\int _{t_1}^{t_2}\int _{B_1}p^2v_t^2\,dx\,dt +\frac{\varepsilon \delta }{2}\sup _{t\le \xi \le t+1}\int _{B_1}p^2v^2\,dx\nonumber \\\le & {} \frac{1}{\varepsilon }F^2(t) +\frac{\varepsilon \delta }{2}\Psi (t). \end{aligned}$$
(84)

Since \(\Psi \) is decreasing, we can use Lemma 3 to conclude that

$$\begin{aligned} \int _{t_1}^{t_2}\int _{B_1}p^2v^2\,dx\,dt \le \frac{2}{\delta \beta _0m_0}F^2(t). \end{aligned}$$
(85)

Integrating (34) from \(t_1\) to \(t_2\), and observing (75), (80), (84), and (85), we have

$$\begin{aligned} \int _{t_1}^{t_2}\Psi (t)\,dt \le C(\varepsilon ) F^2(t)+\varepsilon \left( \frac{8}{\delta ^2}+\frac{3\delta }{4} \right) \Psi (t). \end{aligned}$$
(86)

Using the mean value theorem for integrals, there exists \(\tau ^*\in [t_1,t_2]\) such that

$$\begin{aligned} \frac{1}{2}\Psi (\tau ^*)\le (t_2-t_1)\Psi (\tau ^*) =\int _{t_1}^{t_2}\Psi (t)\,dt. \end{aligned}$$
(87)

Combining (86) with (87), we infer

$$\begin{aligned} \Psi (\tau ^*) \le C(\varepsilon ) F^2(t)+2\varepsilon \left( \frac{8}{\delta ^2}+\frac{3\delta }{4} \right) \Psi (t). \end{aligned}$$
(88)

Taking the same way of (50) and (58), we infer

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \Psi (t)+\frac{3\delta }{4} \int _{B_1}p^2v_t^2\,dx +\frac{\delta }{4}M(\lambda (t))\int _{B_1}|\nabla v|^2\,dx\nonumber \\&+\frac{\delta \beta _0}{4}M(\lambda (t))\int _{B_1} p^2v^2\,dx +M(\lambda (t))\left\{ \int _{B_1}\left[ \left( \frac{N(N-2)}{4} -\beta _1|x|^2\right) p^2vv_t\right. \right. \nonumber \\&+(N-2)(\langle \nabla v,xpv_t\rangle +\langle \nabla v_t,xpv\rangle ) \Big ]\,dx\Bigg \}=0. \end{aligned}$$
(89)

Integrating (89) from t to \(\tau ^*\) and, after this, adding the term

$$\begin{aligned} \frac{3\delta }{4} \int _{t}^{\tau ^*}\int _{B_1}p^2v_t^2\,dx\,d\tau +\frac{\delta }{8}\int _{t}^{\tau ^*} M(\lambda (\tau ))\int _{B_1}|\nabla v|^2\,dx\,d\tau \end{aligned}$$

in both sides of the resultant equation, we have

$$\begin{aligned}&\frac{1}{2}\Psi (t) +\frac{3\delta }{4} \int _{t}^{\tau ^*}\int _{B_1}p^2v_t^2\,dx\,d\tau +\frac{\delta }{8}\int _{t}^{\tau ^*} M(\lambda (\tau ))\int _{B_1}|\nabla v|^2\,dx\,d\tau \nonumber \\&=\frac{1}{2}\Psi (\tau ^*) +\frac{3\delta }{2} \int _{t}^{\tau ^*}\int _{B_1}p^2v_t^2\,dx\,d\tau +\frac{3\delta }{8}\int _{t}^{\tau ^*} M(\lambda (t))\int _{B_1}|\nabla v|^2\,dx\,d\tau \nonumber \\&+\frac{\delta \beta _0}{4} \int _{t}^{\tau ^*}M(\lambda (\tau ))\int _{B_1} p^2v^2\,dx\,d\tau +\int _{t}^{\tau ^*}M(\lambda (\tau ))\Big \{\int _{B_1}\Big [\left( \beta _0-\beta _1|x|^2\right) p^2vv_t\nonumber \\&+(N-2)(\langle \nabla v,xpv_t\rangle +\langle \nabla v_t,xpv\rangle ) \Big ]\,dx\Big \}\,d\tau . \end{aligned}$$
(90)

Analogously to (62), we have

$$\begin{aligned}&\frac{1}{2} \Psi (t) +\frac{1}{\delta }\int _t^{\tau ^*}\left( \frac{\delta ^2}{2} -\frac{2 C_2^2|\lambda (t)|^{2q}}{m_0}\right) \int _{B_1}p^2v_t^2\,dx\,d\tau +\frac{\delta m_0 \beta _0}{4}\int _t^{\tau ^*}\int _{B_1} p^2v^2\,dx\,d\tau \nonumber \\&\le \frac{1}{2} \Psi (\tau ^*) +\frac{3\delta }{2}\int _t^{\tau ^*}\int _{B_1}p^2v_t^2\,dx\,d\tau +\frac{\delta \beta _0}{4}\int _t^{\tau ^*}M(\lambda (\tau ))\int _{B_1} p^2v^2\,dx\,d\tau \nonumber \\&+\frac{3\delta }{8}\int _t^{\tau ^*}M(\lambda (\tau ))\int _{B_1}|\nabla v|^2\,dx\,d\tau . \end{aligned}$$
(91)

Combining (75), (79), (83), (84), (88), and (91), we conclude that

$$\begin{aligned} \left[ \frac{1}{2}-\varepsilon C\right] \Psi (t)\le C F^2(t), \end{aligned}$$
(92)

for all \(t\ge 0\). Taking \(\varepsilon >0\) small enough, we conclude that (74) holds. Therefore, from Nakao’s lemma we obtain that \(\Psi \) decay exponentially, i.e., Theorem 1 is proved. \(\Box \)

4 Proof of the existence and uniqueness theorem

We use the Faedo–Galerkin method. Let \((w_j)_{j\in {\mathbb {N}}}\) be a bases in \(H_0^1(B_1)\cap H^2(\Omega )\). For each \(m\in {\mathbb {N}}\), we denote \(U_m\) the m-dimensional subspaces spanned by the first m vectors of \((w_j)_{j\in {\mathbb {N}}}\). Let \(T > 0\) be any fixed positive number. For each \(m \in {\mathbb {N}}\), we are looking for a \(0< T_m\le T\) and \(v_m:B_1\times [0,T_m]\rightarrow {\mathbb {R}}\) such that

$$\begin{aligned} v_m(x,t)=\sum _{i=1}^m\alpha _{im}(t)w_i(x), \end{aligned}$$

and it satisfies the approximate problem

$$\begin{aligned}&(p^2 v_m''(t),w_j)+M(\lambda _m(t))\left[ (\nabla v_m(t),\nabla w_j)+\beta _0(p^2v_m(t),w_j)\right] \nonumber \\&+(\delta p^2v_m'(t),w_j)=0, \end{aligned}$$
(93)
$$\begin{aligned}&v_m(0)=v_{0m}=\sum _{i=1}^mv_0^iw_i\rightarrow v_0 \text{ in } H_0^1(B_1)\cap H^2(B_1), \end{aligned}$$
(94)
$$\begin{aligned}&v_m'(0)=v_{1m}=\sum _{i=1}^mv_1^iw_i\rightarrow v_1 \text{ in } H_0^1(\Omega ), \end{aligned}$$
(95)

where \('=\frac{d}{dt}\), \(1 \le j \le m\), \(v_0^i,v_1^i\), \(i=1,\ldots , m\), are known scalars, and

$$\begin{aligned} \lambda _m(t)=\int _{B_1}\Big (|\nabla v_m|^2-(N-2)\langle \nabla v_m,xpv_m\rangle +\beta _1 p^2v_m^2|x|^2\Big )\,dx, \end{aligned}$$

where \(\beta _1=\left( \frac{N-2}{2}\right) ^2\). From Ordinary Differential Equations Theory (for instance, see [13]), it is possible to prove that (93)–(95) has a local solution.

From (93) we have the following approximate equation

$$\begin{aligned}&(p^2 v_m''(t),w)+M(\lambda _m(t))[(\nabla v_m(t),\nabla w)+\beta _0(p^2v_m(t),w)]\nonumber \\&\qquad +(\delta p^2v_m'(t),w)=0, \end{aligned}$$
(96)

for all \(w\in U_m\).

Estimate 1

Initially, it is necessary to observe that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Psi _m(t) +\frac{\delta }{4}\int _{B_1}p^2|v_m'|^2\,dx +\frac{\beta _0m_0\delta }{4}\int _{B_1}p^2v_m^2\,dx \le 0, \end{aligned}$$
(97)

for all \(t\le T_m\), where

$$\begin{aligned} \Psi _m(t) =\int _{B_1}p^2 |v_m'|^2\,dV +{\overline{M}}\left( \lambda _m(t)\right) +\frac{\delta }{2} \int _{B_1}p^2v_m'v_m\,dx +\frac{\delta ^2}{4} \int _{B_1}p^2v_m^2\,dx. \end{aligned}$$
(98)

Indeed, taking in (96) \(w=v_m'\), we have

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _{B_1}p^2|v_m'|^2\,dx +\frac{1}{2}M(\lambda _m(t))\frac{d}{dt} \left[ \int _{B_1}\Big (|\nabla v_m|^2+\beta _0p^2v_m^2\Big )\,dx \right] \nonumber \\&\qquad +\delta \int _{B_1}p^2|v_m'|^2\,dx=0. \end{aligned}$$
(99)

We observe that (99) is similar to (48) with v replaced by \(v_m\). Moreover, (97) is similar to (47). Therefore, to prove that (97) holds, it is enough to follow the steps of the proof of Lemma 3.

On the other hand, observing (64), we have,

$$\begin{aligned} \Psi _m(t)\ge \frac{1}{2}\int _{B_1}p^2 |v_m'|^2\,dV +{\overline{M}}\left( \lambda _m(t)\right) +\frac{\delta ^2}{8} \int _{B_1}p^2v_m^2\,dx. \end{aligned}$$
(100)

Thus, integrating (97) from 0 to t and observing (100), we have

$$\begin{aligned} \frac{1}{2}\int _{B_1}p^2 |v_m'|^2\,dV +{\overline{M}}\left( \lambda _m(t)\right) +\frac{\delta ^2}{8}\int _{B_1}p^2v_m^2\,dx \le \Psi _m(0). \end{aligned}$$
(101)

Therefore,

$$\begin{aligned} \int _{B_1}p^2 |v_m'|^2\,dV +{\overline{M}}\left( \lambda _m(t)\right) +\int _{B_1}p^2v_m^2\,dx \le L_1, \end{aligned}$$
(102)

for all \(t\in [0,T_m)\), where \(L_1=\frac{\Psi (0)}{\min \left\{ \frac{1}{2},\frac{\delta ^2}{8}\right\} }\), which is the Estimate 1. This estimate is enough to extend the approximate solution to whole \(t\ge 0\). This gives us that (102) holds with \(T_m\) replaced by \(T<\infty \). Thus, we have that

$$\begin{aligned} (pv_m)_{m\in {\mathbb {N}}} \quad \text{ is } \text{ bounded } \text{ in } L^{\infty }(0,T;L^2(B_1)) \end{aligned}$$

and

$$\begin{aligned} (pv_m')_{m\in {\mathbb {N}}} \quad \text{ is } \text{ bounded } \text{ in } L^{\infty }(0,T;L^2(B_1)). \end{aligned}$$

Estimate 2

From (102) we can estimate \(\Vert \nabla v_m(t)\Vert _2\). Indeed, we observe that

$$\begin{aligned}&\int _{B_1}|\nabla v_m|^2\,dx =\lambda _m(t) +\int _{B_1}\left( (N-2)\langle \nabla v_m,xpv_m\rangle -\beta _1p^2v_m^2|x|^2\right) \,dx\nonumber \\&\qquad \le \frac{{\overline{M}}(\lambda _m(t))}{m_0} +\frac{1}{2}\int _{B_1}|\nabla v_m|^2\,dx +3\beta _1\int _{B_1}p^2v_m^2\,dx. \end{aligned}$$
(103)

From (102) and (103), we infer

$$\begin{aligned} \int _{B_1}|\nabla v_m|^2\,dx \le 2\left( \frac{1}{m_0}+3\beta _1 \right) L_1:= L_2, \end{aligned}$$
(104)

for all \(t\in [0,T]\). Therefore,

$$\begin{aligned} (v_m)_{m\in {\mathbb {N}}} \quad \text{ is } \text{ bounded } \text{ in } L^{\infty }(0,T;H_0^1(B_1)). \end{aligned}$$

Estimate 3

Multiplying (96) by \(\frac{1}{M(\lambda _m(t))}\), differentiating the resultant equation with respect to t, and taking \(w=v_m''\), we have

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx +\int _{B_1}|\nabla v_m'|^2\,dx +\beta _0\int _{B_1}p^2|v_m'|^2 \right] \nonumber \\&\qquad +\delta \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx\nonumber \\&\quad =\frac{1}{2}\int _{B_1}\frac{p^2|v_m''|^2M'(\lambda _m(t))\lambda _m'(t)}{M^2(\lambda _m(t))}\,dx\nonumber \\&\qquad +\delta \int _{B_1}\frac{p^2v_m'v_m''M'(\lambda _m(t))\lambda _m'(t)}{M^2(\lambda _m(t))}\,dx. \end{aligned}$$
(105)

Using Hölder’s inequality and Lemma 1, we have

$$\begin{aligned} |\lambda _m'(t)|= & {} \Big |\int _{B_1} \Big (2\langle \nabla v_m,\nabla v_m'\rangle -(N-2)(\langle xpv_m,\nabla v_m'\rangle \nonumber \\&\qquad +\langle xp v_m',\nabla v_m\rangle )+\beta _1|x|^2p^2v_mv_m'\Big )\,dx \Big |\nonumber \\\le & {} C_7 \left( \int _{B_1} |\nabla v_m|^2\,dx \right) ^{\frac{1}{2}} \left( \int _{B_1} |\nabla v_m'|^2\,dx \right) ^{\frac{1}{2}}, \end{aligned}$$
(106)

where

$$\begin{aligned} C_7=2+2C_H(n-2)+C_H^2\beta _1. \end{aligned}$$

From this and using Estimate 2, we obtain

$$\begin{aligned} |\lambda _m'(t)| \le C_7\sqrt{L_2} \left( \int _{B_1} |\nabla v_m'|^2\,dx \right) ^{\frac{1}{2}}. \end{aligned}$$
(107)

Similarly to (68), it is possible to verify that

$$\begin{aligned} 0\le \lambda _m(t) \le \left( \frac{\delta ^2m_0}{8 C_1^2C_7^2} \right) ^{\frac{1}{2q}}, \end{aligned}$$
(108)

for all \(t\in [0,T]\). Thus, observing (108) and the definition of \(M_1\) in (36), we obtain

$$\begin{aligned}&\frac{1}{2}\int _{B_1}\frac{p^2|v_m''|^2M'(\lambda _m(t))\lambda _m'(t)}{M^2(\lambda _m(t))}\,dx\nonumber \\&\qquad \le \frac{M_1C_7\sqrt{L_2}}{2m_0} \left( \int _{B_1} |\nabla v_m'|^2\,dx \right) ^{\frac{1}{2}} \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx \end{aligned}$$
(109)

and

$$\begin{aligned}&\delta \int _{B_1}\frac{p^2v_m'v_m''M'(\lambda _m(t))\lambda _m'(t)}{M^2(\lambda _m(t))}\,dx\nonumber \\&\le \frac{\delta }{8}\int _{B_1}p^2|v_m'|^2\,dx +\frac{2\delta L_2 M_1^2C_7^2}{m_0^3}\int _{B_1}|\nabla v_m'|^2\,dx\int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx. \end{aligned}$$
(110)

Substituting (109) and (110) into (105), we infer

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx +\int _{B_1}|\nabla v_m'|^2\,dx +\beta _0\int _{B_1}p^2|v_m'|^2 \right] \nonumber \\&\qquad +\delta \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx\nonumber \\&\le +\frac{\delta }{8}\int _{B_1}p^2|v_m'|^2\,dx +C_8 \left( \int _{B_1} |\nabla v_m'|^2\,dx \right) ^{\frac{1}{2}} \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx\nonumber \\&\qquad +\delta C_9 \int _{B_1}|\nabla v_m'|^2\,dx \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx, \end{aligned}$$
(111)

where

$$\begin{aligned} C_8=\frac{M_1\sqrt{L_2}C_6}{2m_0} \quad \text{ and }\quad C_9=\frac{2M_1^2L_2C_6^2}{m_0^3}. \end{aligned}$$
(112)

Adding (97) with (111), we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} {\widetilde{\Psi }}_m(t) +\frac{\delta }{8} \int _{B_1}p^2|v_m'|^2\,dx +\left( \frac{\delta }{2} -\Lambda _m(t) \right) \int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx \le 0, \end{aligned}$$
(113)

where

$$\begin{aligned} {\widetilde{\Psi }}_m(t) =\Psi _m(t) +\int _{B_1}\frac{p^2|v_m''|^2}{M(\lambda _m(t))}\,dx +\int _{B_1}|\nabla v_m'|^2\,dx +\beta _0\int _{B_1}p^2|v_m'|^2\,dx \end{aligned}$$

and

$$\begin{aligned} \Lambda _m(t) =C_8\left( \int _{B_1} |\nabla v_m'|^2\,dx \right) ^{\frac{1}{2}} +\delta C_9\int _{B_1}|\nabla v_m'|^2\,dx. \end{aligned}$$

We are going to prove that

$$\begin{aligned} \Lambda _m(t)<\frac{\delta }{8}, \end{aligned}$$
(114)

for all \(t\in [0,T]\).

We have

$$\begin{aligned} \Lambda _m(t) \le C_8{\widetilde{\Psi }}_m^{\frac{1}{2}}(t) +\delta C_9{\widetilde{\Psi }}_m(t), \end{aligned}$$
(115)

for all \(t\in [0,T]\). Using (115) and the Assumption 3, we infer

$$\begin{aligned} \Lambda _m(0) \le C_8{\widetilde{\Psi }}_m^{\frac{1}{2}}(0) +\delta C_9{\widetilde{\Psi }}_m(0) <\frac{\delta }{8}. \end{aligned}$$
(116)

We suppose that (114) does not hold. From (116) and the continuity of the function \(t\mapsto \Lambda _m(t)\), there exists \(t^*>0\) such that

$$\begin{aligned} \Lambda _m(t)<\frac{\delta }{8}, \end{aligned}$$
(117)

for all \(t\in [0,t^*)\) and

$$\begin{aligned} \Lambda _m(t^*)=\frac{\delta }{8}. \end{aligned}$$
(118)

Integrating (113) from 0 to \(t^*\) and observing (117), we have

$$\begin{aligned} {\widetilde{\Psi }}_m(t^*) \le {\widetilde{\Psi }}_m(0). \end{aligned}$$
(119)

The estimate (115), (119), and the Assumption 3 give us that

$$\begin{aligned} \Lambda _m(t^*)\le C_8{\widetilde{\Psi }}_m^{\frac{1}{2}}(0) +\delta C_{9}{\widetilde{\Psi }}_m(0) <\frac{\delta }{8}, \end{aligned}$$
(120)

which is a contradiction with (118). Thus, (114) holds.

Therefore, integrating (113) from 0 to \(t<T\), and observing (114), we conclude that

$$\begin{aligned} {\widetilde{\Psi }}_m(t) \le {\widetilde{\Psi }}_m(0)\le L_3, \end{aligned}$$
(121)

for all \(t\in [0,T]\), which is the Estimate 3. Thus,

$$\begin{aligned} (v_m')_{m\in {\mathbb {N}}} \quad \text{ is } \text{ bounded } \text{ in } L^{\infty }(0,T;H_0^1(B_1)) \end{aligned}$$

and

$$\begin{aligned} (pv_m'')_{m\in {\mathbb {N}}} \quad \text{ is } \text{ bounded } \text{ in } L^{\infty }(0,T;L^2(B_1)). \end{aligned}$$

The presence of singularities does not allow to estimate the norm of \((v_m)_{m\in {\mathbb {N}}}\) in \(L^{\infty }(0,T;H_0^1(\Omega )\cap H^2(\Omega ))\) as in \({\mathbb {R}}^N\) case. Thus, the Estimates 1, 2, and 3 are not enough to pass to the limit in approximate equation. To overcome this difficulty it is necessary to make one more estimate.

Estimate 4

Multiplying (96) by \(\frac{1}{M(\lambda _m(t))}\), we obtain

$$\begin{aligned} \frac{(p^2 v_m''(t),w) +(\delta p^2v_m'(t),w)}{M(\lambda _m(t))} +(\nabla v_m(t),\nabla w) +\beta _0(p^2v_m(t),w)=0. \end{aligned}$$
(122)

Let m and n be natural numbers such that \(m>n\). Defining \(z_m=v_m-v_n\), we have

$$\begin{aligned}&\frac{(p^2z_m''(t),w) +(\delta p^2z_m'(t),w)}{M(\lambda _m(t))} +(\nabla z_m(t),\nabla w) +\beta _0(p^2z_m(t),w)\nonumber \\&\qquad =\frac{M(\lambda _m(t))-M(\lambda _n(t))}{M(\lambda _m(t))M(\lambda _n(t))} \left[ (p^2 v_m''(t),w) +(\delta p^2v_m'(t),w) \right] . \end{aligned}$$
(123)

Taking \(w=z_m'\), we infer

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \int _{B_1}\frac{p^2|z_m'|^2}{M(\lambda _m(t))}\,dx +\int _{B_1}|\nabla z_m|^2\,dx +\beta _0\int _{B_1}p^2|z_m|^2\,dx \right] \nonumber \\&\qquad +\delta \int _{B_1}\frac{p^2|z_m'|^2}{M(\lambda _m(t))}\,dx\nonumber \\&=-\frac{M'(\lambda _m(t))\lambda _m'(t)}{2M(\lambda _m(t))} \int _{B_1}\frac{p^2|z_m'|^2}{M(\lambda _m(t))}\,dx\nonumber \\&\qquad +\frac{M(\lambda _m(t))-M(\lambda _n(t))}{M(\lambda _m(t))M(\lambda _n(t))} \left[ (p^2 v_m''(t),z_m') +(\delta p^2v_m'(t),z_m') \right] . \end{aligned}$$
(124)

Observing the calculus of \(\lambda _m'(t)\) in (107) and the Estimates 1, 2, and 3, we have

$$\begin{aligned} \frac{M'(\lambda _m(t))\lambda _m'(t)}{2M(\lambda _m(t))} \le C, \end{aligned}$$
(125)

for all \(m\in {\mathbb {N}}\) and for all \(t\in [0,T]\).

We also observe that

$$\begin{aligned}&|M(\lambda _m(t))-M(\lambda _n(t))| =\left| \int _{\lambda _m(t)}^{\lambda _n(t)}M'(s)\,ds \right| \nonumber \\&\le \max _{0\le s\le 2(L_1+L_2)}\{|M'(s)|\}|\lambda _m(t)-\lambda _n(t)|. \end{aligned}$$
(126)

Observing the definition of \(\lambda _m(t)\), we have

$$\begin{aligned}&|\lambda _m(t)-\lambda _n(t)| \le C\Big [\int _{B_1}(|\nabla v_m|+|\nabla v_n|)|\nabla z_m|\,dx\\&+\beta _1\int _{B_1}(p|x||v_m|+p|x||v_n|)p|x||z_m|\,dx\\&+(N-2)\int _{B_1}|\nabla z_m|p|x||v_m|\,dx +(N-2)\int _{B_1}|\nabla v_n|p|x||z_m|\,dx \Big ]. \end{aligned}$$

From this, using the Hölder inequality, Lemma 1, and the Estimates 1, 2, and 3, we obtain

$$\begin{aligned} |\lambda _m(t)-\lambda _n(t)| \le C\left( \int _{B_1}|\nabla z_m|^2\,dx \right) ^{\frac{1}{2}}. \end{aligned}$$
(127)

Combining (126) with (127), we infer

$$\begin{aligned} |M(\lambda _m(t))-M(\lambda _n(t))| \le C\left( \int _{B_1}|\nabla z_m|^2\,dx \right) ^{\frac{1}{2}}. \end{aligned}$$
(128)

From (124), (125), (128), and using Hölder inequality and Lemma 1, and the Estimates 1, 2, and 3, we conclude

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \int _{B_1}\frac{p^2|z_m'|^2}{M(\lambda _m(t))}\,dx +\int _{B_1}|\nabla z_m|^2\,dx +\beta _0\int _{B_1}p^2|z_m|^2 \right] \nonumber \\&\qquad +\delta \int _{B_1}\frac{p^2|z_m'|^2}{M(\lambda _m(t))}\,dx\nonumber \\&\quad \le C\Big [\int _{B_1}\frac{p^2|z_m'|^2}{M(\lambda _m(t))}\,dx +\int _{B_1}|\nabla z_m|^2\,dx\nonumber \\&\qquad +\frac{N(N-2)}{4}\int _{B_1}\frac{p^2|z_m|^2}{M(\lambda _m(t))}\,dx \Big ]. \end{aligned}$$
(129)

Using Lemma 1 and Gronwall’s inequality, we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt} \left[ \int _{B_1}\frac{p^2|z_m'|^2}{M(\lambda _m(t))}\,dx +\int _{B_1}|\nabla z_m|^2\,dx +\beta _0\int _{B_1}p^2|z_m|^2\,dx \right] \nonumber \\&\le C(T)\left[ \int _{B_1}|\nabla z_m'(0)|^2\,dx +\int _{B_1}|\nabla z_m(0)|^2\,dx \right] . \end{aligned}$$
(130)

which is the Estimate 4. Therefore, (94), (95), and (130) give us that

$$\begin{aligned}&(v_m)_{m\in {\mathbb {N}}} \quad \text{ is } \text{ a } \text{ Cauchy } \text{ sequence } \text{ in } C^0([0,T];H_0^1(B_1)),\\&(pv_m)_{m\in {\mathbb {N}}} \quad \text{ is } \text{ a } \text{ Cauchy } \text{ sequence } \text{ in } C^0([0,T];L^2(B_1)), \end{aligned}$$

and

$$\begin{aligned}&(pv_m')_{m\in {\mathbb {N}}} \quad \text{ is } \text{ a } \text{ Cauchy } \text{ sequence } \text{ in } C^0([0,T];L^2(B_1)). \end{aligned}$$

Pass to the limit. Estimates 1–4 yield subsequences, that we still denote in the same way, and a function v such that

$$\begin{aligned} v_m\rightarrow v \text{ in } C^0([0,T];H_0^1(B_1)),\, v_m'{\mathop {\rightharpoonup }\limits ^{*}} v' \text{ in } L^{\infty }(0,T;H_0^1(B_1)), \end{aligned}$$
(131)
$$\begin{aligned} pv_m{\mathop {\rightharpoonup }\limits ^{*}} pv \text{ in } C^0([0,T];L^2(B_1)),\, pv_m'\rightarrow pv' \text{ in } C^0([0,T];L^2(B_1)),\, \end{aligned}$$
(132)
$$\begin{aligned} p v_m''{\mathop {\rightharpoonup }\limits ^{*}} pv'' \text{ in } L^{\infty }(0,T;L^2(B_1)). \end{aligned}$$
(133)

From (131), (132), and observing the definition of \(\lambda _m(t)\), we infer

$$\begin{aligned} \lambda _m(\cdot )\rightarrow \lambda (\cdot ) \text{ in } C^0([0,T]). \end{aligned}$$
(134)

This convergence and the continuity of M allow us to conclude that

$$\begin{aligned} M(\lambda _m(\cdot ))\rightarrow M(\lambda (\cdot )) \text{ in } C^0([0,T]). \end{aligned}$$
(135)

The convergences (131)–(133), and (135) are enough to pass to the limit in the approximate equation (96) and to conclude that v is a unique solution of (22)–(24).

\(\Box \)

Summarizing the results of Theorems 1 and 2, we have the following result:

Corollary 1

Assume that Assumptions 1, 2, and 3 are in place, then there exist a solution v of (22)–(24) in the class (43) which decay exponentially.