On a Class of Nonlinear Viscoelastic Kirchhoff Plates: Well-Posedness and General Decay Rates

Abstract

This paper is concerned with well-posedness and energy decay rates to a class of nonlinear viscoelastic Kirchhoff plates. The problem corresponds to a class of fourth order viscoelastic equations of $p$-Laplacian type which is not locally Lipschitz. The only damping effect is given by the memory component. We show that no additional damping is needed to obtain uniqueness in the presence of rotational forces. Then, we show that the general rates of energy decay are similar to ones given by the memory kernel, but generally not with the same speed, mainly when we consider the nonlinear problem with large initial data.

1 Introduction

This paper is motivated by models of Kirchhoff plates subject to a viscoelastic damping

$$\begin{aligned} u_{tt} - \sigma \Delta u_{tt} + \mu (0)\Delta ^2 u + \int _{-\infty }^t \mu '(t-s) \Delta ^2 u(s) \, ds = \mathcal {F}, \end{aligned}$$

(1.1)

where $\sigma >0$ is the uniform plate thickness, the kernel $\mu >0$ corresponds to the viscoelastic flexural rigidity, and $\mathcal {F}=\mathcal {F}(x,t,u, u_t,\ldots )$ represents additional damping and forcing terms. The unknown function $u=u(x,t)$ represents the transverse displacement of a plate filament with prescribed history $u_0(x,t), \ t\le 0.$ The derivation of the linear mathematical model (1.1) with $\mathcal {F}=0$ is given in Lagnese [15] and Lagnese and Lions [16], by assuming viscoelastic stress-strain laws on an isotropic material occupying a region of $\mathbb {R}^3$ and constant Poisson’s ratio.

Lagnese [15], Chapter 6] studied the behavior of the energy associated to the linear model (1.1) in a bounded domain $\Omega \subset \mathbb {R}^2,$ by introducing boundary feedback laws which induce further dissipation in the system, geometrical descriptions of $\partial \Omega$, and also

$$\begin{aligned} \mu \in C^2[0,\infty ), \quad \mu (t)>0, \quad \mu '(t)<0, \quad \mu ''(t)\ge 0, \quad \mu (\infty )>0, \end{aligned}$$

see also Lagnese [14]. Muñoz Rivera and Naso [29] considered an abstract model which encompasses equation (1.1) in the cases $\mathcal {F}=-u_t$ or else $\mathcal {F}=\Delta u_t$. They showed that the associated semigroup is not exponential stable in the first case (weak damping) whereas in the second case (strong damping) the corresponding semigroup is exponential stable. We note that in both cases $\mathcal {F}$ introduces an additional dissipation to the system.

More recently, Jorge Silva and Ma [30, 31] investigated the asymptotic behavior of a $N$-dimensional system like (1.1) with $\sigma =0$ (without rotational inertia), by considering $\mathcal {F}=\Delta _p u - f(u) + h(x) + \Delta u_t,$ where $\Delta _p u:= \text{ div }(|\nabla u|^{p-2}\nabla u), \ p\ge 2.$ Then (1.1) becomes to

$$\begin{aligned} u_{tt}+ \mu (0)\Delta ^2 u - \Delta _p u + f(u) + \int _{-\infty }^t \mu '(t-s) \Delta ^2 u(s) \, ds - \Delta u_t = h(x), \end{aligned}$$

In such case the strong damping plays an important role to obtain global well-posedness (mainly uniqueness) in higher dimensions $N\ge 3$ due to the presence of the $p$-Laplacian term $\Delta _p u$.

If we take $u(\cdot , t)=0$ for $t\le 0$, $\mu (0)=1$ and $g(t)=-\mu '(t),$ then (1.1) can be rewritten as follows,

$$\begin{aligned} u_{tt} - \sigma \Delta u_{tt} + \Delta ^2 u - \int _{0}^t g(t-s) \Delta ^2 u(s) \, ds = \mathcal {F}. \end{aligned}$$

(1.2)

Barreto et al. [4] investigated problem (1.2) in a bounded domain $\Omega \subset \mathbb {R}^2$ with mixed boundary condition, suitable geometrical hypotheses on $\partial \Omega$, and $\mathcal {F}=0$. They established that the energy decays to zero with the same rate of the kernel $g$ such as exponential and polynomial decay. To do so in the second case they made assumptions on $g, \, g'$ and $g''$ which means that $g\approx (1+t)^{-p}$ for $p>2$. Then they obtained the same decay rate for the energy. However, their approach can not be applied to prove similar results for $1<p\le 2.$

Concerning $N$-dimensional systems which cover the system (1.2) with $\sigma =0$, both Cavalcanti et al. [7] and Andrade et al. [2] investigated the global existence, uniqueness and stabilization of energy. By taking a bounded or unbounded open set $\Omega$ and $\mathcal {F}=-M\left( \int _{\Omega } |\nabla u| \, dx \right) u_t$ as a kind of non-degenerated weak damping, where $M(s)>m_0>0$ for all $s\ge 0$, the authors showed in [7] that the energy goes to zero exponentially provided that $g$ goes to zero at the same form. In [2] the authors studied the same concepts by considering a bounded domain and $\mathcal {F}=\Delta _p u-f(u)+\Delta u_t ,$ but replacing the fourth order memory term in (1.2) by a weaker memory of the form $\int _{0}^{t} g(t-s) \Delta u(s) \, ds$. It is worth noting that in both cases, respectively, the weak or strong damping constitutes an important role to obtain uniqueness and energy decay. We also refer to the paper by Alabau-Boussouira et al. [1] where exponential and polynomial decay were obtained for (1.2) by taking $\sigma =0$ and $\mathcal {F}$ of lower order of type $\mathcal {F}=\nabla h(u)$, but for two different boundary conditions.

If we consider (1.2) with the Laplace operator instead of the bi-harmonic one we get the model

$$\begin{aligned} u_{tt} - \sigma \Delta u_{tt} - \Delta u + \int _{0}^t g(t-s) \Delta u(s) \, ds = \mathcal {F}, \end{aligned}$$

(1.3)

which corresponds to a viscoelastic wave equation. Equation (1.3) and related quasilinear problems with $|u_t|^{\rho }u_{tt}$ instead, $\rho >0,$ have been extensively studied by many researches with possible external forces $\mathcal {F}$ like source $f_1(u)$ and damping $f_2(u_t)$. See for instance [5, 6, 12, 13, 18, 19, 21–23, 25, 32, 33] and the references therein.

In 2008 Messaoudi [21, 22] established a general decay of the energy solution to a viscoelastic equation corresponding to (1.3) with $\sigma =0$, by taking $\mathcal {F}=0$ and $\mathcal {F}=|u|^{\gamma }u, \ \gamma >0$. More precisely, he considered the following decay condition on the memory kernel

$$\begin{aligned} g'(t) \, \le \, -\xi (t) g(t), \quad \forall \ t>0, \end{aligned}$$

(1.4)

under proper conditions on $\xi (t)>0$, and proved general decay of energy such as

$$\begin{aligned} E(t) \le c_0 \, e^{-c_1 \int _0^t \xi (s) \, ds}, \quad \forall \ t\ge 0, \end{aligned}$$

(1.5)

for some $c_0,c_1>0$ depending on the weak initial data. Ever since several authors have used this condition to obtain arbitrary decay of energy for problems related to (1.3). See for instance the papers by Han and Wang [12, 13], Liu [18], Liu and Sun [20], Park and Park [25]. It is worth pointing that in all papers mentioned above when authors deal with nonlinear systems then $c_1$ is a proportional constant to $E(0)$ (denoted here by $c_1\sim E(0)$), but it is not specified how this occurs. See also Messaoudi et al. [17, 24], Tatar [32] and Wu [33] for other kinds of interesting arbitrary decay rates in viscoelastic wave models related to (1.3). More recently, sharp decay rates for a viscoelastic von Karman plate model were shown in Cavalcanti et al. [8] when the initial energy is taken in bounded sets of the weak phase space.

There are also previous and recent works which encompass viscoelastic wave equations in a history framework and only employ memory dissipation to treat asymptotic behavior of solutions. We refer, for instance, the papers by Dafermos [9], Giorgi et al. [11], Muñoz Rivera and Salvatierra [28], and Pata [26, 27], Fabrizio et al. [10] and Araújo [3].

Our main goal in the present paper is to discuss the well-posedness and the asymptotic behavior of energy to the following nonlinear viscoelastic Kirchhoff plate equation

$$\begin{aligned} u_{tt} - \sigma \Delta u_{tt} + \Delta ^2 u - \text{ div }F(\nabla u) - \int _0^t g(t-s) \Delta ^2 u(s) \, ds =0 \quad \text{ in } \quad \Omega \times \mathbb {R}^{+} , \end{aligned}$$

(1.6)

with simply supported boundary condition

$$\begin{aligned} u= \Delta u =0 \quad \text{ on } \quad \partial \Omega \times \mathbb {R}^{+} , \end{aligned}$$

(1.7)

and initial conditions

$$\begin{aligned} u(\, \cdot \, ,0)=u_0 \quad \text{ and } \quad u_t(\, \cdot \, ,0)=u_1 \quad \text{ in } \quad \Omega , \end{aligned}$$

(1.8)

where $\Omega$ is a bounded domain of $\mathbb {R}^N$ with smooth boundary $\partial \Omega ,$ $\sigma \ge 0$, $F:\mathbb {R}^N\rightarrow \mathbb {R}^N$ is a vector field and $g:[0,\infty )\rightarrow \mathbb {R}^{+}$ is a real function. The hypotheses are given later.

Essentially, we consider the model (1.2) under the presence of a nonlinear perturbation of lower order $\mathcal {F}=\text{ div }F(\nabla u)$ which is not a locally Lipschitz operator on the weak phase spaces $\mathcal {H}$ and $\mathcal {W}$, whose definitions are given later (see (2.1) in Sect. 2). Our main results are Theorems 2.3, 2.8 and 2.10. Making a comparison with the above related papers on the viscoelastic plate model (1.2) our main results yield the following improvements and contributions:

• 1. The only damping effect is caused by the memory term. Besides, our condition on the kernel $g$ like (1.4) is less restrictive than those used in [2, 4, 7, 29]. Nevertheless, our general decay of energy [see (2.16) and (2.19)] generalizes all results on stability obtained in [2, 4, 7, 29]. We also specify how the decay rate depends on the initial data.

• 2. The $p$-Laplacian term $\Delta _p u$ is considered as a particular case. We show that in the presence of the rotational inertia term ($\sigma >0$) the well-posedness of (1.6)–(1.8) is achieved without strong damping term. Moreover, for $\sigma \ge 0$ all results on stability are shown by exploiting only the memory dissipation. No additional weak or strong dissipation is necessary. Therefore our results improve those ones given in [2, 7, 30].

• 3. No further $C^2$-smoothness is imposed on the relaxation function $g$ as regarded e.g. in [4, 7]. Moreover, the decay rate $(1+t)^{-p}$ holds for every $p>1$ when $g$ has a polynomial behavior. We note that the case $1<p\le 2$ was not approached in [4].

• 4. The parameter $\sigma$ is related to the uniform plate thickness and the results on stability hold by moving $\sigma \in [0,\infty )$ uniformly. Further, there is no result by now which treats the asymptotic behavior for plates with perturbation of $p$-Laplacian type just using dissipation from the memory. We also exemplify other interesting types of decay rates of energy beyond exponential and polynomial ones.

Remark 1.1

It is worth pointing out that the parameter $\sigma$ changes the character of the system (1.6) depending whether it is null or not. In the case of choosing $\sigma >0,$ the term $-\sigma \Delta u_{tt}$ acts as a regularizing term by allowing us to consider stronger solutions and uniqueness (see Theorem 2.3). This is possible because the rotational inertia term gives a way to control the nonlinear perturbation $\text{ div }F(\nabla u)$. On the other hand, if we consider $\sigma =0$ in (1.6) then we can also check the existence of weak solutions (see Theorem 2.10 $(i)$) but uniqueness and stronger solutions are not provided once the term $\text{ div }F(\nabla u)$ spoils the estimates along with lack of regularity for $u_t$. In spite of having two different systems according to parameter $\sigma$ all results on stability hold in both cases [see Theorems 2.8 and 2.10 $(ii)$]. In the second case (when $\sigma =0$) the stability is obtained first for approximate solutions and then for weak solutions by taking $\liminf$ on the approximate energy.

The rest of the paper is organized as follows. In Sect. 2 we fix some notations and present our assumptions and main results. Section 3 is devoted to show that problem (1.6)–(1.8) is well posed. Section 4 is dedicated to the proof of the energy decay. Finally, Sect. 4 consists in an appendix where we first give some examples for different rates of decay. Then we provide some properties and examples for the vector field $F$.

2 Assumptions and Main Results

We begin by introducing the following Hilbert spaces

$$\begin{aligned} \mathcal {V}_0=L^2(\Omega ), \quad \mathcal {V}_1=H^1_0(\Omega ), \quad \mathcal {V}_2=H^2(\Omega )\cap H^1_0(\Omega ), \end{aligned}$$

and

$$\begin{aligned} \mathcal {V}_3=\{u\in H^3(\Omega )\cap H^1_0(\Omega ) ; \ \Delta u \in H^1_0(\Omega )\}, \end{aligned}$$

with norms

$$\begin{aligned} \Vert u\Vert _{\mathcal {V}_0}=\Vert u\Vert _2, \quad \Vert u\Vert _{\mathcal {V}_1}=\Vert \nabla u\Vert _2, \quad \Vert u\Vert _{\mathcal {V}_2}=\Vert \Delta u\Vert _2, \quad \text{ and } \quad \Vert u\Vert _{\mathcal {V}_3}=\Vert \nabla \Delta u\Vert _2, \end{aligned}$$

respectively. As usual, $\Vert \cdot \Vert _p$ means the $L^p$-norm as well as $(\cdot ,\cdot )$ denotes either the $L^2$-inner product or else a duality pairing between a Banach space $V$ and its dual $V'$. The constants $\lambda _0,\lambda _1,\lambda _2,\lambda >0$ represent the embedding constants

$$\begin{aligned} \lambda _0 \Vert u\Vert _2^2 \le \Vert \nabla u\Vert _2^2, \quad \lambda _1 \Vert u\Vert _2^2 \le \Vert \Delta u\Vert _2^2, \quad \lambda _2 \Vert \nabla u\Vert _2^2 \le \Vert \Delta u\Vert _2^2, \quad \lambda =\frac{1}{\lambda _1}+\frac{1}{\lambda _2}, \end{aligned}$$

for $u\in \mathcal {V}_2$. We also consider the following phase spaces with their respective norms

$$\begin{aligned} \mathcal {H}=\mathcal {V}_2\times \mathcal {V}_1&\text{ with }&||(u,v)||^{ 2}_{\mathcal {H}}=||\Delta u||^{2}_{2}+||\nabla v||^{2}_{2}, \nonumber \\ \mathcal {H}_1=\mathcal {V}_3\times \mathcal {V}_2&\text{ with }&||(u,v)||^{2}_{\mathcal {H}_{1}}=||\nabla \Delta u||^{2}_{2}+||\Delta v||^{2}_{2}, \nonumber \\ \mathcal {W}=\mathcal {V}_2\times \mathcal {V}_0&\text{ with }&||(u,v)||^{2}_{\mathcal {W}}=||\Delta u||^{2}_{2}+||v||^{2}_{2}. \end{aligned}$$

(2.1)

2.1 The Problem with Rotational Inertia

Let us first consider (1.6) with $\sigma >0.$ Without loss of generality we can take $\sigma =1.$ Setting $I=[0,T]$ with $T>0$ arbitrary, weak solutions are defined as follows.

Given initial data $(u_0,u_1)\in \mathcal {H}$, we call a function $U:=(u,u_t)\in C(I, \mathcal {H})$ a weak solution of the problem (1.6)–(1.8) on $I$ if $U(0)=(u_0,u_1)$ and, for every $\omega \in \mathcal {V}_2$,

$$\begin{aligned}&\frac{d}{dt}\Big [(u_t(t),\omega )+(\nabla u_t(t),\nabla \omega )\Big ]+(\Delta u(t),\Delta \omega ) \\&\quad +\left( F(\nabla u(t)),\nabla \omega \right) -\int _0^t g(t-s) (\Delta u(s),\Delta \omega ) \, ds=0 \ \text{ a.e. } \text{ in } I. \end{aligned}$$

The energy corresponding to the problem with rotational inertia is defined as

$$\begin{aligned} E(t) = \frac{1}{2} \Vert u_t(t) \Vert _2^2 + \frac{1}{2} \Vert \nabla u_t(t) \Vert _2^2 + \frac{h(t)}{2}\Vert \Delta u(t) \Vert _2^2 + \frac{1}{2}(g\square \Delta u)(t) + \int _{\Omega } {f}(\nabla u(t)) \, dx, \end{aligned}$$

(2.2)

where $h(t)$ is given below in (2.7) and

$$\begin{aligned} (g\square \, w)(t):=\int _{0}^{t}g(t-s) \Vert w(t)-w(s) \Vert _2^2 ds. \end{aligned}$$

Now let us precise the hypotheses on $g$ and $F$.

Assumption A.1

The $C^1$-function $g:[0,\infty )\rightarrow \mathbb {R}^+$ satisfies

$$\begin{aligned} l:=1-\int _0^\infty { g(s) \, ds}>0 \quad \text{ and } \quad g'(t)\le 0, \quad \forall \ t\ge 0. \end{aligned}$$

(2.3)

Assumption A.2

$F:\mathbb {R}^N\rightarrow \mathbb {R}^N$ is a $C^1$-vector field given by $F=(F_1,\ldots ,F_N)$ such that

$$\begin{aligned} |\nabla F_j(z)|\le k_j\left( 1+|z|^{(p_j-1)/2}\right) , \quad \forall \; z\in \mathbb {R}^N, \end{aligned}$$

(2.4)

where, for every $j=1,\ldots ,N,$ we consider $k_j>0$ and $p_j$ satisfying

$$\begin{aligned} p_j\ge 1 \quad \text{ if } \quad N=1,2 \quad \text{ and } \quad 1 \le p_j \le \frac{N+2}{N-2} \quad \text{ if } \quad N \ge 3. \end{aligned}$$

(2.5)

Moreover, $F$ is a conservative vector field with $F=\nabla f$, where $f:\mathbb {R}^N\rightarrow \mathbb {R}$ is a real valued function satisfying

$$\begin{aligned} -\alpha _0-\frac{\alpha l}{2}|z|^{2}\le f(z)\le F(z)\cdot z + \frac{\alpha l}{2}|z|^{2}, \quad \forall \; z\in \mathbb {R}^N, \end{aligned}$$

(2.6)

with $\alpha _0\ge 0$ and $\alpha \in [0,\lambda _2).$

Remark 2.1

From the choice of $l$ and $\alpha$ we have

$$\begin{aligned} h(t):= 1-\int _0^tg(s)\,ds\ge l, \quad t\ge 0, \quad \text{ and } \quad \beta :=l\left( 1-\frac{\alpha }{\lambda _2}\right) >0. \end{aligned}$$

(2.7)

Also, applying (2.5) it follows from Sobolev embedding that

$$\begin{aligned} \mathcal {V}_2 \hookrightarrow W_0^{1,p_j+1}(\Omega ), \quad \forall \ j=1,\ldots ,N. \end{aligned}$$

(2.8)

Thereby, the constants $\mu _{p_1},\ldots ,\mu _{p_N}>0$ represent the embedding constants for

$$\begin{aligned} \Vert \nabla u\Vert _{p_j+1} \le \mu _{p_j}\Vert \Delta u\Vert _2, \quad j=1,\ldots ,N. \end{aligned}$$

Remark 2.2

Without loss of generality we can consider $F(0)=0$. Indeed, if $F(0)=F_0\ne 0,$ then we define $G(z)=F(z)-F_0$ so that $G$ satisfies $G(0)=0$, $|\nabla G_j(z)|=|\nabla F_j(z)|, \ j=1,\ldots ,N,$ and $G(z)=\nabla \widetilde{f}(z),$ where $\widetilde{f}(z)=f(z)-F_0 \cdot z.$ Also, it is easy to check that $\widetilde{f}$ and $G(z)$ satisfy (2.6) for some constants $\widetilde{\alpha }_0\ge 0$, $\widetilde{\alpha }\in [0,\lambda _2).$ Therefore, $G$ is a $C^1$-conservative vector field satisfying (2.4)–(2.6). In the Sect. 4 we give some examples of vector fields satisfying such properties.

Our first two main results establish the Hadamard well-posedness of (1.6)–(1.8) with respect to weak solutions, and a general decay rate of the energy.

Theorem 2.3

(Well-Posedness) Under Assumptions A.1 and A.2 we have:

• (i) If $(u_0,u_1) \in \mathcal {H}_1$, then problem (1.6)–(1.8) has a stronger weak solution satisfying

  $$\begin{aligned} u\in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_3), \quad u_t \in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_2), \quad (I-\Delta )u_{tt}\in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_1'). \end{aligned}$$

  (2.9)

• (ii) If $(u_0,u_1) \in \mathcal {H}$, then problem (1.6)–(1.8) has a weak solution satisfying

  $$\begin{aligned} u\in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_2), \quad u_t \in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_1), \quad (I-\Delta )u_{tt}\in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_2'). \end{aligned}$$

  (2.10)

• (iii) In both cases we have continuous dependence on initial data in $\mathcal {H}$, that is, given $U_0=(u_{0},u_{1}), \, V_0=(v_{0},v_{1})\in \mathcal {H}$, let us consider the corresponding weak solutions $U=(u,u_{t}), \, V=(v,v_{t})$ of the problem (1.6)–(1.8). Then

  $$\begin{aligned} \Vert U(t)-V(t)||_{\mathcal {H}} \, \le \, C_T ||U_0-V_0||_{\mathcal {H}} , \quad \forall \ t\in I, \end{aligned}$$

  (2.11)

  for some constant $C_T=C(\Vert U_0\Vert _{\mathcal {H}}, \Vert V_0\Vert _{\mathcal {H}},T)>0$. In particular, problem (1.6)–(1.8) has a unique weak solution.

Remark 2.4

The proof of existence is given by the Faedo-Galerkin method. We first prove the existence of stronger (weak) solutions and then the existence of a weak solution is given by density arguments. The uniqueness follows as a consequence of the continuous dependence of stronger and weak solutions. The proofs are given in Sect. 3.

Lemma 2.5

Under the assumptions of Theorem 2.3 the energy $E(t)$ satisfies

$$\begin{aligned} \frac{d}{dt}E(t) \, = \, \frac{1}{2}(g'\square \Delta u)(t) - \frac{g(t)}{2}\Vert \Delta u(t) \Vert _2^2, \quad \forall \ t>0. \end{aligned}$$

(2.12)

Remark 2.6

From conditions (2.3) and (2.12) it follows that $t\mapsto E(t)$ is nonincreasing. Since $g(t)\le g(0),$ for each $t\ge 0,$ if we take $g(0)=0$ then $g, \,g'\equiv 0$ and Lemma 2.5 implies that $E(t)$ is constant. That is, the system (1.6)–(1.8) is conservative. This motivates us to define the following decay condition on the memory kernel $g(t)$.

Assumption A.3

$g(0)>0$, and there exist a constant $\xi _0\ge 0$ and a $C^1$-function $\xi :[0,\infty )\rightarrow \mathbb {R}^+$ such that

$$\begin{aligned} g'(t) \, \le \, -\xi (t) g(t), \quad \forall \ t>0, \end{aligned}$$

(2.13)

and

$$\begin{aligned} \xi (t)>0, \quad \, \xi '(t)\le 0, \quad \, \left| \frac{\xi '(t)}{\xi (t)}\right| \le \xi _0, \quad \forall \ t\ge 0. \end{aligned}$$

(2.14)

Remark 2.7

The first two conditions in (2.14) allow us to conclude that

$$\begin{aligned} \xi (t)\le \xi (0):=\xi _1>0, \quad \forall \ t\ge 0. \end{aligned}$$

Also, condition (2.13) implies that the memory kernel has the uniform decay

$$\begin{aligned} g(t) \le g(0) e^{-\int _0^t \xi (s) \, ds}, \quad \forall \ t\ge 0. \end{aligned}$$

(2.15)

Then our second main result is given by the following

Theorem 2.8

Under the assumptions of Theorem 2.3, let $(u,u_t)$ be the weak solution of problem (1.6)–(1.8) with given initial data $(u_0,u_1)\in \mathcal {H}$. If we additionally assume Assumption A.3 and $\alpha _0=0$ in (2.6), then

$$\begin{aligned} E(t) \le c \, e^{-\gamma \int _0^t \xi (s) \, ds}, \quad \forall \ t\ge 0, \end{aligned}$$

(2.16)

where $c=3 E(0) \, e^{\gamma \int _0^1 \xi (s) \, ds}>0,$ and

$$\begin{aligned} \displaystyle \gamma \sim \frac{k}{ 1+[E(0)]^{\frac{p-1}{2}} } \ \ \text{ with } \ \ k>0 \ \ \text{ and } \ \ p=\left\{ \begin{array}{rl} \displaystyle \max _{j=1,\ldots ,N}\{p_j\} &{} \text{ if } \quad E(0)\ge 1,\\ \displaystyle \min _{j=1,\ldots ,N}\{p_j\} &{} \text{ if } \quad E(0) < 1. \end{array}\right. \end{aligned}$$

Remark 2.9

Theorem 2.8 is proved in Sect. 4. Concerning to estimate (2.16) it is worth pointing out two issues: $(i)$ when every component of $F=(F_1,\ldots ,F_N)$ is linear, namely, when $p_1,\ldots ,p_N=1$ in (2.4) and so $p=1$, then the general estimate (2.16) is similar to the decay of the memory kernel $g$ and provides us several kinds of decay according to the feature of $\xi (t)$ independently of the size of the initial data; $(ii)$ otherwise, in the presence of the nonlinear perturbation $F$, then estimate (2.16) can be very slow for large initial data even if the memory kernel $g$ decays quickly. In the Sect. 4 we consider some concrete examples for function $\xi$.

2.2 The Problem Without Rotational Inertia

Let us now consider (1.6) with $\sigma =0.$ Let us also take $I=[0,T]$ with $T>0$.

Given initial data $(u_0,u_1)\in \mathcal {W}$, we say a function $U:=(u,u_t)\in C(I, \mathcal {W})$ is a weak solution of (1.6)–(1.8) on $I$ if $U(0)=(u_0,u_1)$ and, for every $\omega \in \mathcal {V}_2$,

$$\begin{aligned}&\frac{d}{dt}(u_t(t),\omega )+(\Delta u(t),\Delta \omega ) +\left( F(\nabla u(t)),\nabla \omega \right) \\&\quad -\int _0^t g(t-s) (\Delta u(s),\Delta \omega ) \, ds=0 \ \text{ a.e. } \text{ in } I. \end{aligned}$$

Now the energy associated to the problem without rotational inertia is given by

$$\begin{aligned} \mathcal {E}(t) = \frac{1}{2} \Vert u_t(t) \Vert _2^2 + \frac{h(t)}{2}\Vert \Delta u(t) \Vert _2^2 + \frac{1}{2}(g\square \Delta u)(t) + \int _{\Omega } {f}(\nabla u(t)) \, dx. \end{aligned}$$

(2.17)

Our third main result is the following theorem.

Theorem 2.10

Under Assumptions A.1 and A.2, we have:

• (i) If $(u_0,u_1) \in \mathcal {W}$, then problem (1.6)–(1.8) has a weak solution in the class

  $$\begin{aligned} u\in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_2), \quad u_t \in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_0), \quad u_{tt}\in L^{\infty }_{loc}(\mathbb {R}^+,\mathcal {V}_2'). \end{aligned}$$

  (2.18)

• (ii) Besides, if Assumption A.3 holds and $\alpha _0=0$ in (2.6), then $\mathcal {E}(t)$ also satisfies

  $$\begin{aligned} \mathcal {E}(t) \le c \, e^{-\gamma \int _0^t \xi (s) \, ds}, \quad \forall \ t\ge 0, \end{aligned}$$

  (2.19)

  where $c>0$ and $\gamma >0$ are given in terms of $\mathcal {E}(0)$ as in Theorem 2.8.

Remark 2.11

To prove Theorem 2.10 (i) one also uses the Faedo–Galerkin method. Only one a priori estimate is necessary to get a weak solution satisfying (2.18). Since we do not have regularity for $u_t$ the estimate (2.19) is shown first for Galerkin’s approximate solutions. Then the proof of Theorem 2.10 (ii) will hold true by passing the $\liminf$ on the Galerkin’s approximate energy. The details of the proof are very similar to those ones used in the proofs of Theorems 2.3 and 2.8. Thus we omit them here but an idea is presented in Remarks 3.2 and 4.6.

3 Well-Posedness

In this section we prove Theorem 2.3. We start with the following approximate problem

$$\begin{aligned}&(u^n_{tt}(t),\omega _j)+(\nabla u^n_{tt}(t),\nabla \omega _j)+(\Delta u^n(t),\Delta \omega _j) \end{aligned}$$

(3.1)

$$\begin{aligned}&\quad +\left( F(\nabla u^n(t)),\nabla \omega _j\right) -\int _0^t g(t-s)(\Delta u^n(s),\Delta \omega _j) \, ds=0, \nonumber \\&\quad u^n(0)=u^n_0 \quad \text{ and } \quad u^n_t(0)=u^n_1, \end{aligned}$$

(3.2)

for $j=1,\ldots ,n,$ which has a local solution

$$\begin{aligned} u^n(t)=\sum _{j=1}^n y_{jn}(t)\omega _j \, \in \, [\omega _1, \ldots , \omega _n], \end{aligned}$$

on $[0,t_n), \ n\in \mathbb {N},$ given by ODE theory, where $(\omega _j)_{j\in \mathbb {N}}$ is an orthonormal basis in $\mathcal {V}_0$ given by eigenfunctions of $\Delta ^2$ with boundary condition (1.7). The a priori estimates below imply that the local solution can be extended to the interval $[0,T]$ and allow us to concluded the existence of a weak solution.

Proof of Theorem 2.3 $(i)$. Let us take regular initial data $(u_0,u_1)\in \mathcal {H}_1:=\mathcal {V}_3\times \mathcal {V}_2$. Then we consider the approximate problem (3.1)–(3.2) with

$$\begin{aligned} u^n_0 \, \rightarrow \, u_0 \quad \text{ in } \,\, \mathcal {V}_3 \quad \text{ and } \quad u^n_1 \, \rightarrow \, u_1 \quad \text{ in } \quad \mathcal {V}_2. \end{aligned}$$

(3.3)

A Priori Estimate I Replacing $w_j$ by $u^n_t(t)$ in (3.1) and since it hold

$$\begin{aligned}&\int _0^t g(t-s)(\Delta u^n(s), \Delta u^n_t(t)) \, ds \nonumber \\&\quad = - \frac{1}{2} \frac{d}{dt} \left\{ \left( g \square \Delta u^n \right) (t) - \left( \int _0^t g(s) \, ds \right) \Vert \Delta u^n(t)\Vert _2^2 \right\} \nonumber \\&\qquad + \, \frac{1}{2} \left( g' \square \Delta u^n \right) (t) - \frac{1}{2}g(t)\Vert \Delta u^n(t)\Vert _2^2, \end{aligned}$$

(3.4)

and

$$\begin{aligned} \int _{\Omega } F(\nabla u^n(t)) \cdot \nabla u^n_t(t) \, dx= & {} \int _{\Omega } \nabla f(\nabla u^n(t)) \cdot \nabla u^n_t(t) \, dx \nonumber \\= & {} \frac{d}{dt} \int _{\Omega } f(\nabla u^n(t)) \, dx, \end{aligned}$$

(3.5)

it follows that

$$\begin{aligned} \frac{d}{dt}E^n(t) \, = \, \frac{1}{2} \left( g'\square \Delta u^n \right) (t) - \frac{1}{2} g(t) \Vert \Delta u^n(t)\Vert _2^2, \quad t>0, \end{aligned}$$

(3.6)

where

$$\begin{aligned} E^n(t)= & {} \frac{1}{2} \Vert u^n_t(t) \Vert _2^2 + \frac{1}{2} \Vert \nabla u^n_t(t) \Vert _2^2 + \frac{h(t)}{2}\Vert \Delta u^n(t) \Vert _2^2 \\&+\, \frac{1}{2}(g\square \Delta u^n)(t) + \int _{\Omega } {f}(\nabla u^n(t)) \, dx. \end{aligned}$$

From (2.7) and first condition in (2.6) we get

$$\begin{aligned} E(t)+\alpha _0|\Omega | \, \ge \, \frac{1}{2}\Vert \nabla u^n_t(t) \Vert _2^2 +\frac{\beta }{2} \Vert \Delta u^n(t) \Vert _2^2 \, \ge \, 0, \end{aligned}$$

Besides, assumption (2.3) along with (3.6) imply that $E^n(t)\le E^n(0).$ Consequently,

$$\begin{aligned} \frac{1}{2} \Vert \nabla u^n_t(t) \Vert _2^2 + \frac{\beta }{2} \Vert \Delta u^n(t) \Vert _2^2 \le E^n(t) +\alpha _0|\Omega | \le E^n(0) + \alpha _0|\Omega |. \end{aligned}$$

From (3.3), second condition in (2.6), (4.28) and Hölder’s inequality we conclude

$$\begin{aligned} \Vert \nabla u^n_t(t) \Vert _2^2 + \Vert \Delta u^n(t) \Vert _2^2 \le M_1, \quad \forall \ t\in [0,T], \ \forall \ n\in \mathbb {N}, \end{aligned}$$

(3.7)

where $M_1=M_1(\Vert \nabla u_1 \Vert _2, \Vert \Delta u_0\Vert _2,|\Omega |)>0$.

A Priori Estimate II Replacing $w_j$ by $-\Delta u^n_t(t)$ in (3.1), since (3.4) holds with $\nabla \Delta$ in the place of $\Delta$, and also

$$\begin{aligned} \int _{\Omega } F(\nabla u^n(t)) \cdot \nabla \Delta u^n_t(t) \, dx \, = \, \frac{d}{dt} \int _{\Omega } F(\nabla u^n(t)) \cdot \nabla \Delta u^n(t) \, dx + J_F, \end{aligned}$$

with $J_F$ given by

$$\begin{aligned} J_F \, = \, - \int _{\Omega } \left[ \big ( \nabla F_1(\nabla u^n(t))\cdot \nabla u^n_t(t), \ldots , \nabla F_N(\nabla u^n(t))\cdot \nabla u^n_t(t) \big )\right] \cdot \nabla \Delta u^n(t) \, dx, \end{aligned}$$

then we infer

$$\begin{aligned} \frac{d}{dt}F^n(t) = \frac{1}{2} \left( g'\square \nabla \Delta u^n \right) (t) - \frac{1}{2} g(t) \Vert \nabla \Delta u^n(t)\Vert _2^2 +J_F \le J_F, \quad t>0,\quad \end{aligned}$$

(3.8)

where

$$\begin{aligned} F^n(t) \, = \, \frac{1}{2} \Vert \nabla u^n_t(t) \Vert _2^2 + \frac{1}{2} \Vert \Delta u^n_t(t) \Vert _2^2 + \frac{h(t)}{2} \Vert \nabla \Delta u^n(t) \Vert _2^2 + \frac{1}{2} \left( g \square \nabla \Delta u^n \right) (t) -I_F \end{aligned}$$

with

$$\begin{aligned} I_F= \int _{\Omega } F(\nabla u^n(t)) \cdot \nabla \Delta u^n(t) \, dx. \end{aligned}$$

Let us estimate the right hand side of (3.8). Using assumption (2.4), generalized Hölder inequality, and (2.8) we get

$$\begin{aligned} | J_F |\le & {} \sum _{j=1}^{N} \int _{\Omega } |\nabla F_j(\nabla u^n(t))| \, |\nabla u^n_t(t)| \, |\nabla \Delta u^n(t)| \, dx \\\le & {} \sum _{j=1}^{N}k_j \int _{\Omega } \left( 1+|\nabla u^n(t)|^{(p_j-1)/2}\right) \, |\nabla u^n_t(t)| \, |\nabla \Delta u^n(t)| \, dx \\\le & {} \sum _{j=1}^{N}k_j\mu _{p_j}\left( |\Omega |^{\frac{p_j-1}{2(p_j+1)}}+\Vert \nabla u^n(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}} \right) \Vert \Delta u^n_t(t)\Vert _2 \Vert \nabla \Delta u^n(t)\Vert _2. \end{aligned}$$

From estimate (3.7) and using again (2.8) we obtain

$$\begin{aligned} \sum _{j=1}^{N}k_j\mu _{p_j}\left( |\Omega |^{\frac{p_j-1}{2(p_j+1)}}+\Vert \nabla u^n(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}} \right) \le C<\infty . \end{aligned}$$

From this and Young’s inequality there exists a constant $C_1=C_1\left( ||\nabla u_1||_2, ||\Delta u_0||_2\right) >0$ such that

$$\begin{aligned} | J_F | \, \le \, {C_1} \left( \Vert \Delta u^n_t(t) \Vert _2^2 + \Vert \nabla \Delta u^n(t) \Vert _2^2\right) . \end{aligned}$$

(3.9)

Inserting (3.9) in (3.8) and integrating from $0$ to $t\le T$, yields

$$\begin{aligned} F^{n}(t) \le F^{n}(0) + {C_1} \int _0^t\left( \Vert \Delta u^n_t(s) \Vert _2^2 + \Vert \nabla \Delta u^n(s) \Vert _2^2\right) ds, \quad t\ge 0. \end{aligned}$$

(3.10)

On the other hand, from (4.28) in the appendix with $F(0)=0$ and Hölder’s inequality, we have

$$\begin{aligned} \left| I_F \right|\le & {} \int _{\Omega } | F(\nabla u^n(t))| \, |\nabla \Delta u^n(t)| \, dx \\\le & {} K \sum _{j=1}^{N}\int _{\Omega } \left( |\nabla u^n(t)|+|\nabla u^n(t)|^{(p_j+1)/2} \right) \, |\nabla \Delta u^n(t)| \, dx \\\le & {} K \left( N \Vert \nabla u^n(t)\Vert _{2} + \sum _{j=1}^{N} \Vert \nabla u^n(t)\Vert _{p_j+1}^{\frac{p_j+1}{2}} \right) \Vert \nabla \Delta u^n(t)\Vert _2. \end{aligned}$$

Moreover, the estimates (3.7) and (2.8) imply

$$\begin{aligned} K \left( N \Vert \nabla u^n(t)\Vert _{2} + \sum _{j=1}^{N} \Vert \nabla u^n(t)\Vert _{p_j+1}^{\frac{p_j+1}{2}} \right) \le C<\infty . \end{aligned}$$

Using Young’s inequality there exists a constant $C_2=C_2\left( ||\nabla u_1||_2, ||\Delta u_0||_2\right) >0$ such that

$$\begin{aligned} |I_F| \, \le \, C_2 + \frac{l}{4} \Vert \nabla \Delta u^n(t) \Vert _2^2. \end{aligned}$$

Since $h(t)\ge l$, then

$$\begin{aligned} \frac{h(t)}{2} \Vert \nabla \Delta u^n(t) \Vert _2^2 -I_F \, \ge \, \frac{l}{4} \Vert \nabla \Delta u^n(t) \Vert _2^2 - C_2, \end{aligned}$$

and, consequently,

$$\begin{aligned} \frac{l}{4} \Vert \Delta u^n_t(t) \Vert _2^2+ \frac{l}{4} \Vert \nabla \Delta u^n(t) \Vert _2^2 \, \le \, F^n(t) + C_2. \end{aligned}$$

(3.11)

Combining (3.10) and (3.11) we arrive at

$$\begin{aligned}&\Vert \Delta u^n_t(t) \Vert _2^2+ \Vert \nabla \Delta u^n(t) \Vert _2^2 \\&\quad \le \frac{4}{l}(C_2+ F^{n}(0)) + \frac{4C_1}{l} \int _0^t\left( \Vert \Delta u^n_t(s) \Vert _2^2 + \Vert \nabla \Delta u^n(s) \Vert _2^2\right) ds. \end{aligned}$$

Taking into account (3.3) and applying Gronwall’s inequality, we finally conclude

$$\begin{aligned} \Vert \Delta u^n_t(t) \Vert _2^2 + \Vert \nabla \Delta u^n(t) \Vert _2^2 \le M_2, \quad \forall \ t\in [0,T], \ \forall \ n\in \mathbb {N}, \end{aligned}$$

(3.12)

where $M_2=M_2(||\Delta u_1||_2, ||\nabla \Delta u_0||_2, |\Omega |, T)>0$.

The estimates (3.7) and (3.12) are sufficient to pass to the limit in the approximate problem (3.1)–(3.2) and to obtain a stronger weak solution

$$\begin{aligned} (u,u_t)\in C\left( [0,T], \mathcal {H}\right) \, \cap \, L^{\infty }(0,T;\mathcal {H}_1), \quad T>0, \end{aligned}$$

(3.13)

satisfying

$$\begin{aligned} (I- \Delta ) u_{tt} = - \Delta ^2 u + \text{ div }F(\nabla u) + \int _0^t g(t-s) \Delta ^2 u(s) \, ds \quad \text{ in } \quad L^\infty \left( 0,T; \mathcal {V}_1'\right) . \end{aligned}$$

(3.14)

This finishes the proof of the existence of regular weak solutions. $\square$

Remark 3.1

Unless for the term which involves $\text{ div }F(\nabla u)$, the limit on the other terms in the approximate system can be done in a usual way. With respect to this term we only need to apply estimates (3.7) and (4.27) along with the Aubin–Lions Lemma. Then it will hold later in the case of weak solutions, see for instance [2, 30]. $\square$

Proof of Theorem 2.3 (iii) (stronger weak solutions) We first show that solution in (3.13) satisfies the continuous dependence property (2.11).

Let us consider two stronger weak solutions $U=(u,u_{t}), \, V=(v,v_{t})$ of the problem (1.6)–(1.8) corresponding to initial data $U_0=(u_{0},u_{1}), \, V_0=(v_{0},v_{1})\in \mathcal {H}_1$, respectively. By setting $w=u-v$, then function $(w,w_{t})=U-V$ solves the equation

$$\begin{aligned} w_{tt} - \Delta w_{tt} + \Delta ^2 w - \int _0^t g(t-s) \Delta ^2 w(s) \, ds = \text{ div }F(\nabla u)-\text{ div }F(\nabla v)\quad \quad \end{aligned}$$

(3.15)

in $L^\infty \left( 0,T; \mathcal {V}_1'\right) ,$ with initial data $(w(0),w_t(0)) =U_0-V_0.$

Since $w_t(t)\in \mathcal {V}_2\hookrightarrow \mathcal {V}_1,$ then multiplying Eq. (3.15) by $w_{t}(t)$ and integrating over $\Omega$, we get

$$\begin{aligned} \frac{d}{dt} W(t) \, = \, \frac{1}{2} \left( g'\square \Delta w \right) (t) - \frac{1}{2} g(t) \Vert \Delta w(t)\Vert _2^2 + L_F \, \le \, L_F, \quad t>0, \end{aligned}$$

(3.16)

where

$$\begin{aligned} W(t) = \frac{1}{2} \Vert w_t(t) \Vert _2^2 + \frac{1}{2} \Vert \nabla w_t(t) \Vert _2^2 + \frac{h(t)}{2}\Vert \Delta w(t) \Vert _2^2 + \frac{1}{2}(g\square \Delta w)(t), \quad t\ge 0, \end{aligned}$$

and

$$\begin{aligned} L_F \, = \, -\int _{\Omega } [F(\nabla u(t))-F(\nabla v(t))]\cdot \nabla w_{t}(t) \, dx. \end{aligned}$$

Estimate (4.27) from the appendix, the generalized Hölder inequality, and (2.8) imply

$$\begin{aligned} | L_F |\le & {} \int _{\Omega } |F(\nabla u(t))-F(\nabla v(t))| \, |\nabla w_t(t)| \, dx \\\le & {} \! K \sum _{j=1}^{N} \int _{\Omega } \left( 1+|\nabla u(t)|^{(p_j-1)/2}+|\nabla v(t)|^{(p_j-1)/2}\right) \, |\nabla w(t)| \, |\nabla w_t(t)| \, dx \\\le & {} \! K \sum _{j=1}^{N}\mu _{p_j}\left( |\Omega |^{\frac{p_j-1}{2(p_j+1)}}\!+\!\Vert \nabla u(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}} \!+\!\Vert \nabla v(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}}\right) \Vert \Delta w(t)\Vert _2 \Vert \nabla w_t(t)\Vert _2. \end{aligned}$$

From (2.8) and (3.13) we obtain

$$\begin{aligned} K \sum _{j=1}^{N}\mu _{p_j}\left( |\Omega |^{\frac{p_j-1}{2(p_j+1)}}+\Vert \nabla u(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}} +\Vert \nabla v(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}}\right) \le C<\infty , \end{aligned}$$

and making use of Young inequality there exists a constant $C_3\!=\!C_3\left( ||\nabla u_1||_2, ||\Delta u_0||_2\right) >0$ such that

$$\begin{aligned} | L_F | \, \le \, {C_3} \left( \Vert \Delta w(t)\Vert _2^2 + \Vert \nabla w_t(t)\Vert _2^2\right) . \end{aligned}$$

(3.17)

Inserting (3.17) into (3.16) and integrating from $0$ to $t\le T$, one has

$$\begin{aligned} W(t) \le W(0) + {C_3} \int _0^t \left( \Vert \Delta w(s)\Vert _2^2 + \Vert \nabla w_t(s)\Vert _2^2\right) ds, \quad t\ge 0. \end{aligned}$$

(3.18)

On the other hand it is easy to check that

$$\begin{aligned} \Vert \Delta w(t)\Vert _2^2 + \Vert \nabla w_t(t)\Vert _2^2 \le \frac{2}{l} W(t), \quad t\ge 0, \end{aligned}$$

(3.19)

and

$$\begin{aligned} W(0) \le \frac{1}{2}\left( 1+\frac{1}{\lambda _0}\right) \left( \Vert \Delta w(0)\Vert _2^2 + \Vert \nabla w_t(0)\Vert _2^2\right) . \end{aligned}$$

(3.20)

Combining (3.18)–(3.20) and applying Gronwall’s inequality we conclude

$$\begin{aligned} \left( \Vert \Delta w(t)\Vert _2^2 + \Vert \nabla w_t(t)\Vert _2^2\right) \, \le \, C_T^{2} \left( \Vert \Delta w(0)\Vert _2^2 + \Vert \nabla w_t(0)\Vert _2^2\right) , \quad \forall \ t\in [0,T], \end{aligned}$$

(3.21)

for some constant $C_T=C(\Vert U_0\Vert _{\mathcal {H}}, \Vert V_0\Vert _{\mathcal {H}},T)>0$. This shows that the estimate (2.11) is guaranteed for regular solutions since we have $(w,w_{t})=U-V$. $\square$

Proof of Theorem 2.3 $(ii)$ Let us take initial data $(u_0,u_1)\in \mathcal {H}$. Then there exists a sequence $(u_0^n,u_1^n)\in \mathcal {H}_1$ such that

$$\begin{aligned} u^n_0 \, \rightarrow \, u_0 \,\, \text{ in } \,\, \mathcal {V}_2 \quad \text{ and } \quad u^n_1 \, \rightarrow \, u_1 \quad \text{ in } \quad \mathcal {V}_1. \end{aligned}$$

(3.22)

For each regular initial data $(u_0^n,u_1^n), \, n\in \mathbb {N},$ there exists a regular solution $(u^n,u^n_t)$ satisfying (3.13)–(3.14). Taking the multiplier $u^n_t(t)$ in (3.14) and proceeding analogously as in (3.4)–(3.6) then estimate (3.7) holds true. This implies

$$\begin{aligned} (u^{n},u^{n}_t)&\mathop {\rightharpoonup }\limits ^{*}&(u,u_t) \quad \text{ in } \quad L^{\infty }(0,T;\mathcal {H}). \end{aligned}$$

(3.23)

Besides, if we consider $m,n\in \mathbb {N}, \, m\ge n,$ and $w=u^m-u^n$, then function $(w,w_t)$ satisfies

$$\begin{aligned} w_{tt} - \Delta w_{tt} + \Delta ^2 w - \int _0^t g(t-s) \Delta ^2 w(s) \, ds = \text{ div }F(\nabla u^m)-\text{ div }F(\nabla u^n) \end{aligned}$$

in $L^\infty \left( 0,T; \mathcal {V}_1'\right) ,$ with initial data $(w(0),w_t(0)) =(u^m_0-u^n_0,u^m_1-u^n_1).$ Taking the multiplier $w_t(t)$ and using analogous arguments as given in (3.16)–(3.20), the estimate (3.21) also holds. This means that

$$\begin{aligned}&\Vert \Delta (u^m(t)-u^n(t))\Vert _2^2 + \Vert \nabla (u^m_t(t)-u^n_t(t))\Vert _2^2\\&\quad \le C \left( \Vert \Delta (u^m_0-u^n_0)\Vert _2^2 + \Vert \nabla (u^m_1-u^n_1)\Vert _2^2\right) , \end{aligned}$$

for any $t\in [0,T],$ and some constant $C=C(\Vert (u_0,u_1)\Vert _{\mathcal {H}},T)>0$. From (3.13) and (3.22), and since $C$ is a constant depending only on the initial data in $\mathcal {H}$, we infer

$$\begin{aligned} (u^{n},u^{n}_t)\rightarrow & {} (u,u_t) \quad \text{ in } \quad C([0,T],\mathcal {H}). \end{aligned}$$

(3.24)

Finally, we note that the limits (3.23) and (3.24) are enough to pass to the limit in the approximate problem (3.1)–(3.2) and to obtain a weak solution $(u,u_t)\in C\left( [0,T], \mathcal {H}\right)$ satisfying (2.10) and

$$\begin{aligned} (I- \Delta ) u_{tt} = - \Delta ^2 u + \text{ div }F(\nabla u) + \int _0^t g(t-s) \Delta ^2 u(s) \, ds \quad \text{ in } \quad L^\infty \left( 0,T; \mathcal {V}_2'\right) . \end{aligned}$$

This concludes the proof on existence of weak solutions.$\square$

Remark 3.2

The proof of Theorem 2.10 (i) can be concluded by using similar arguments, namely, there exists a sequence of solutions to the approximate problem (3.1)–(3.2) without rotational inertia term satisfying

$$\begin{aligned} u^{n}&\mathop {\rightharpoonup }\limits ^{*} u \quad \text{ in } \quad L^{\infty }(0,T;\mathcal {V}_2), \nonumber \\ u^{n}_t&\mathop {\rightharpoonup }\limits ^{*} u_t \quad \text{ in } \quad L^{\infty }(0,T;\mathcal {V}_0), \nonumber \\ u^{n}&\rightarrow u \quad \text{ in } \quad L^{2}(0,T;\mathcal {V}_1), \end{aligned}$$

(3.25)

where we replace (3.3) by

$$\begin{aligned} u^n_0 \, \rightarrow \, u_0 \;\; \text{ in } \;\; \mathcal {V}_2 \quad \; \text{ and } \; \quad u^n_1 \, \rightarrow \, u_1 \;\; \text{ in } \;\; \mathcal {V}_0. \end{aligned}$$

(3.26)

Thereby, we can pass the limit on the corresponding approximate problem to obtain

$$\begin{aligned} u_{tt} = - \Delta ^2 u + \text{ div }F(\nabla u) + \int _0^t g(t-s) \Delta ^2 u(s) \, ds \quad \text{ in } \quad L^\infty \left( 0,T; \mathcal {V}_2'\right) . \end{aligned}$$

$\square$

Proof of Theorem 2.3 $(iii)$ (weak solutions) Given initial data $U_0=(u_{0},u_{1}), V_0=(v_{0},v_{1})\in \mathcal {H}$, let us consider the corresponding initial regular data $U^n_0=(u^n_{0},u^n_{1}), \, V^n_0=(v^n_{0},v^n_{1})\in \mathcal {H}_1$ such that

$$\begin{aligned} (U^n_0,V^n_0)\rightarrow & {} (U_0,V_0) \quad \text{ in } \quad \mathcal {H}\times \mathcal {H}, \end{aligned}$$

(3.27)

and the respective regular solutions $U^n=(u^n,u^n_{t}), \, V^n=(v^n,v^n_{t})$ converging to the weak solutions $U=(u,u_{t}), \, V=(v,v_{t})$ as in (3.24), namely

$$\begin{aligned} (U^{n},V^{n})\rightarrow & {} (U,V) \quad \text{ in } \quad C([0,T],\mathcal {H}\times \mathcal {H}). \end{aligned}$$

(3.28)

Since (2.11) holds for stronger weak solutions we have

$$\begin{aligned} \Vert U^n(t)-V^n(t)||_{\mathcal {H}} \, \le \, C_T ||U^n_0-V^n_0||_{\mathcal {H}} , \quad t\in [0,T], \ n\in \mathbb {N}, \end{aligned}$$

(3.29)

for some constant $C_T=C(\Vert U_0\Vert _{\mathcal {H}}, \Vert V_0\Vert _{\mathcal {H}},T)>0.$

Therefore, (2.11) is given for weak solutions after passing (3.29) to the limit when $n\rightarrow \infty$ and applying (3.27)–(3.28). In particular, we have uniqueness of solution in both cases. This completes the proof of Theorem 2.3. $\square$

4 Uniform Decay of the Energy

Our methods on stability are similar to (but not equal to) those for viscoelastic wave equations, see for instance [13, 18, 21, 22, 25, 32]. The proofs of Lemma 2.5 and Theorem 2.8 are given first for regular solutions. Then by standard density arguments the conclusion of Theorem 2.8 also holds for weak solutions.

Proof of Lemma 2.5 By taking the multiplier $u_t$ with (1.6) and using the identities (3.4)–(3.5) for the solution, then the energy defined in (2.2) satisfies (2.12). Therefore, the proof of Lemma 2.5 follows readily. $\square$

Before proving Theorem 2.8 we need to state some technical lemmas.

Lemma 4.1

Under the assumptions of Theorem 2.3 we have

$$\begin{aligned} \frac{d}{dt}E(t) \, \le \, \frac{1}{2}(g'\square \Delta u)(t)\, \le \, 0, \quad \forall \ t>0. \end{aligned}$$

(4.1)

Proof

Inequality (4.1) is an immediate consequence of Lemma 2.5. $\square$

Let us first define the functionals

$$\begin{aligned} \phi _{\chi }(t)= & {} \int _{\Omega } \left( \int _0^t g(t-s)|\chi (t)-\chi (s)| \,ds \right) ^2 dx, \\ \psi _{\chi }(t)= & {} \int _{\Omega } \left( -\int _0^t g'(t-s)|\chi (t)-\chi (s)| \, ds \right) ^2 dx, \\ \zeta _{\chi }(t)= & {} \int _{\Omega } \left( \int _0^t g(t-s)|\chi (s)| \, ds \right) ^2 dx. \end{aligned}$$

Lemma 4.2

Under the assumptions of Theorem 2.3 we have:

• (a) $\displaystyle \phi _{u}(t) \le \frac{(1-l)}{\lambda _1}(g\square \Delta u)(t), \quad \forall \ t\ge 0.$

• (b) $\displaystyle \phi _{\nabla u}(t) \le \frac{(1-l)}{\lambda _2}(g\square \Delta u)(t), \quad \forall \ t\ge 0.$

• (c) $\displaystyle \phi _{\Delta u}(t) \le {(1-l)}(g\square \Delta u)(t), \quad \forall \ t\ge 0.$

• (d) $\displaystyle \psi _{u}(t) \le \frac{g(0)}{\lambda _1}(-g'\square \Delta u)(t), \quad \forall \ t\ge 0.$

• (e) $\displaystyle \psi _{\nabla u}(t) \le \frac{g(0)}{\lambda _2}(-g'\square \Delta u)(t), \quad \forall \ t\ge 0.$

• (f) $\displaystyle \zeta _{\Delta u}(t) \le 2(1-l)(g\square \Delta u)(t) + 2(1-l)^2 \Vert \Delta u(t)\Vert ^2_2, \quad \forall \ t\ge 0.$

Proof

To prove the items (a)–(e) it is enough to apply Hölder’s inequality along with the embeddings $\mathcal {V}_2\hookrightarrow \mathcal {V}_1 \hookrightarrow \mathcal {V}_0$ and the first condition in (2.7). Moreover, from (2.7) and item $(c)$ of Lemma 4.2, we prove the item $(f)$ as follows.

$$\begin{aligned} \zeta _{\Delta u}(t)\le & {} \int _{\Omega } \left( \int _0^t g(t-s)\big (|\Delta u(t) - \Delta u(s)| + |\Delta u(t)| \big ) \, ds \right) ^2 dx \\\le & {} \int _{\Omega } \left( \int _0^t g(t-s)|\Delta u(t) - \Delta u(s)| \, ds + \int _0^t g(t-s)\, ds |\Delta u(t)| \right) ^2 dx \\\le & {} 2 \phi _{\Delta u}(t) + 2\left( \int _0^t g(s) \, ds\right) ^2 \Vert \Delta u(t) \Vert ^2_2 \\\le & {} 2(1-l)(g\square \Delta u)(t) + 2(1-l)^2 \Vert \Delta u(t)\Vert ^2_2. \end{aligned}$$

$\square$

Let us now define the functionals

$$\begin{aligned} {G}(t) \, = \, E(t) + \epsilon _1 \Phi (t) + \epsilon _2 \Psi (t), \quad t\ge 0, \end{aligned}$$

(4.2)

where $\epsilon _1, \, \epsilon _2>0$ will be fixed later and

$$\begin{aligned} \Phi (t)= & {} \xi (t) \int _{\Omega } \big (u_t(t)-\Delta u_t(t)\big ) u(t) \,dx, \end{aligned}$$

(4.3)

$$\begin{aligned} \Psi (t)= & {} -\xi (t) \int _{\Omega } \big (u_t(t)-\Delta u_t(t)\big ) \left( \int _0^t g(t-s)(u(t)-u(s)) ds \right) dx. \quad \end{aligned}$$

(4.4)

Lemma 4.3

Under the assumptions of Theorem 2.8 there exists a constant $c_0>0$ such that $\Phi (t)$ given in (4.3) satisfies

$$\begin{aligned} \frac{d}{dt}\Phi (t)\le & {} c_0\xi (t)\Big [ \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t) \Vert _2^2+(g\square \Delta u)(t)\Big ] \nonumber \\&- \, \xi (t)\left[ \frac{\beta _1}{2}\Vert \Delta u(t) \Vert _2^2 + E(t)\right] , \quad \forall \ t>0, \end{aligned}$$

(4.5)

where $\beta _1=\beta /2>0$.

Proof

Differentiating $t\mapsto \Phi (t)$, using Eq. (1.6) and integrating by parts we get

$$\begin{aligned} \frac{d}{dt}\Phi (t)= & {} \xi (t)\Big [ \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t) \Vert _2^2 \Big ] + \xi '(t) J_1 +\xi (t) J_2 \nonumber \\&- \, \xi (t)\left[ \Vert \Delta u(t) \Vert _2^2+\int _{\Omega }F(\nabla u(t))\cdot \nabla u(t) \, dx \right] , \end{aligned}$$

(4.6)

where

$$\begin{aligned} J_1= & {} (u_t(t),u(t)) + (\nabla u_t(t), \nabla u(t) ), \\ J_2= & {} \int _0^t g(t-s) (\Delta u(s), \Delta u(t)) \, ds. \end{aligned}$$

Now, applying Young’s inequality with $\eta _1>0$ and $\eta _2>0,$ it is easy to check that

$$\begin{aligned} |J_1|\le & {} \eta _1\lambda \Vert \Delta u(t)\Vert ^2_2 +\frac{1}{4\eta _1} \left( \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t) \Vert _2^2\right) , \\ |J_2|\le & {} \left( \int _0^t g(s) \, ds\right) \Vert \Delta u(t)\Vert ^2_2 + \eta _2 \Vert \Delta u(t)\Vert ^2_2+ \frac{1}{4\eta _2}(g\square \Delta u)(t). \end{aligned}$$

Inserting these two last estimates into (4.6), using the third condition in (2.14), Adding and subtracting $\xi (t)E(t)$, we obtain

$$\begin{aligned} \frac{d}{dt}\Phi (t)\le & {} \xi (t)\left( \frac{3}{2}+\frac{\xi _0}{4\eta _1}\right) \Big [ \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t) \Vert _2^2 \Big ] + \xi (t)\left( \frac{1}{2}+\frac{1}{4\eta _2}\right) (g\square \Delta u)(t) \\&- \, \xi (t) \frac{h(t)}{2}\Vert \Delta u(t) \Vert _2^2 + \xi (t) \int _{\Omega } \big [f(\nabla u(t)) - F(\nabla u(t))\cdot \nabla u(t) \big ] dx \\&+ \, \xi (t) (\lambda \xi _0\eta _1+\eta _2) \Vert \Delta u(t) \Vert _2^2 -\xi (t) E(t). \end{aligned}$$

Now applying assumption (2.6) and condition (2.7), we have

$$\begin{aligned} \frac{d}{dt}\Phi (t)\le & {} \xi (t)\left( \frac{3}{2}+\frac{\xi _0}{4\eta _1}\right) \Big [ \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t) \Vert _2^2 \Big ] + \xi (t)\left( \frac{1}{2}+\frac{1}{4\eta _2}\right) (g\square \Delta u)(t) \nonumber \\&- \, \xi (t) \left( \frac{\beta }{2}-\lambda \xi _0\eta _1-\eta _2\right) \Vert \Delta u(t) \Vert _2^2 -\xi (t) E(t). \end{aligned}$$

(4.7)

Since $\beta _1=\frac{\beta }{2}>0$, so choosing $\eta _2=\eta _1 \le \frac{\beta _1}{2(1+\lambda \xi _0)}$, and setting $c_0=\max \big \{\frac{3}{2}+\frac{\xi _0}{4\eta _1},\frac{1}{2}+ \frac{1}{4\eta _1}\big \}$ in (4.7), we conclude that (4.5) holds true. This completes the proof of Lemma 4.3. $\square$

Lemma 4.4

Under the assumptions of Theorem 2.8, and given any $\delta >0,$ then there exists a constant $c_{\delta }>0$ such that $\Psi$ defined in (4.4) satisfies

$$\begin{aligned} \frac{d}{dt}\Psi (t)\le & {} \left( \delta (1+\xi _0) \!-\! \int _0^t g(s) \, ds\right) \xi (t)\Big [ \Vert u_t(t) \Vert _2^2 \!+\! \Vert \nabla u_t(t)\Vert ^2_2\Big ] \!+\! 4\delta \xi (t) \Vert \Delta u(t) \Vert _2^2 \nonumber \\&+ \,\, c_{\delta }\left( 1+[E(0)]^{\frac{p-1}{2}}\right) \xi (t)(g\square \Delta u)(t) + c_{\delta }(-g'\square \Delta u)(t), \quad \forall \ t>0, \nonumber \\ \end{aligned}$$

(4.8)

where

$$\begin{aligned} p=\left\{ \begin{array}{rl} \displaystyle \max \{p_1,\ldots ,p_N\} &{} \text{ if } \quad E(0)\ge 1,\\ \displaystyle \min \{p_1,\ldots ,p_N\} &{} \text{ if } \quad E(0) < 1. \end{array}\right. \end{aligned}$$

Proof

Differentiating $\Psi$, using Eq. (1.6) and integrating by parts we get

$$\begin{aligned} \frac{d}{dt}\Psi (t)= & {} -\left( \int _0^t g(s) \, ds \right) \xi (t) \Big [ \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t) \Vert _2^2 \Big ] + \sum _{j=1}^{6} I_j+I_F, \quad \quad \end{aligned}$$

(4.9)

where

$$\begin{aligned} I_1= & {} -\xi '(t)\int _{\Omega } u_t(t) \left( \int _0^t g(t-s) ( u(t) - u(s) ) \, ds \right) dx, \\ I_2= & {} -\xi '(t)\int _{\Omega } \nabla u_t(t) \left( \int _0^t g(t-s) ( \nabla u(t) - \nabla u(s) ) \, ds \right) dx, \\ I_3= & {} \xi (t)\int _{\Omega } \Delta u(t) \left( \int _0^t g(t-s) ( \Delta u(t) - \Delta u(s) ) \, ds \right) dx, \\ I_4= & {} -\xi (t)\int _{\Omega } \left( \int _0^tg(t-s)\Delta u(s) \, ds\right) \left( \int _0^t g(t-s) ( \Delta u(t) - \Delta u(s) ) \, ds \right) dx, \\ I_5= & {} \xi (t)\int _{\Omega } u_t(t) \left( -\int _0^t g'(t-s) ( u(t) - u(s) ) \, ds \right) dx, \\ I_6= & {} \xi (t)\int _{\Omega } \nabla u_t(t) \left( -\int _0^t g'(t-s) ( \nabla u(t) - \nabla u(s) ) \, ds \right) dx, \\ I_F= & {} \xi (t)\int _{\Omega } F(\nabla u(t)) \left( \int _0^t g(t-s) ( \nabla u(t) - \nabla u(s) ) \, ds \right) dx, \end{aligned}$$

Now let us estimate $I_j,$ $j=1,\ldots ,6,$ and $I_F$. From Young’s inequality with $\delta >0$, item $(a)$ of Lemma 4.2 and assumption (2.14) we obtain

$$\begin{aligned} |I_1|\le & {} \left| \frac{\xi '(t)}{\xi (t)}\right| \xi (t)\left( \delta \Vert u_t(t) \Vert _2^2 + \frac{1}{4\delta }\phi _{u}(t) \right) \nonumber \\\le & {} \delta \xi _0\xi (t)\Vert u_t(t) \Vert _2^2 + \frac{\xi _0}{4\delta \lambda _1}(1-l)\xi (t)(g\square \Delta u)(t) \end{aligned}$$

(4.10)

Analogously, but using items $(b)$ and $(c)$ of Lemma 4.2 instead of $(a)$, we have

$$\begin{aligned} |I_2|\le & {} \delta \xi _0\xi (t)\Vert \nabla u_t(t) \Vert _2^2 + \frac{\xi _0}{4\delta \lambda _2} (1-l)\xi (t)(g\square \Delta u)(t), \end{aligned}$$

(4.11)

$$\begin{aligned} |I_3|\le & {} \delta \xi (t)\Vert \Delta u(t) \Vert _2^2 + \frac{1}{4\delta }(1-l)\xi (t)(g\square \Delta u)(t). \end{aligned}$$

(4.12)

Again from Young’s inequality with $\delta >0$, items $(c),$ $(f)$ and $(d)$ of Lemma 4.2, we deduce

$$\begin{aligned} |I_4|\le & {} \delta \xi (t) \zeta _{\Delta u}(t) + \frac{1}{4\delta }\xi (t)\phi _{\Delta u}(t) \nonumber \\\le & {} 2\delta (1-l)^2\xi (t) \Vert \Delta u(t)\Vert ^2_2 +\Big (2\delta +\frac{1}{4\delta }\Big )(1-l)\xi (t)(g\square \Delta u)(t), \end{aligned}$$

(4.13)

and

$$\begin{aligned} |I_5|\le & {} \delta \xi (t) \Vert u_t(t)\Vert _2^2 + \frac{1}{4\delta }\xi (t)\psi _{u}(t) \nonumber \\\le & {} \delta \xi (t) \Vert u_t(t)\Vert ^2_2 + g(0)\frac{\xi _1}{4\delta \lambda _1}(-g'\square \Delta u)(t). \end{aligned}$$

(4.14)

Similarly with item $(e)$ in the place of $(d)$ in Lemma 4.2, we also have

$$\begin{aligned} |I_6| \, \le \, \delta \xi (t) \Vert \nabla u_t(t)\Vert ^2_2 + g(0)\frac{\xi _1}{4\delta \lambda _2}(-g'\square \Delta u)(t). \end{aligned}$$

(4.15)

Now with respect to $I_F$ we have

$$\begin{aligned} |I_F| \, \le \, \xi (t)\int _0^t g(t-s) \underbrace{\left( \int _{\Omega } |F(\nabla u(t))| | \nabla u(t) - \nabla u(s) | \, dx \right) }_{:=I^1_F}ds. \end{aligned}$$

Applying (4.28) from the appendix with $F(0)=0$, Hölder’s inequality, Young’s inequality with $\delta >0$, and since $\mathcal {V}_2 \hookrightarrow W_0^{1,p_j+1}(\Omega )\hookrightarrow \mathcal {V}_1, \ j=1,\ldots ,N$, we infer

$$\begin{aligned} I^1_F\le & {} K \int _{\Omega } \left( \sum _{j=1}^{N}\left( 1+|\nabla u(t)|^{(p_j-1)/2} \right) \right) |\nabla u(t)||\nabla u(t) - \nabla u(s)| \, dx \\\le & {} K\sum _{j=1}^{N}\left( |\Omega |^{\frac{p_j-1}{2(p_j+1)}}+\Vert \nabla u(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}} \right) \Vert \nabla u(t)\Vert _{p_j+1} \Vert \nabla u(t)-\nabla u(s)\Vert _2\\\le & {} \Vert \Delta u(t)\Vert _{2} \left[ \displaystyle \frac{K}{\lambda _2}\sum _{j=1}^{N}\mu _{p_j}\left( |\Omega |^{\frac{p_j-1}{2(p_j+1)}}+\Vert \nabla u(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}} \right) \right] \Vert \Delta u(t)-\Delta u(s)\Vert _2 \\\le & {} \delta \Vert \Delta u(t)\Vert _{2}^2 \!+\!\frac{1}{4\delta } \underbrace{\left[ \frac{K}{\lambda _2}\sum _{j=1}^{N}\mu _{p_j}\left( |\Omega |^{\frac{p_j-1}{2(p_j+1)}}\!+\!\Vert \nabla u(t)\Vert _{p_j+1}^{\frac{p_j-1}{2}} \right) \right] ^2}_{:=I^2_F}\Vert \Delta u(t)\!-\!\Delta u(s)\Vert ^2_2. \end{aligned}$$

Since $\Vert \nabla u(t)\Vert _{p_j+1}\le \mu _{p_j} \Vert \Delta u(t)\Vert _{2}$ and $\frac{\beta }{2}\Vert \Delta u(t)\Vert _{2}^2\le E(t)\le E(0)$ for any $t>0,$ then

$$\begin{aligned} I^2_F\le & {} \frac{2K^2}{\lambda _2^2}\left( \sum _{j=1}^{N}\mu _{p_j} |\Omega |^{\frac{p_j-1}{2(p_j+1)}}\right) ^2 + \frac{2K^2}{\lambda _2^2}\left( \sum _{j=1}^{N}\mu _{p_j}^{\frac{p_j+1}{2}}\left( \frac{2}{\beta }\right) ^{\frac{p_j-1}{4}} [E(0)]^{\frac{p_j-1}{4}} \right) ^2 \\\le & {} \mu _1 + \mu _2 [E(0)]^{\frac{p-1}{2}}, \end{aligned}$$

where we consider

$$\begin{aligned} p:= & {} \left\{ \begin{array}{rl} \displaystyle \max \{p_1,\ldots ,p_N\} &{} \text{ if } \quad E(0)\ge 1,\\ \displaystyle \min \{p_1,\ldots ,p_N\} &{} \text{ if } \quad E(0) < 1, \end{array}\right. \\ \mu _1:= & {} \frac{2K^2}{\lambda _2^2}\left( \sum _{j=1}^{N}\mu _{p_j} |\Omega |^{\frac{p_j-1}{2(p_j+1)}}\right) ^2 \quad \text{ and } \quad \mu _2:=\frac{2K^2}{\lambda _2^2}\left( \sum _{j=1}^{N}\mu _{p_j}^{\frac{p_j+1}{2}}\left( \frac{2}{\beta }\right) ^{\frac{p_j-1}{4}}\right) ^2. \end{aligned}$$

Thus,

$$\begin{aligned} I^1_F \le \delta \Vert \Delta u(t)\Vert _{2}^2 +\frac{1}{4\delta } \left( \mu _1 + \mu _2 [E(0)]^{\frac{p-1}{2}}\right) \Vert \Delta u(t)-\Delta u(s)\Vert ^2_2 \end{aligned}$$

from where it follows that

$$\begin{aligned} |I_F| \, \le \, \delta \xi (t)\Vert \Delta u(t)\Vert _{2}^2 + \frac{1}{4\delta }\left( \mu _1 + \mu _2 [E(0)]^{\frac{p-1}{2}}\right) \xi (t)(g\square \Delta u)(t). \end{aligned}$$

(4.16)

Inserting (4.10)–(4.16) into (4.9), and since $1-l<1,$ yields

$$\begin{aligned} \frac{d}{dt}\Psi (t)\le & {} \left( \delta (1+\xi _0) -\int _0^t g(s) \, ds \right) \xi (t) \Big [ \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t) \Vert _2^2 \Big ] \\&+ \, 4\delta \xi (t) \Vert \Delta u(t) \Vert _2^2 + g(0)\frac{\xi _1\lambda }{4\delta }(-g'\square \Delta u)(t) \\&+ \, \frac{1}{4\delta }\left( 2+\xi _0\lambda +8\delta ^2 +\mu _1 + \mu _2 [E(0)]^{\frac{p-1}{2}}\right) \xi (t)(g\square \Delta u)(t). \end{aligned}$$

Therefore, inequality (4.8) follows by taking $c_{\delta }=\frac{1}{4\delta }\max \{2+\xi _0\lambda +8\delta ^2+\mu _1, \mu _2, g(0)\xi _1\lambda \}$. This concludes the proof of Lemma 4.4. $\square$

Lemma 4.5

Under the assumptions of Theorem 2.8 and fixing any $t_0>0$, then

$$\begin{aligned} \frac{d}{dt} G(t) \, \le \, -\epsilon _1 \xi (t) E(t), \quad \forall \ t\ge t_0, \end{aligned}$$

(4.17)

for some positive constant $\epsilon _1\sim \frac{c_1}{1+[E(0)]^{\frac{p-1}{2}}},$ with $c_1>0$ independent of the initial data.

Proof

From definition of $G(t)$ in (4.2), and Lemmas 4.1, 4.3 and 4.4, we get

$$\begin{aligned} \frac{d}{dt}G(t)\le & {} \left( \epsilon _1 c_0 +\epsilon _2\Big (\delta b_1 - \int _0^t g(s) \, ds\Big )\right) \xi (t)\Big [ \Vert u_t(t) \Vert _2^2 + \Vert \nabla u_t(t)\Vert ^2_2\Big ] \nonumber \\&-\, \left( \epsilon _1\frac{\beta _1}{2}-4\delta \epsilon _2\right) \xi (t) \Vert \Delta u(t) \Vert _2^2 -\epsilon _1\xi (t)E(t) \nonumber \\&+ \, \left( \epsilon _1c_0+\epsilon _2c_{\delta }\left( 1+[E(0)]^{\frac{p-1}{2}}\right) \right) \xi (t)(g\square \Delta u)(t) \nonumber \\&+ \left( \frac{1}{2}-\epsilon _2c_{\delta }\right) (g'\square \Delta u)(t), \end{aligned}$$

(4.18)

for any $\delta , \epsilon _1, \epsilon _2>0,$ where we denote $b_1=1+\xi _0>0.$ By fixing any $t_0>0$ we note that

$$\begin{aligned} \int _0^t g(s) \, ds \ge \int _0^{t_0} g(s) \, ds :=g_0>0, \quad \forall \ t\ge t_0. \end{aligned}$$

From this and condition (2.13) we can rewrite (4.18) as follows

$$\begin{aligned} \frac{d}{dt}G(t)\le & {} -\big (\epsilon _2(g_0 -\delta b_1)-\epsilon _1 c_0\big ) \xi (t)\Big [ \Vert u_t(t)\Vert _2^2 + \Vert \nabla u_t(t)\Vert ^2_2\Big ] \nonumber \\&- \left( \epsilon _1\frac{\beta _1}{2}-4\delta \epsilon _2\right) \xi (t) \Vert \Delta u(t) \Vert _2^2 -\epsilon _1\xi (t) E(t) \nonumber \\&+ \left( \frac{1}{2}-\epsilon _1c_0-2\epsilon _2c_{\delta }\left( 1+[E(0)]^{\frac{p-1}{2}}\right) \right) (g'\square \Delta u)(t), \end{aligned}$$

(4.19)

for every $t\ge t_0$. Now we first choose $0<\delta \le \min \{\frac{g_0}{2b_1},\frac{g_0\beta _1 }{32c_0}\}.$ Thus

$$\begin{aligned} g_0-\delta b_1\ge \frac{g_0 }{2} \quad \text{ and } \quad 4\delta \le \frac{g_0\beta _1}{8c_0}. \end{aligned}$$

(4.20)

Once fixed $\delta >0$ we pick out $\epsilon _1>0$ and $\epsilon _2>0$ small enough such that

$$\begin{aligned} \frac{1}{2}\epsilon _2 \, < \, \frac{2c_0}{g_0}\epsilon _1 \, < \, \epsilon _2 \, < \, \min \left\{ \frac{1}{2g_0},\frac{1}{8c_{\delta }\left( 1+[E(0)]^{\frac{p-1}{2}}\right) }\right\} . \end{aligned}$$

(4.21)

Therewith (4.20)–(4.21) imply that

$$\begin{aligned}&\epsilon _2(g_0 -\delta b_1)-\epsilon _1 c_0 >0, \quad \epsilon _1\frac{\beta _1}{2}-4\delta \epsilon _2> 0, \nonumber \\&\quad \frac{1}{2}-\epsilon _1c_0 -2\epsilon _2c_{\delta }\left( 1+[E(0)]^{\frac{p-1}{2}}\right) >0. \end{aligned}$$

Therefore, since $g'\le 0,$ we obtain from (4.19) that estimate (4.17) holds true for some positive constant $\epsilon _1\sim {c_1}/{\left( 1+[E(0)]^{\frac{p-1}{2}}\right) },$ where $c_1>0$ is independent of the initial data. This completes the proof of Lemma 4.5. $\square$

With the lemmas above we have obtained the main ingredients to prove Theorem 2.8.

Proof of Theorem 2.8 First of all we note that condition $\alpha _0=0$ in (2.6) along with (2.7) ensure that the energy $E(t)$ is nonnegative for all $t\ge 0.$ In addition, using Hölder and Young inequalities, Lemma 4.2 (a)–(b), and (2.6) with $\alpha _0=0$, it is easy to check that

$$\begin{aligned} \frac{1}{2} \, E(t) \, \le \, G(t) \, \le \, \frac{3}{2} \, E(t), \quad \forall \ t\ge 0, \end{aligned}$$

(4.22)

where we take $\epsilon _1$ and $\epsilon _2$ such that

$$\begin{aligned} 0<\epsilon _1, \, \epsilon _2 < {\beta _1} \min \left\{ \frac{1}{2\xi _1},\frac{1}{\lambda \xi _1}\right\} . \end{aligned}$$

(4.23)

Now let us fix $t_0=1$ and choose $\epsilon _1>0$ and $\epsilon _2>0$ so that (4.21) and (4.23) are satisfied. Then from Lemma 4.5 and (4.22) we have

$$\begin{aligned} \frac{d}{dt} G(t) \le - \epsilon _1 \xi (t) E(t) \le -\frac{2\epsilon _1}{3}\xi (t) G(t), \quad \forall \ t\ge 1. \end{aligned}$$

A straightforward computation implies that

$$\begin{aligned} G(t) \le G(1) e^{-\gamma \int _1^t \xi (s) \, ds}, \quad \forall \ t\ge 1, \end{aligned}$$

where $\gamma =\frac{2\epsilon _1}{3}\sim {k}/{\left( 1+[E(0)]^{\frac{p-1}{2}}\right) },$ for some $k>0$. Applying again (4.22) and since $E(t)$ is nonincreasing we obtain

$$\begin{aligned} E(t) \le 3 E(1) \, e^{-\gamma \int _1^t \xi (s) \, ds} \le \left( 3 E(0) \, e^{\gamma \int _0^1 \xi (s) \, ds} \right) \, e^{-\gamma \int _0^t \xi (s) \, ds}, \end{aligned}$$

from where it follows that

$$\begin{aligned} E(t) \le c\, e^{-\gamma \int _0^t \xi (s) \, ds}, \quad \forall \ t\ge 1, \end{aligned}$$

(4.24)

where $c=3 E(0) \, e^{\gamma \int _0^1 \xi (s) \, ds},$ and $\gamma \sim {k}/{\left( 1+[E(0)]^{\frac{p-1}{2}}\right) }$ for some positive constant $k$.

On the other hand, since $0<\xi (t)\le \xi _1>0$ for any $t\ge 0,$ we see that

$$\begin{aligned} 1< e^{\gamma \int _t^1 \xi (s) \, ds}= \left( e^{\gamma \int _0^1 \xi (s) \, ds}\right) e^{-\gamma \int _0^t \xi (s) \, ds}< e^{\gamma \xi _1}< \infty , \quad \forall \ t\in [0,1]. \end{aligned}$$

Then, since $E(0) \,e^{\gamma \int _0^1 \xi (s) \, ds}< c$, we get

$$\begin{aligned} E(t) \le E(0) \le c \, e^{-\gamma \int _0^t \xi (s) \, ds}, \quad \forall \ t\in [0,1]. \end{aligned}$$

(4.25)

Therefore, the uniform decay (2.16) is ensured by estimates (4.24)–(4.25). The proof of Theorem 2.8 is now complete. $\square$

Remark 4.6

The proof of Theorem 2.10 (ii) can be done with minor changes on the above calculations. In fact, neglecting terms which appear due to the presence of rotational inertia term it can be proved in a similar way that the approximate energy

$$\begin{aligned} \mathcal {E}^n(t) = \frac{1}{2} \Vert u^n_t(t) \Vert _2^2 + \frac{h(t)}{2}\Vert \Delta u^n(t) \Vert _2^2 + \frac{1}{2}(g\square \Delta u^n)(t) + \int _{\Omega } {f}(\nabla u^n(t)) \, dx \end{aligned}$$

satisfies (2.19) for Galerkin solutions given in (3.25). In addition, (3.25) implies that $\mathcal {E}^n(t)$ is lower semi-continuous on the weak star topology whereas (3.26) ensures that $\mathcal {E}^n(0)$ converges in the strong topology, namely,

$$\begin{aligned} \mathcal {E}(t) \le \liminf _{n\rightarrow \infty }\mathcal {E}^n(t) \quad \text{ and } \quad \mathcal {E}^n(0)\rightarrow \mathcal {E}(0). \end{aligned}$$

This is sufficient to conclude that (2.19) holds true. $\square$

5 Appendix

5.1 Rates of Energy Decay

We emphasize below that (2.16) provides several rates of energy decay according to the function $\xi$. In fact, the Examples 4.7 and 4.8 are motivated by [18, 21, 22, 24, 32, 33]. Besides, the Examples 4.9 and 4.10 are new and illustrate the wide variety of distinct decay rates provided by (2.16).

Example 4.7

Let us consider $\xi (t)= \kappa \ln (a+1)$, where $a>0$ and $\kappa >0$. Then $\xi$ satisfies (2.14). The integrability condition (2.3) is satisfied if $g(0)<\kappa \ln (a+1)$. Applying (2.16) results in

$$\begin{aligned} E(t) \le c \, (a+1)^{-\kappa \gamma t}, \quad \ t\ge 0. \end{aligned}$$

For $a=e-1$ we recover the standard exponential type.

Example 4.8

Let us take a rational function $\xi (t)=\frac{\kappa }{t+1},$ with $\kappa >1.$ It is also easy to check that condition (2.14) holds true. From (2.16) we have the following polynomial-type decay

$$\begin{aligned} E(t) \le \frac{c}{(t+1)^{\kappa \gamma }}, \quad \ t\ge 0. \end{aligned}$$

The case $1\le \kappa \gamma \le 2$ is allowed depending on the size of $\gamma >0,$ compare the remarks following (1.2). However, once fixed initial conditions, the interesting case consists in considering polynomial decay of higher order which is possible by taking large values for $\kappa$. In this case, the condition $g(0)<\kappa -1$, observing (2.15), ensures the integrability condition on $g$ required in (2.3).

Example 4.9

Let us now take $\xi (t)=\kappa \frac{1+\ln [\ln (t+1)+1]}{t+1}$, with $\kappa >0$. Then $\xi$ satisfies the conditions in (2.14). From (2.16) we get the decay type

$$\begin{aligned} E(t)\le \frac{c}{[\ln (t+1)+1]^{\kappa \gamma [\ln (t+1)+1]}}, \quad \ t\ge 0. \end{aligned}$$

The integrability condition (2.3) is satisfied if $g(0) \int _0^\infty \,[\ln (t+1)+1]^{-\kappa [\ln (t+1)+1]}\,dt <1.$

Example 4.10

We also consider $\xi (t)=\kappa \frac{1+2\ln (t+e^{1/2})}{t+e^{1/2}}, \ t\ge 0,$ where $\kappa >1$. Then $\xi$ fulfills condition (2.14). The integrability condition on $g$ stated in (2.3) is satisfied if $g(0) < (\kappa -1)e^{-\frac{\kappa +2}{4}}$. From (2.16) we conclude that the energy has a decay of the form

$$\begin{aligned} E(t)\le \frac{c \, e^{3\kappa /4}}{(t+e^{1/2})^{\kappa \gamma (1+\ln (t+e^{1/2}))}}, \quad \ t\ge 0. \end{aligned}$$

5.2 The Vector Field $F$

In this section we show an interesting property to vector fields which satisfy a condition like (2.4). This is given as an application of the Mean Value Inequality.

Lemma 4.1

Let $F:\mathbb {R}^N\rightarrow \mathbb {R}^N$ be a $C^1$-vector field given by $F=(F_1,\ldots ,F_N)$.

• (a) If there exist positive constants $k_1,\ldots ,k_N$ and $q_1,\ldots ,q_N$ such that

  $$\begin{aligned} |\nabla F_j(z)|\le k_j(1+|z|^{q_j}), \quad \forall \ z\in \mathbb {R}^N, \ \forall \ j=1,\ldots ,N. \end{aligned}$$

  (4.26)

  Then, there exists a constant $K=K(k_j, q_j,N)>0, \ j=1,\ldots ,N,$ such that

  $$\begin{aligned} |F(x)-F(y)| \, \le \, K \sum _{j=1}^{N}\left( 1+|x|^{q_j}+|y|^{q_j}\right) |x-y|, \quad \forall \ x,y\in \mathbb {R}^N. \end{aligned}$$

  (4.27)

• (b) In particular, we have

  $$\begin{aligned} |F(x)| \, \le \, |F(0)| + K \sum _{j=1}^{N}\left( 1+|x|^{q_j}\right) |x|, \quad \forall \ x\in \mathbb {R}^N. \end{aligned}$$

  (4.28)

Proof

$(a)$ From condition (4.26) it is not so difficult to check that $F':\mathbb {R}^N \rightarrow \mathcal {L}(\mathbb {R}^N)$ satisfies

$$\begin{aligned} ||F'(z)||_{\mathcal {L}(\mathbb {R}^N)} \, \le \, N \sum _{j=1}^{N} k_j(1+|z|^{q_j}), \quad \forall \ z\in \mathbb {R}^N. \end{aligned}$$

Given $x,y\in \mathbb {R}^N$, we consider $z\in [x,y]\subset \mathbb {R}^N$ written as

$$\begin{aligned} z=(1-\theta )y+\theta x, \quad \theta =\theta (x,y)\in [0,1]. \end{aligned}$$

Thus,

$$\begin{aligned} |z|^{q_j} \, \le \, 2^{q_j}(|x|^{q_j}+|y|^{q_j}), \quad \forall \ j=1,\ldots ,N, \end{aligned}$$

from where it follows that

$$\begin{aligned} ||F'(z)||_{\mathcal {L}(\mathbb {R}^N)} \, \le \, K \sum _{j=1}^{N} (1+|x|^{q_j}+|y|^{q_j}), \quad \forall \ z\in [x,y], \end{aligned}$$

where $K=N\displaystyle \max _{1\le j\le N} \{2^{q_j}k_j\}.$ Applying the Mean Value Inequality we obtain (4.27).

$(b)$ It is suffices to define $G(z)=F(z)-F(0).$ $\square$

5.3 Examples for $F$

We finally give examples of vector fields satisfying conditions like (2.4) and (2.6). More generally, we show below some applications of conservative $C^1$-vector fields $F=\nabla f$ such that (4.26) holds and also

$$\begin{aligned} -a_0-a_1|z|^{2}\le f(z)\le F(z)\cdot z + a_2|z|^{2}, \quad \forall \; z\in \mathbb {R}^N, \end{aligned}$$

(4.29)

for some nonnegative constants $a_0,a_1,a_3\ge 0.$

Example 4.11

Let us first consider

$$\begin{aligned} F : \mathbb {R}^{N}&\longrightarrow \mathbb {R}^{N} \\ z&\longmapsto F(z)=|z|^{q}z, \quad q\ge 0. \end{aligned}$$

Denoting by $F(z)=(F_1(z),\ldots ,F_N(z))$ and $z=(z_1,\ldots ,z_N)\in \mathbb {R}^N,$ then

$$\begin{aligned} F_{j}(z)=|z|^{q}z_{j}, \quad j=1,\ldots ,N. \end{aligned}$$

If we consider $0\ne z\in \mathbb {R}^N$ and $i,j=1,\ldots ,N,$ we have

$$\begin{aligned} \begin{array}{lc} \displaystyle \frac{\partial }{\partial z_{i}}F_{j}(z)=q|z|^{q-2}z_{i}z_{j} &{} \text{ for } \ i\ne j, \\ \displaystyle \frac{\partial }{\partial z_{j}} F_{j}(z)=q|z|^{q-2}z^{2}_{j}+|z|^{q} &{} \text{ for } \ i= j. \end{array} \end{aligned}$$

(4.30)

It is also easy to check by definition that

$$\begin{aligned} \frac{\partial }{\partial z_{i}} F_{j}({0})=0=\lim _{z\rightarrow {0}}\displaystyle \frac{\partial }{\partial z_{i}} F_{j}(z), \quad i,j=1,\ldots ,N. \end{aligned}$$

Thus, the components $F_1,\ldots ,F_N$ are $C^1$-functions in $\mathbb {R}^N$. Besides, from (4.30) we get

$$\begin{aligned} \left| \frac{\partial }{\partial z_{i}}F_{j}(z)\right| \le (1+q)|z|^q, \quad i,j=1,\ldots ,N. \end{aligned}$$

This is suffices to ensure that (4.26) holds true. Moreover, we note that $F=\nabla f$, where

$$\begin{aligned} f : \mathbb {R}^{N}&\longrightarrow \mathbb {R} \\ z&\longmapsto f(z)=\frac{1}{q+2}|z|^{q+2}, \end{aligned}$$

and then condition (4.29) is readily verified for any $a_0,a_1,a_2\ge 0$.

Therefore, this vector field generates the following operator

$$\begin{aligned} \text{ div }F(\nabla u) \, = \, \text{ div }\left( |\nabla u|^{q} \nabla u\right) , \end{aligned}$$

which consists a $p$-Laplacian one by taking $p=2q+1$ satisfying (2.5).

Example 4.12

The above argument can be applied to the vector field $F=\nabla f$, where

$$\begin{aligned} f(z)=\frac{k}{q+2}|z|^{q+2}+\tau \cdot z, \ \ z=(z_1,\ldots ,z_N)\in \mathbb {R}^N, \end{aligned}$$

with $q\ge 0,$ $k>0,$ and $\tau =(\tau _1,\ldots ,\tau _N)\in \mathbb {R}^N.$ Thus, condition (4.26) follows analogously as above whereas (4.29) is fulfilled with $a_0= \frac{|\tau |^2}{2}$ $a_1=\frac{1}{2}$, and any $a_2\ge 0$.

Example 4.13

Another case of $p$-Laplacian operator arises when we consider the vector field $F=(F_1,\ldots ,F_N)$ whose components $F_j, \, j=1,\ldots ,N,$ are given by

$$\begin{aligned} F_{j}(z)=|z_j|^{p-2}z_{j}, \quad \forall \ z=(z_1,\ldots ,z_N)\in \mathbb {R}^N, \end{aligned}$$

where $p\ge 2.$ In this case

$$\begin{aligned} \text{ div }F(\nabla u) \, = \, \sum _{j=1}^{N}\frac{\partial }{\partial x_j}\left( \left| \frac{\partial u}{\partial x_j}\right| ^{p-2} \frac{\partial u}{\partial x_j}\right) . \end{aligned}$$

Example 4.14

To illustrate another vector field, different one of $p$-Laplacian type, we consider $F=\nabla f$, where the potential function is given by

$$\begin{aligned} f(z)=\ln \left( \sqrt{|z|^{2}+1}\right) , \ \ z=(z_1,\ldots ,z_N)\in \mathbb {R}^N. \end{aligned}$$

In such case we have

$$\begin{aligned} F(z) \, = \, \frac{z}{|z|^2+1}, \quad \forall \ z\in \mathbb {R}^N, \end{aligned}$$

which vanishes when $z\rightarrow \infty .$ It is easy to check that $F$ and $f$ satisfy (4.26) and (4.29).