2201/2201.00983.md

Global existence and decay estimates for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity

Bhargav Kumar Kakumani ^*†, Suman Prabha Yadav ^‡§

Abstract

In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Using the Faedo-Galerkin method we establish the global existence of the solution of the problem and we also prove few general decay rate results.

Keywords: Viscoelasticity, Global existence, Decay estimates, Convexity, Logarithmic nonlinearity.

AMS Subject Classification 2010: 35A01, 35L55, 74D10, 93D20.

1 Introduction

This work deals with the existence and decay of solutions to the following plate problem:

$$\{\begin{array}{l}\mathrm{\mid }{u}_t{\mathrm{\mid }}^{\rho }{u}_{tt}+{\mathrm{\Delta }}^{2}u+{\mathrm{\Delta }}^2{u}_{tt}+u-{\int }_0^{t}b(t-s){\mathrm{\Delta }}^{2}u(s)ds\\ +h(u_t)=ku\ln \mathrm{\mid }u\mathrm{\mid },(x,t)\in \mathrm{\Omega }\times (0,\mathrm{\infty }),\\ u(x,t)=\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }(x,t)=0,\text{in }\mathrm{\partial }\mathrm{\Omega }\times (0,\mathrm{\infty }),\\ u(x,0)=u_0(x),u_t(x,0)=u_1(x)\text{in }\mathrm{\Omega }\end{array}(1)\backslash begin\{cases\}\; |u\_t|^\backslash rho\; u\_\{tt\}\; +\; \backslash Delta^2\; u\; +\; \backslash Delta^2\; u\_\{tt\}\; +\; u\; -\; \backslash int\_0^t\; b(t-s)\; \backslash Delta^2\; u(s)\; ds\; \backslash \backslash \; \backslash quad\; +\; h(u\_t)\; =\; ku\; \backslash ln\; |u|,\; \backslash quad\; (x,\; t)\; \backslash in\; \backslash Omega\; \backslash times\; (0,\; \backslash infty),\; \backslash \backslash \; u(x,\; t)\; =\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash nu\}(x,\; t)\; =\; 0,\; \backslash quad\; \backslash text\{in\; \}\; \backslash partial\backslash Omega\; \backslash times\; (0,\; \backslash infty),\; \backslash \backslash \; u(x,\; 0)\; =\; u\_0(x),\; \backslash quad\; u\_t(x,\; 0)\; =\; u\_1(x)\; \backslash quad\; \backslash text\{in\; \}\; \backslash Omega\; \backslash end\{cases\}\; \backslash quad\; (1)$$

where $\Omega \subset \mathbb{R}^2$ is a bounded domain, $\nu$ is the unit outer normal to $\partial\Omega$ , $k$ and $\rho$ are positive constants and $0 < \rho \leq \frac{2}{n-2}$ if $n \geq 3$ . We use the Lebesgue space $L^2(\Omega)$ and $H_0^2(\Omega)$ with their usual scalar product and norms. Through out this paper, we consider the following hypotheses:

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^*bhargav@hyderabad.bits-pilani.ac.in

^†Department of Mathematics, BITS-Pilani, Hyderabad Campus, Hyderabad, India.

^‡p20200454@hyderabad.bits-pilani.ac.in

^§Department of Mathematics, BITS-Pilani, Hyderabad Campus, Hyderabad, India.(H1) Let $b : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ is a $C^1$ -nonincreasing function satisfying

$$b(0)>0,1-{\int }_0^{t}b(\tau )d\tau =l>0.(2)b(0)\; >\; 0,\; 1\; -\; \backslash int\_0^t\; b(\backslash tau)\; d\backslash tau\; =\; l\; >\; 0.\; \backslash quad\; (2)$$

(H2) Assume that there exist a nonincreasing positive differentiable function $\xi$ such that $\xi(0) > 0$ and $\xi : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ . Further assume that there exists a $C^1$ function $B : (0, \infty) \rightarrow (0, \infty)$ which is linear or strictly convex $C^2$ function and strictly increasing on $(0, r_1]$ , $r_1 \leq b(0)$ , $B(0) = B'(0) = 0$ and $B$ satisfies

$$b^{\mathrm{\prime }}(t)\le -\xi (t)B(b(t)),\mathrm{\forall }t\ge 0.(3)b\text{'}(t)\; \backslash leq\; -\backslash xi(t)B(b(t)),\; \backslash quad\; \backslash forall\; t\; \backslash geq\; 0.\; \backslash quad\; (3)$$

(H3) Let $h : \mathbb{R} \rightarrow \mathbb{R}$ is a nondecreasing continuous function such that there exist a $h_1 \in C^1(\mathbb{R}^+)$ with $h_1(0) = 0$ which is strictly increasing function and $h_1$ satisfies

where $c_1, c_2, \epsilon$ are positive constants. Moreover, define $H$ to be a strictly convex $C^2$ function in $(0, r_2]$ for some $r_2 \geq 0$ such that $H(s) = \sqrt{s}h_1(\sqrt{s})$ when $h_1$ is nonlinear.

(H4) The constant $k$ in (1) is such that

where $c_p$ is the smallest positive number satisfying

$$\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2\le c_p\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_2^2,u\in H_0^2(\mathrm{\Omega }),(5)\backslash |\backslash nabla\; u\backslash |\_2^2\; \backslash leq\; c\_p\; \backslash |\backslash Delta\; u\backslash |\_2^2,\; \backslash quad\; u\; \backslash in\; H\_0^2(\backslash Omega),\; \backslash quad\; (5)$$

where $|\cdot|2 = |\cdot|{L^2(\Omega)}$ .

Throughout this article, we use $c$ to denote a generic positive constant.

Remark 1.1. Hypothesis (H3) implies that $\tau h(\tau) > 0$ , $\forall \tau > 0$ .

Remark 1.2. *If $B$ is a strictly convex $C^2$ function and strictly increasing on $(0, r]$ for some $r > 0$ , then we can extend $B$ to $\bar{B}$ . Moreover, $\bar{B}$ is also a strictly convex $C^2$ function and Strictly increasing on $(0, \infty)$ (see [16]). Similarly, we denote the extension of $H$ to be $\bar{H}$ .*Plate problems have been broadly explored by mathematicians and other scientists. This type of problems have a lot of applications in different areas of science and engineering such as material engineering, mechanical engineering, nuclear physics and optics.

Let us discuss some work related to the plate problems. In [1], the authors treated the following problem

where $h$ is a function having a polynomial growth near the origin, and they have established few energy decay results. Decay results for arbitrary growth of the damping term have been considered and studied for the first time in the work of Lasiecka and Tataru (see [9]). They have found that the energy decays as fast as the solution of an associated differential equation whose coefficients depend on the damping term. In [27], Liu considered the following problem

and they have proved a general decay result that depends on the behavior of $g, \alpha$ and $h$ without imposing any restrictive growth assumption on the damping term at origin. For more results in the direction of the plate problems, see [5, 8, 11, 10, 12, 23, 21, 26] and the references there in.

Now, let us review some work with a logarithmic term that are related to the problem (1). Cazenave and Haraux [22] studied the following problem:

$$u_{tt}-\mathrm{\Delta }u=u\ln \mathrm{\mid }u{\mathrm{\mid }}^k,\text{in }{\mathbb{R}}^3(6)u\_\{tt\}\; -\; \backslash Delta\; u\; =\; u\; \backslash ln\; |u|^k,\; \backslash quad\; \backslash text\{in\; \}\; \backslash mathbb\{R\}^3\; \backslash quad\; (6)$$

and established the existence and uniqueness of the solution for the Cauchy problem. Gorka [19] obtained the global existence of weak solutions in the one-dimensional case by using compactness arguments, for all $(u_0, u_1) \in H_0^1([a, b]) \times L^2([a, b])$ , to the initial-boundary value problem (6). The authors in [13] considered the one dimensional Cauchy problem for equation (6) and they have proved the existence of classical solutions and also they have investigated the weak solutions. Birula and Micielski [6, 7] considered

which is a relativistic version of logarithmic quantum mechanics. Moreover, it can also be obtained for the p-adic string equation by taking the limit as $p \rightarrow 1$ (see [20, 25]). Mohammad M. Al-Gharabli [17] considered equation (1) with out damping term and they have established the existence of solution and proved the decay rates and stability result. Mohammad M. Al-Gharabli et.al. (in [3]) have considered the viscoelastic problem with variable exponent and logarithmic nonlinearities:

$$u_{tt}-\mathrm{\Delta }u+u+{\int }_0^{t}b(t-s)\mathrm{\Delta }u(s)ds+\mathrm{\mid }{u}_t{\mathrm{\mid }}^{\gamma (\cdot )-2}{u}_t=u\ln \mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u\_\{tt\}\; -\; \backslash Delta\; u\; +\; u\; +\; \backslash int\_0^t\; b(t-s)\backslash Delta\; u(s)ds\; +\; |u\_t|^\{\backslash gamma(\backslash cdot)-2\}\; u\_t\; =\; u\; \backslash ln\; |u|^\backslash alpha$$

and they have established a global existence result using the well-depth method and then they have also established explicit and general decay results under a wide class of relaxation functions. Gongwei Liu ([15]) considered the differential equation

$$u_{tt}+{\mathrm{\Delta }}^{2}u+\mathrm{\mid }{u}_t{\mathrm{\mid }}^{m-2}{u}_t=u\mathrm{\mid }u{\mathrm{\mid }}^{p-2}\log \mathrm{\mid }u{\mathrm{\mid }}^k(x,t)\in \mathrm{\Omega }\times {\mathbb{R}}^{+}u\_\{tt\}\; +\; \backslash Delta^2\; u\; +\; |u\_t|^\{m-2\}\; u\_t\; =\; u\; |u|^\{p-2\}\; \backslash log\; |u|^k\; \backslash quad\; (x,\; t)\; \backslash in\; \backslash Omega\; \backslash times\; \backslash mathbb\{R\}^+$$

with the boundary conditions given in (1). They have established the local existence result by the fixed point techniques. The global existence and decay estimate of the solution at sub-critical initial energy is obtained, and they additionally prove that the solution with negative initial energy blows up in finite time under some suitable conditions. Moreover, they find out the blow-up in finite time of solution at the arbitrarily high initial energy for linear damping (i.e., $m = 2$ ). In [4], Adel M. Al-Mahdi considered viscoelastic plate equation with infinite memory and logarithmic nonlinearity:

$$\{\begin{array}{l}\mathrm{\mid }{u}_t{\mathrm{\mid }}^{\rho }{u}_{tt}+{\mathrm{\Delta }}^{2}u+{\mathrm{\Delta }}^2{u}_{tt}+u-{\int }_0^{\mathrm{\infty }}b(s){\mathrm{\Delta }}^{2}u(t-s)ds=\alpha u\ln \mathrm{\mid }u\mathrm{\mid },\\ u(x,t)=\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }(x,t)=0,\text{in }\mathrm{\partial }\mathrm{\Omega }\times (0,\mathrm{\infty }),\\ u(x,-t)=u_0(x),u_t(x,0)=u_1(x)\text{in }\mathrm{\Omega }\mathrm{.}\end{array}\text{in }\mathrm{\Omega }\times (0,\mathrm{\infty }),\backslash begin\{cases\}\; |u\_t|^\backslash rho\; u\_\{tt\}\; +\; \backslash Delta^2\; u\; +\; \backslash Delta^2\; u\_\{tt\}\; +\; u\; -\; \backslash int\_0^\backslash infty\; b(s)\backslash Delta^2\; u(t-s)ds\; =\; \backslash alpha\; u\; \backslash ln\; |u|,\; \backslash \backslash \; u(x,\; t)\; =\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash nu\}(x,\; t)\; =\; 0,\; \backslash quad\; \backslash text\{in\; \}\; \backslash partial\backslash Omega\; \backslash times\; (0,\; \backslash infty),\; \backslash \backslash \; u(x,\; -t)\; =\; u\_0(x),\; \backslash quad\; u\_t(x,\; 0)\; =\; u\_1(x)\; \backslash quad\; \backslash text\{in\; \}\; \backslash Omega.\; \backslash end\{cases\}\; \backslash quad\; \backslash text\{in\; \}\; \backslash Omega\; \backslash times\; (0,\; \backslash infty),$$

By imposing minimal conditions on the relaxation function the authors in [4] established an explicit and general decay rate results. See [2, 14] (and the references there in) for more results in this direction.

In this article, we are engaged with the global existence and stability of the plate problem (1) with kernels $b$ having an arbitrary growth at infinity. This article organized as follows. In Section 2, we establish the local existence of the solutions to the problem (1). The global existence is proved in Section 3. Finally, in the last section we derive few stability results.

2 Local existence

In this section, we state and prove the local existence result for the problem (1). The energy associated with problem (1) is$$\begin{aligned} E(t) &= \frac{1}{\rho+2} |u_t|_{\rho+2}^{\rho+2} + \frac{1}{2} \left[ (1 - \int_0^t b(s) ds) |\Delta u|_2^2 + |\Delta u_t|2^2 \right. \ &\quad \left. - k \int{\Omega} u^2 \ln |u| dx + |u|_2^2 + b \circ \Delta u \right] + \frac{k}{4} |u|_2^2. \end{aligned} \tag{7}$$

where the product $\circ$ is defined by

$$(b\circ \mathrm{\Delta }u)(t)={\int }_0^{t}b(t-s)\mathrm{\parallel }\mathrm{\Delta }u(s)-\mathrm{\Delta }u(t){\mathrm{\parallel }}_2^{2}ds\mathrm{.}(b\; \backslash circ\; \backslash Delta\; u)(t)\; =\; \backslash int\_0^t\; b(t-s)\; \backslash |\backslash Delta\; u(s)\; -\; \backslash Delta\; u(t)\backslash |\_2^2\; ds.$$

Direct differentiation of (7) with respect to $t$ and using (1) we observe that

Lemma 2.1. (Logarithmic Sobolev inequality) Let $u \in H_0^1(\Omega)$ and $a > 0$ be any number. Then

$$\begin{array}{cccc}& {\int }_{\mathrm{\Omega }}{u}^2\ln \mathrm{\mid }u\mathrm{\mid }dx\le \frac{1}{2}\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\ln \mathrm{\parallel }u{\mathrm{\parallel }}_2^2+\frac{a^2}{2\pi }\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2-(1+\ln a)\mathrm{\parallel }u{\mathrm{\parallel }}_2^2& & \text{(9)}\end{array}\backslash int\_\{\backslash Omega\}\; u^2\; \backslash ln\; |u|\; dx\; \backslash leq\; \backslash frac\{1\}\{2\}\; \backslash |u\backslash |\_2^2\; \backslash ln\; \backslash |u\backslash |\_2^2\; +\; \backslash frac\{a^2\}\{2\backslash pi\}\; \backslash |\backslash nabla\; u\backslash |\_2^2\; -\; (1\; +\; \backslash ln\; a)\; \backslash |u\backslash |\_2^2\; \backslash tag\{9\}$$

Corollary 2.1. Let $u \in H_0^2(\Omega)$ and $a > 0$ be any number. Then

$$\begin{array}{cccc}& {\int }_{\mathrm{\Omega }}{u}^2\ln \mathrm{\mid }u\mathrm{\mid }dx\le \frac{1}{2}\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\ln \mathrm{\parallel }u{\mathrm{\parallel }}_2^2+\frac{c_p{a}^2}{2\pi }\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_2^2-(1+\ln a)\mathrm{\parallel }u{\mathrm{\parallel }}_2^2& & \text{(10)}\end{array}\backslash int\_\{\backslash Omega\}\; u^2\; \backslash ln\; |u|\; dx\; \backslash leq\; \backslash frac\{1\}\{2\}\; \backslash |u\backslash |\_2^2\; \backslash ln\; \backslash |u\backslash |\_2^2\; +\; \backslash frac\{c\_p\; a^2\}\{2\backslash pi\}\; \backslash |\backslash Delta\; u\backslash |\_2^2\; -\; (1\; +\; \backslash ln\; a)\; \backslash |u\backslash |\_2^2\; \backslash tag\{10\}$$

Lemma 2.2. Let $\epsilon_0 \in (0, 1)$ , then there exists $d_{\epsilon_0} > 0$ such that

$$\begin{array}{cccc}& s\mathrm{\mid }\ln s\mathrm{\mid }\le s^2+d_{{ϵ}_0}{s}^{1-{ϵ}_0}\mathrm{\forall }s>0& & \text{(11)}\end{array}s\; |\backslash ln\; s|\; \backslash leq\; s^2\; +\; d\_\{\backslash epsilon\_0\}\; s^\{1-\backslash epsilon\_0\}\; \backslash forall\; s\; >\; 0\; \backslash tag\{11\}$$

Definition 2.1. A function

$$u\in C^1([0,T],H_0^2(\mathrm{\Omega }))u\; \backslash in\; C^1([0,\; T],\; H\_0^2(\backslash Omega))$$

is called a weak solution of (1) on $[0, T]$ if, for any $t \in [0, T]$ and $\forall w \in H_0^2(\Omega)$ , $u$ satisfies

Theorem Assume that the hypothesis $(H1) - (H4)$ hold. Let $(u_0, u_1) \in H_0^2(\Omega) \times H_0^2(\Omega)$ . Then the problem (1) has weak solution on $[0, T]$ .

Proof. To prove the existence of a solution to the problem (1), we use the Faedo-Galerkin approximations. Let $(w_j)_{j=1}^\infty$ be an orthogonal basis of the separable space $H_0^2(\Omega)$ . Let $V_m = \text{span}(w_1, w_2, \dots, w_m)$ and let the projections of the initial data on the finite dimensional subspace $V_m$ be given by

$$u_0^m(x)=\sum _{j=1}^m{\alpha }_j{w}_j(x),u_1^m(x)=\sum _{j=1}^m{\beta }_j{w}_j(x)\mathrm{.}(13)u\_0^m(x)\; =\; \backslash sum\_\{j=1\}^m\; \backslash alpha\_j\; w\_j(x),\; \backslash quad\; u\_1^m(x)\; =\; \backslash sum\_\{j=1\}^m\; \backslash beta\_j\; w\_j(x).\; \backslash quad\; (13)$$

We search for an approximation solution

$$u_0^m(x)=\sum _{j=1}^m{g}_j^m(t)w_j(x),(14)u\_0^m(x)\; =\; \backslash sum\_\{j=1\}^m\; g\_j^m(t)\; w\_j(x),\; \backslash quad\; (14)$$

of the approximate problem in $V_m$ :

$$\{\begin{array}{l}{\int }_{\mathrm{\Omega }}[\mathrm{\mid }{u}_t^m{\mathrm{\mid }}^{\rho }{u}_{tt}^{m}w+\mathrm{\Delta }{u}^m\mathrm{\Delta }w+\mathrm{\Delta }{u}_{tt}^m\mathrm{\Delta }w+u^{m}w+h(u_t^m)w\\ -{\int }_0^{t}b(t-s)\mathrm{\Delta }{u}^m(s)\mathrm{\Delta }wds]dx=k{\int }_{\mathrm{\Omega }}w{u}^m\ln \mathrm{\mid }{u}^m\mathrm{\mid }dx,\mathrm{\forall }w\in V_m,\\ u^m(0):=u_0^m={\sum }_{j=1}^m(u_0,w_j)w_j,\\ u_t^m(0):=u_1^m={\sum }_{j=1}^m(u_1,w_j)w_j\mathrm{.}\end{array}(15)\backslash left\backslash \{\; \backslash begin\{array\}\{l\}\; \backslash int\_\{\backslash Omega\}\; \backslash left[\; |u\_t^m|^\backslash rho\; u\_\{tt\}^m\; w\; +\; \backslash Delta\; u^m\; \backslash Delta\; w\; +\; \backslash Delta\; u\_\{tt\}^m\; \backslash Delta\; w\; +\; u^m\; w\; +\; h(u\_t^m)\; w\; \backslash right.\; \backslash \backslash \; \backslash quad\; \backslash left.\; -\; \backslash int\_0^t\; b(t-s)\; \backslash Delta\; u^m(s)\; \backslash Delta\; w\; ds\; \backslash right]\; dx\; =\; k\; \backslash int\_\{\backslash Omega\}\; w\; u^m\; \backslash ln\; |u^m|\; dx,\; \backslash quad\; \backslash forall\; w\; \backslash in\; V\_m,\; \backslash \backslash \; u^m(0)\; :=\; u\_0^m\; =\; \backslash sum\_\{j=1\}^m\; (u\_0,\; w\_j)\; w\_j,\; \backslash \backslash \; u\_t^m(0)\; :=\; u\_1^m\; =\; \backslash sum\_\{j=1\}^m\; (u\_1,\; w\_j)\; w\_j.\; \backslash end\{array\}\; \backslash right.\; \backslash quad\; (15)$$

This gives a system of ordinary differential equation (ODE's) for the unknown functions $g_j^m(t)$ . Using the standard existence theory for ODE's, one can obtain functions

$$g_j:[0,t_m)\to \mathbb{R},j=1,2,\dots ,m,g\_j\; :\; [0,\; t\_m)\; \backslash rightarrow\; \backslash mathbb\{R\},\; \backslash quad\; j\; =\; 1,\; 2,\; \backslash dots,\; m,$$

which satisfy (15) in a maximal interval $[0, t_m)$ , $t_m \in (0, T]$ . Later, we show that $t_m = T$ and the local solution is uniformly bounded which is independent of $m$ and $t$ . To do this, substitute $w = u_t^m$ in (15) and using integration by parts to obtain

$$\frac{d}{dt}{E}^m(t)\le \frac{1}{2}(b^{\mathrm{\prime }}\circ \mathrm{\Delta }{u}^m)-{\int }_{\mathrm{\Omega }}{u}_t^{m}h(u_t^m)dx\le 0,(16)\backslash frac\{d\}\{dt\}\; E^m(t)\; \backslash leq\; \backslash frac\{1\}\{2\}\; (b\text{'}\; \backslash circ\; \backslash Delta\; u^m)\; -\; \backslash int\_\{\backslash Omega\}\; u\_t^m\; h(u\_t^m)\; dx\; \backslash leq\; 0,\; \backslash quad\; (16)$$

where

from (16), we have

$$E^m(t)\le E^m(0),\mathrm{\forall }t\ge 0.E^m(t)\; \backslash leq\; E^m(0),\; \backslash forall\; t\; \backslash geq\; 0.$$

the logarithmic Sobolev inequality together with last inequality, we observe that

Choose $a$ such that

then $a$ satisfies

$$l-\frac{k{a}^2{c}_p}{2\pi }>0,(20)l\; -\; \backslash frac\{ka^2c\_p\}\{2\backslash pi\}\; >\; 0,\; \backslash quad\; (20)$$

and

$$\frac{k+2}{2}+k(1+\ln a)>0.(21)\backslash frac\{k+2\}\{2\}\; +\; k(1\; +\; \backslash ln\; a)\; >\; 0.\; \backslash quad\; (21)$$

So, we obtain

And we know that

$$u^m(\mathrm{.},t)=u^m(\mathrm{.},0)+{\int }_0^t\frac{\mathrm{\partial }{u}^m}{\mathrm{\partial }s}(\mathrm{.},s)ds\mathrm{.}u^m(.,\; t)\; =\; u^m(.,\; 0)\; +\; \backslash int\_0^t\; \backslash frac\{\backslash partial\; u^m\}\{\backslash partial\; s\}(.,\; s)\; ds.$$

Using Cauchy Schwarz inequality, observe that

therefore inequality (22) gives

$$\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2\le 2\mathrm{\parallel }{u}^m(0){\mathrm{\parallel }}_2^2+2Tc(1+{\int }_0^t\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2\ln \mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^{2}ds),(24)\backslash |u^m\backslash |\_2^2\; \backslash leq\; 2\; \backslash |u^m(0)\backslash |\_2^2\; +\; 2Tc\; \backslash left(\; 1\; +\; \backslash int\_0^t\; \backslash |u^m\backslash |\_2^2\; \backslash ln\; \backslash |u^m\backslash |\_2^2\; ds\; \backslash right),\; \backslash quad\; (24)$$if we substitute $c_1 = \max{2Tc, 2|u(0)|_2^2}$ , then (24) leads to

$$\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2\le 2c_1(1+{\int }_0^t\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2\ln (\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2)ds),\backslash |u^m\backslash |\_2^2\; \backslash leq\; 2c\_1\; \backslash left(\; 1\; +\; \backslash int\_0^t\; \backslash |u^m\backslash |\_2^2\; \backslash ln(\backslash |u^m\backslash |\_2^2)\; ds\; \backslash right),$$

since $c_1 \geq 0$ , we get

$$\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2\le 2c_1(1+{\int }_0^t(c_1+\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2)\ln (c_1+\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2)ds)\mathrm{.}(25)\backslash |u^m\backslash |\_2^2\; \backslash leq\; 2c\_1\; \backslash left(\; 1\; +\; \backslash int\_0^t\; (c\_1\; +\; \backslash |u^m\backslash |\_2^2)\; \backslash ln(c\_1\; +\; \backslash |u^m\backslash |\_2^2)\; ds\; \backslash right).\; \backslash quad\; (25)$$

When Logarithmic Gronwall inequality applied to (25), we get the following estimate:

$$\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2\le 2c_1{e}^{2c_{1}T}=c_2\mathrm{.}\backslash |u^m\backslash |\_2^2\; \backslash leq\; 2c\_1\; e^\{2c\_1\; T\}\; =\; c\_2.$$

Hence, from inequality (22) it follows that

$$(b\circ \mathrm{\Delta }{u}^m)(t)+\mathrm{\parallel }{u}_t^m{\mathrm{\parallel }}_{\rho +2}^{\rho +2}+\mathrm{\parallel }\mathrm{\Delta }{u}_t^m{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\Delta }{u}^m{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2\le c(1+c_2\ln c_2)\le c_3\mathrm{.}(b\; \backslash circ\; \backslash Delta\; u^m)(t)\; +\; \backslash |u\_t^m\backslash |\_\{\backslash rho+2\}^\{\backslash rho+2\}\; +\; \backslash |\backslash Delta\; u\_t^m\backslash |\_2^2\; +\; \backslash |\backslash Delta\; u^m\backslash |\_2^2\; +\; \backslash |u^m\backslash |\_2^2\; \backslash leq\; c(1\; +\; c\_2\; \backslash ln\; c\_2)\; \backslash leq\; c\_3.$$

This implies

$$\underset{t\in (0,t_m)}{\sup }[(g\circ \mathrm{\Delta }{u}^m)(t)+\mathrm{\parallel }{u}_t^m{\mathrm{\parallel }}_{\rho +2}^{\rho +2}+\mathrm{\parallel }\mathrm{\Delta }{u}_t^m{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\Delta }{u}^m{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{u}^m{\mathrm{\parallel }}_2^2]\le c_3\mathrm{.}(26)\backslash sup\_\{t\; \backslash in\; (0,\; t\_m)\}\; \backslash left[\; (g\; \backslash circ\; \backslash Delta\; u^m)(t)\; +\; \backslash |u\_t^m\backslash |\_\{\backslash rho+2\}^\{\backslash rho+2\}\; +\; \backslash |\backslash Delta\; u\_t^m\backslash |\_2^2\; +\; \backslash |\backslash Delta\; u^m\backslash |\_2^2\; +\; \backslash |u^m\backslash |\_2^2\; \backslash right]\; \backslash leq\; c\_3.\; \backslash quad\; (26)$$

So, we have

$$\{\begin{array}{l}{u}^m\text{ is uniformly bounded in }{L}^{\mathrm{\infty }}(0,T;H_0^2(\mathrm{\Omega })),\\ u_t^m\text{ is uniformly bounded in }{L}^{\mathrm{\infty }}(0,T;L^{\rho +2}(\mathrm{\Omega }))\cap L^{\mathrm{\infty }}(0,T;H_0^2(\mathrm{\Omega })),\end{array}(27)\backslash begin\{cases\}\; u^m\; \backslash text\{\; is\; uniformly\; bounded\; in\; \}\; L^\backslash infty(0,\; T;\; H\_0^2(\backslash Omega)),\; \backslash \backslash \; u\_t^m\; \backslash text\{\; is\; uniformly\; bounded\; in\; \}\; L^\backslash infty(0,\; T;\; L^\{\backslash rho+2\}(\backslash Omega))\; \backslash cap\; L^\backslash infty(0,\; T;\; H\_0^2(\backslash Omega)),\; \backslash end\{cases\}\; \backslash quad\; (27)$$

therefore, these satisfies a subsequence of $(u_m)$ , such that

$$\{\begin{array}{l}{u}^m\stackrel{\ast }{\to }u\text{ in }{L}^{\mathrm{\infty }}(0,T;H_0^2(\mathrm{\Omega })),\\ u_t^m\rightharpoonup u_t\text{ in }{L}^{\mathrm{\infty }}(0,T;L^{\rho +2}(\mathrm{\Omega }))\cap L^{\mathrm{\infty }}(0,T;H_0^2(\mathrm{\Omega })),\\ u^m\rightharpoonup u_t\text{ in }{L}^2(0,T;H_0^2(\mathrm{\Omega })),\\ u^m\rightharpoonup u_t\text{ in }{L}^2(0,T;L^{\rho +2}(\mathrm{\Omega }))\cap L^2(0,T;H_0^2(\mathrm{\Omega })),\end{array}(28)\backslash begin\{cases\}\; u^m\; \backslash xrightarrow\{*\}\; u\; \backslash text\{\; in\; \}\; L^\backslash infty(0,\; T;\; H\_0^2(\backslash Omega)),\; \backslash \backslash \; u\_t^m\; \backslash rightharpoonup\; u\_t\; \backslash text\{\; in\; \}\; L^\backslash infty(0,\; T;\; L^\{\backslash rho+2\}(\backslash Omega))\; \backslash cap\; L^\backslash infty(0,\; T;\; H\_0^2(\backslash Omega)),\; \backslash \backslash \; u^m\; \backslash rightharpoonup\; u\_t\; \backslash text\{\; in\; \}\; L^2(0,\; T;\; H\_0^2(\backslash Omega)),\; \backslash \backslash \; u^m\; \backslash rightharpoonup\; u\_t\; \backslash text\{\; in\; \}\; L^2(0,\; T;\; L^\{\backslash rho+2\}(\backslash Omega))\; \backslash cap\; L^2(0,\; T;\; H\_0^2(\backslash Omega)),\; \backslash end\{cases\}\; \backslash quad\; (28)$$

where $\xrightarrow{}$ represent the weak $$ convergence and $\rightharpoonup$ represent weak convergence. Therefore, the approximate solution is uniformly bounded and it is independent of $m$ and $t$ . Therefore we can extend $t_m$ to $T$ .

Next we prove that $u_{tt}^m$ is bounded in $L^2(0, T; H_0^2(\Omega))$ . To do this, we substitute $w = u_{tt}^m$ in (12). Using (1), we see that

By using the Cauchy-Schwarz' inequality, Young's inequality, and the embedding inequality we obtain,

Using Lemma 2.2 with $\epsilon_0 = \frac{1}{2}$ , the second term in the right hand side of (30) is estimated as follows:

from (30) and (31) we have

Now we prove that $h(u_t^m)$ is bounded in $L^2(0, T; L^2(\Omega))$ . For this purpose, we consider two cases:

Case 1. When $h_1$ is linear, we can directly prove that

$${\int }_0^t{\int }_{\mathrm{\Omega }}{h}^2(u_t^m)dxdt\le C\backslash int\_0^t\; \backslash int\_\{\backslash Omega\}\; h^2(u\_t^m)\; dx\; dt\; \backslash leq\; C$$Case 2. When $h_1$ is nonlinear,
Consider $|s| \leq \epsilon$ , then

$$H(h^2(s))=\mathrm{\mid }h(s)\mathrm{\mid }{h}_1(\mathrm{\mid }h(s)\mathrm{\mid })\le sh(s),H(h^2(s))\; =\; |h(s)|\; h\_1(|h(s)|)\; \backslash leq\; sh(s),$$

this implies that

$$H^{-1}(sh(s))\ge h^2(s),\mathrm{\forall }\mathrm{\mid }s\mathrm{\mid }\le \delta \mathrm{.}H^\{-1\}(sh(s))\; \backslash geq\; h^2(s),\; \backslash quad\; \backslash forall\; |s|\; \backslash leq\; \backslash delta.$$

then using the similar lines from Theorem 3.2 in [18], we get $\int_0^t \int_{\Omega} h^2(u_t^m) dx dt \leq C_T$ . we conclude that $h(u_t^m)$ is bounded in $L^2(0, T; L^2(\Omega))$ .

Integrating (32) from $(0, T)$ , using the hypothesis (H1) and (26), we obtain

From (33) and using the fact that $h(u_t^m)$ is bounded in $L^2(0, T; L^2(\Omega))$ , it is easy to observe that for $\delta$ small enough,

$$u_{tt}^m\text{ is bounded in }{L}^2(0,T;H_0^2(\mathrm{\Omega }))\mathrm{.}(34)u\_\{tt\}^m\; \backslash text\{\; is\; bounded\; in\; \}\; L^2(0,\; T;\; H\_0^2(\backslash Omega)).\; \backslash quad\; (34)$$

Taking $m \rightarrow \infty$ to (12) and from (28) and (34) (thanks to Aubin-Lions' theorem), for all $w \in H_0^2(\Omega)$ and a.e. $t \in (0, T)$ , we see that

This complete the proof of this theorem. $\square$

3 Global existence

In this section, under smallness condition on the initial data, we state and prove global existence result. For the sake of simplicity, we introduce the following functionals:

and

$$I(t):=I(u(t))=\mathrm{\parallel }\mathrm{\Delta }{u}_t{\mathrm{\parallel }}_2^2+(1-{\int }_0^{t}b(s)ds)\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_2^2+\mathrm{\parallel }u{\mathrm{\parallel }}_2^2+(b\circ \mathrm{\Delta }u)-{\int }_{\mathrm{\Omega }}{u}^2\ln \mathrm{\mid }u{\mathrm{\mid }}^k,(36)I(t)\; :=\; I(u(t))\; =\; \backslash |\backslash Delta\; u\_t\backslash |\_2^2\; +\; (1\; -\; \backslash int\_0^t\; b(s)\; ds)\; \backslash |\backslash Delta\; u\backslash |\_2^2\; +\; \backslash |u\backslash |\_2^2\; +\; (b\; \backslash circ\; \backslash Delta\; u)\; -\; \backslash int\_\{\backslash Omega\}\; u^2\; \backslash ln\; |u|^k,\; \backslash quad\; (36)$$

Note: From (7), (35) and (36), it is clear that

$$J(t)=\frac{1}{2}I(t)+\frac{k}{4}\mathrm{\parallel }u{\mathrm{\parallel }}_2^2(37)J(t)\; =\; \backslash frac\{1\}\{2\}\; I(t)\; +\; \backslash frac\{k\}\{4\}\; \backslash |u\backslash |\_2^2\; \backslash quad\; (37)$$

$$E(t)=\frac{1}{\rho +2}\mathrm{\parallel }{u}_t{\mathrm{\parallel }}_{\rho +2}^{\rho +2}+J(t),(38)E(t)\; =\; \backslash frac\{1\}\{\backslash rho+2\}\; \backslash |u\_t\backslash |\_\{\backslash rho+2\}^\{\backslash rho+2\}\; +\; J(t),\; \backslash quad\; (38)$$

Notation: Define $\bar{\rho} = e^{\frac{2Q_0-k}{k}}$ , $d = \frac{1}{2} Q_0 \bar{\rho}^2 - \frac{k}{4} \bar{\rho}^2 \ln \bar{\rho}^2$ and

Lemma 3.1. Let $(u_0, u_1) \in H_0^2(\Omega) \times H_0^2(\Omega)$ and assume that the hypothesis (H1) holds. Further assume that $|u_0| < \bar{\rho}$ and $0 < E(0) < d$ . Then $I(u) > 0 \ \forall t \in [0, T)$ .

Proof. We divide the proof for this lemma into two steps. In step (1), we prove that $|u|_2 < \bar{\rho} \ \forall t \in [0, T)$ and in step (2), we prove $I(t) > 0 \ t \in [0, T)$ .

Step 1. From (7) and (35), and using Logarithmic Sobolev inequality it is easy to see that

Using (20), we obtain

$$E(t)\ge Q(\tilde{\rho })=\frac{1}{2}{Q}_0{\tilde{\rho }}^2-\frac{k}{4}{\tilde{\rho }}^2\ln {\tilde{\rho }}^2(39)E(t)\; \backslash geq\; Q(\backslash tilde\{\backslash rho\})\; =\; \backslash frac\{1\}\{2\}\; Q\_0\; \backslash tilde\{\backslash rho\}^2\; -\; \backslash frac\{k\}\{4\}\; \backslash tilde\{\backslash rho\}^2\; \backslash ln\; \backslash tilde\{\backslash rho\}^2\; \backslash quad\; (39)$$

where $\tilde{\rho} = |u|_2$ . Observe that (39) implies $Q$ is increasing on $(0, \bar{\rho})$ and decreasing on $(\bar{\rho}, \infty)$ . Also, note that $Q(\tilde{\rho}) \rightarrow -\infty$ as $\tilde{\rho} \rightarrow +\infty$ . Let

Suppose that $|u| < \bar{\rho}$ does not hold in $[0, T)$ , then there exist $t_0 \in (0, T)$ and $|u(x, t_0)| = \bar{\rho}$ . Using (39), we get $E(t_0) \geq Q(|u(x, t_0)|_2) = Q(\bar{\rho}) = d$ , which is contradiction to the fact $E(t) \leq E(0) < d \ \forall t > 0$ . Hence $|u|_2 < \bar{\rho} \ \forall t \in [0, T)$ . Hence, $|u| < \bar{\rho}$ for all $t \in [0, T)$ .

Step 2. Using the definition of $I(t)$ and (20), we notice that for $t \in [0, T)$ ,

This complete the proof of this lemma. $\square$

Remark 3.1. Under the assumptions of Lemma (3.1), for $t \in [0, T)$ , we have

$$\mathrm{\parallel }{u}_t{\mathrm{\parallel }}_{\rho +2}^{\rho +2}\le (\rho +2)E(t)\le (\rho +2)E(0),\backslash |u\_t\backslash |\_\{\backslash rho+2\}^\{\backslash rho+2\}\; \backslash leq\; (\backslash rho\; +\; 2)E(t)\; \backslash leq\; (\backslash rho\; +\; 2)E(0),$$

and

$$\mathrm{\parallel }\mathrm{\Delta }{u}_t{\mathrm{\parallel }}^2\le 2E(t)\le 2E(0)\mathrm{.}\backslash |\backslash Delta\; u\_t\backslash |^2\; \backslash leq\; 2E(t)\; \backslash leq\; 2E(0).$$

This shows that the solution is global and bounded in time (in the above mentioned norm).

4 Stability

In this section, we state and prove the decay of the solutions of the problem (1). At first we establish some lemmas which are useful to prove our main results.

Lemma 4.1. Assume that $b$ satisfies the hypothesis (H1). Then, for $u \in H_0^2(\Omega)$ , we have

$${\int }_{\mathrm{\Omega }}{({\int }_0^{t}b(t-s)(u(t)-u(s))ds)}^{2}dx\le c(b\circ \mathrm{\Delta }u)(t),\backslash int\_\{\backslash Omega\}\; \backslash left(\; \backslash int\_0^t\; b(t-s)(u(t)\; -\; u(s))ds\; \backslash right)^2\; dx\; \backslash leq\; c(b\; \backslash circ\; \backslash Delta\; u)(t),$$

and

$${\int }_{\mathrm{\Omega }}{({\int }_0^t{b}^{\mathrm{\prime }}(t-s)(u(t)-u(s))ds)}^{2}dx\le -c(b^{\mathrm{\prime }}\circ \mathrm{\Delta }u)(t),\backslash int\_\{\backslash Omega\}\; \backslash left(\; \backslash int\_0^t\; b\text{'}(t-s)(u(t)\; -\; u(s))ds\; \backslash right)^2\; dx\; \backslash leq\; -c(b\text{'}\; \backslash circ\; \backslash Delta\; u)(t),$$

for some $c > 0$ .

For the proof of Lemma 4.1, refer to [17].

Lemma 4.2. Assume that $h$ satisfies the hypothesis (H3). Then, the solution of (1) satisfies the following estimates:

$${\int }_{\mathrm{\Omega }}{h}^2(u_t)dx\le c{\int }_{\mathrm{\Omega }}{u}_{t}h(u_t)dx\le -c{E}^{\mathrm{\prime }}(t),\text{if }{h}_1\text{ is linear}(40)\backslash int\_\{\backslash Omega\}\; h^2(u\_t)dx\; \backslash leq\; c\; \backslash int\_\{\backslash Omega\}\; u\_t\; h(u\_t)dx\; \backslash leq\; -cE\text{'}(t),\; \backslash quad\; \backslash text\{if\; \}\; h\_1\; \backslash text\{\; is\; linear\}\; \backslash quad\; (40)$$

$${\int }_{\mathrm{\Omega }}{h}^2(u_t)dx\le c{H}^{-1}(G(t))-c{E}^{\mathrm{\prime }}(t),\text{if }{h}_1\text{ is nonlinear}(41)\backslash int\_\{\backslash Omega\}\; h^2(u\_t)dx\; \backslash leq\; cH^\{-1\}(G(t))\; -\; cE\text{'}(t),\; \backslash quad\; \backslash text\{if\; \}\; h\_1\; \backslash text\{\; is\; nonlinear\}\; \backslash quad\; (41)$$

where $G(t) := \frac{1}{|\Omega_1|} \int_{\Omega_1} u_t h(u_t)dx \leq -cE'(t)$ and $\Omega_1 = {x \in \Omega : |u_t| \leq \epsilon}$ for some $c, \epsilon > 0$ .

Lemma 4.3. Assume that $b$ satisfies the hypothesis (H1) and (H2). Then the solution of (1) satisfies the following estimate:

$${\int }_{t_1}^{t}b(s){\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\Delta }u(t)-\mathrm{\Delta }u(t-s){\mathrm{\mid }}^{2}dxds\le \frac{t-t_1}{\delta }{\bar{B}}^{-1}\left(\frac{\delta M(t)}{(t-t_1)\xi (t)}\right),(42)\backslash int\_\{t\_1\}^t\; b(s)\; \backslash int\_\{\backslash Omega\}\; |\backslash Delta\; u(t)\; -\; \backslash Delta\; u(t-s)|^2\; dx\; ds\; \backslash leq\; \backslash frac\{t-t\_1\}\{\backslash delta\}\; \backslash bar\{B\}^\{-1\}\; \backslash left(\; \backslash frac\{\backslash delta\; M(t)\}\{(t-t\_1)\backslash xi(t)\}\; \backslash right),\; \backslash quad\; (42)$$where $\delta \in (0, 1)$ , $\bar{B}$ is an extension of $B$ and

$$M(t):=-{\int }_{t_1}^t{b}^{\mathrm{\prime }}(s){\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\Delta }u(t)-\mathrm{\Delta }u(t-s){\mathrm{\mid }}^{2}dxds\le -c{E}^{\mathrm{\prime }}(t)\mathrm{.}(43)M(t)\; :=\; -\; \backslash int\_\{t\_1\}^t\; b\text{'}(s)\; \backslash int\_\{\backslash Omega\}\; |\backslash Delta\; u(t)\; -\; \backslash Delta\; u(t-s)|^2\; dx\; ds\; \backslash leq\; -cE\text{'}(t).\; \backslash quad\; (43)$$

The proofs of Lemma's 4.2 and 4.3 follows from the similar lines as in [16]. So, we skip the proof's.

Lemma 4.4. Under the hypothesis (H1) – (H4), the functional $\Psi_1(t)$

$${\mathrm{\Psi }}_1(t):=\frac{1}{\rho +1}{\int }_{\mathrm{\Omega }}\mathrm{\mid }{u}_t{\mathrm{\mid }}^{\rho }{u}_{t}udx+{\int }_{\mathrm{\Omega }}\mathrm{\Delta }u\mathrm{\Delta }{u}_{t}dx,\backslash Psi\_1(t)\; :=\; \backslash frac\{1\}\{\backslash rho+1\}\; \backslash int\_\{\backslash Omega\}\; |u\_t|^\backslash rho\; u\_t\; u\; dx\; +\; \backslash int\_\{\backslash Omega\}\; \backslash Delta\; u\; \backslash Delta\; u\_t\; dx,$$

satisfies the estimate:

Proof. Differentiating $\Psi_1$ with respect to $t$ and using (1), we get

At first, we estimate the sixth term in the right hand side of (45). Using the hypothesis (H1) and Lemma 4.1, we have

The last line in the above inequality is obtained by choosing $\eta = l$ . Now, we estimate the last term in (45).

Now choose $\delta = \frac{l}{4c}$ to get (44). Hence proved. $\square$Lemma 4.5. Under the hypothesis (H1) – (H4), the functional $\Psi_2(t)$

$${\mathrm{\Psi }}_2(t):=-{\int }_{\mathrm{\Omega }}({\mathrm{\Delta }}^2{u}_t+\frac{1}{\rho +1}\mathrm{\mid }{u}_t{\mathrm{\mid }}^{\rho }{u}_t){\int }_0^{t}b(t-s)(u(t)-u(s))dsdx,\backslash Psi\_2(t)\; :=\; -\; \backslash int\_\{\backslash Omega\}\; \backslash left(\; \backslash Delta^2\; u\_t\; +\; \backslash frac\{1\}\{\backslash rho+1\}\; |u\_t|^\backslash rho\; u\_t\; \backslash right)\; \backslash int\_0^t\; b(t-s)(u(t)\; -\; u(s))\; ds\; dx,$$

satisfies the estimate:

for some $\epsilon_0 \in (0, 1)$ .

Proof. Differentiating $\Psi_2$ with respect to $t$ and using (1), we get

In order to estimate all the nine terms in the right hand side of (47), we will use Young's inequality, Cauchy-Schwarz' inequality and Lemma 4.1. All the terms except for the third and fifth term, are estimated in the similar lines as in [18]. For the sake of completeness, we will write the final estimates.$$\begin{aligned} \text{(I)} \quad & - \int_{\Omega} u \ln |u|^k \left( \int_0^t b(t-s)(u(t) - u(s)) ds \right) dx \ & \leq \frac{\delta}{4} |\Delta u|_2^2 + \frac{c}{\delta} (b \circ \Delta u)(t) + c(b \circ \Delta u) \frac{1}{1 + \epsilon_0(t)}, \end{aligned}$$

$$\text{(III)}{\int }_{\mathrm{\Omega }}u({\int }_0^{t}b(t-s)(u(t)-u(s))ds)dx\le \frac{\delta }{4}\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_2^2+\frac{c}{\delta }(b\circ \mathrm{\Delta }u)(t),\backslash text\{(III)\}\; \backslash quad\; \backslash int\_\{\backslash Omega\}\; u\; \backslash left(\; \backslash int\_0^t\; b(t-s)(u(t)\; -\; u(s))\; ds\; \backslash right)\; dx\; \backslash leq\; \backslash frac\{\backslash delta\}\{4\}\; \backslash |\backslash Delta\; u\backslash |\_2^2\; +\; \backslash frac\{c\}\{\backslash delta\}\; (b\; \backslash circ\; \backslash Delta\; u)(t),$$

$$\text{(VII)}-{\int }_{\mathrm{\Omega }}\frac{1}{\rho +1}\mathrm{\mid }{u}_t{\mathrm{\mid }}^{\rho }{u}_t^2({\int }_0^{t}b(s)ds)dx=-\frac{1}{\rho +1}({\int }_0^{t}b(s)ds)\mathrm{\parallel }{u}_t{\mathrm{\parallel }}_{\rho +2}^{\rho +2},\backslash text\{(VII)\}\; \backslash quad\; -\; \backslash int\_\{\backslash Omega\}\; \backslash frac\{1\}\{\backslash rho+1\}\; |u\_t|^\backslash rho\; u\_t^2\; \backslash left(\; \backslash int\_0^t\; b(s)\; ds\; \backslash right)\; dx\; =\; -\backslash frac\{1\}\{\backslash rho+1\}\; \backslash left(\; \backslash int\_0^t\; b(s)\; ds\; \backslash right)\; \backslash |u\_t\backslash |\_\{\backslash rho+2\}^\{\backslash rho+2\},$$

and

$$\text{(IX)}-({\int }_0^{t}b(s)ds)\left({\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\Delta }{u}_t{\mathrm{\mid }}^{2}dx\right)=-({\int }_0^{t}b(s)ds)\mathrm{\parallel }\mathrm{\Delta }{u}_t{\mathrm{\parallel }}_2^2\mathrm{.}\backslash text\{(IX)\}\; \backslash quad\; -\; \backslash left(\; \backslash int\_0^t\; b(s)\; ds\; \backslash right)\; \backslash left(\; \backslash int\_\{\backslash Omega\}\; |\backslash Delta\; u\_t|^2\; dx\; \backslash right)\; =\; -\; \backslash left(\; \backslash int\_0^t\; b(s)\; ds\; \backslash right)\; \backslash |\backslash Delta\; u\_t\backslash |\_2^2.$$

Combining all the estimates we will arrive at (46). $\square$Lemma 4.6. Assume that the hypothesis of Lemma 3.1 and the hypothesis of (H1) – (H4) holds. Then there exists $N, \epsilon > 0$ such that the functional

$$L(t)=NE(t)+ϵ{\mathrm{\Psi }}_1(t)+{\mathrm{\Psi }}_2(t)L(t)\; =\; NE(t)\; +\; \backslash epsilon\backslash Psi\_1(t)\; +\; \backslash Psi\_2(t)$$

satisfies,

$$L\sim E,(48)L\; \backslash sim\; E,\; \backslash quad\; (48)$$

and for all $t \geq 0$ , there exists $m > 0$ and $\epsilon_0 \in (0, 1)$ such that

$$L^{\mathrm{\prime }}(t)\le -mE(t)+c(b\circ \mathrm{\Delta }u)(t)+c(b\circ \mathrm{\Delta }u{)}^{\frac{1}{1+{ϵ}_0}}(t)+c{\int }_{\mathrm{\Omega }}{h}^2(u_t)dx,(49)L\text{'}(t)\; \backslash leq\; -mE(t)\; +\; c(b\; \backslash circ\; \backslash Delta\; u)(t)\; +\; c(b\; \backslash circ\; \backslash Delta\; u)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}(t)\; +\; c\; \backslash int\_\{\backslash Omega\}\; h^2(u\_t)\; dx,\; \backslash quad\; (49)$$

Proof. In order to prove (48), we use the Sobolev embedding $H_0^1(\Omega) \hookrightarrow L^{\rho+2}(\Omega)$ , $\int_{\Omega} |u_t|^{2(\rho+1)} dx \leq c|\Delta u_t|^2$ and Remark 3.1. From (49), we observe that

Therefore, by choosing $N$ large enough we obtain (48). For the inequality (49), we follow similar lines given in [17]. $\square$

Remark 4.1. Since,

Hence,

$$(b\circ \mathrm{\Delta }u)(t)\le (b\circ \mathrm{\Delta }u{)}^{\frac{{ϵ}_0}{1+{ϵ}_0}}(t)(b\circ \mathrm{\Delta }u{)}^{\frac{1}{1+{ϵ}_0}}(t)\le c(b\circ \mathrm{\Delta }u{)}^{\frac{1}{1+{ϵ}_0}}(t)\mathrm{.}(50)(b\; \backslash circ\; \backslash Delta\; u)(t)\; \backslash leq\; (b\; \backslash circ\; \backslash Delta\; u)^\{\backslash frac\{\backslash epsilon\_0\}\{1+\backslash epsilon\_0\}\}(t)\; (b\; \backslash circ\; \backslash Delta\; u)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}(t)\; \backslash leq\; c(b\; \backslash circ\; \backslash Delta\; u)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}(t).\; \backslash quad\; (50)$$

Remark 4.2. If $b$ is linear then, we have

Theorem 4.1. Let $(u_0, u_1) \in H_0^2(\Omega) \times H_0^2(\Omega)$ . Assume that the hypothesis (H1) – (H4) holds and $h_1$ is linear. Then for all $t \geq t_0$ , we have

$$\text{if B is linear then }E(t)\le c{(1+{\int }_{t_0}^t{\xi }^{1+{ϵ}_0}(s)ds)}^{-\frac{1}{{ϵ}_0}},\mathrm{\forall }t\ge t_0,(52)\backslash text\{if\; B\; is\; linear\; then\; \}\; E(t)\; \backslash leq\; c\; \backslash left(\; 1\; +\; \backslash int\_\{t\_0\}^t\; \backslash xi^\{1+\backslash epsilon\_0\}(s)\; ds\; \backslash right)^\{-\backslash frac\{1\}\{\backslash epsilon\_0\}\},\; \backslash quad\; \backslash forall\; t\; \backslash geq\; t\_0,\; \backslash quad\; (52)$$

and

$$\text{if B is nonlinear then }E(t)\le c(t-t_0{)}^{\frac{1}{1+ϵ}}{\mathcal{K}}_1^{-1}\left(\frac{c}{(t-t_0{)}^{\frac{1}{1+ϵ}}{\int }_{t_1}^t\xi (s)ds}\right),\mathrm{\forall }t\ge t_1,(53)\backslash text\{if\; B\; is\; nonlinear\; then\; \}\; E(t)\; \backslash leq\; c(t-t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; \backslash mathcal\{K\}\_1^\{-1\}\; \backslash left(\; \backslash frac\{c\}\{(t-t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; \backslash int\_\{t\_1\}^t\; \backslash xi(s)\; ds\}\; \backslash right),\; \backslash quad\; \backslash forall\; t\; \backslash geq\; t\_1,\; \backslash quad\; (53)$$

where $\mathcal{K}_1(t) = t \mathcal{K}'(\epsilon_1 t)$ and $\mathcal{K}(t) = \left( (\bar{B}^{-1})^{\frac{1}{1+\epsilon}} \right)^{-1}(t)$ .

Proof. We will divide the proof of this theorem into two cases.

Case 1: $G$ is linear. Using (49),

Denote $L_1(t) = L(t) + cE(t)$ , then the above inequality becomes

$$L_1^{\mathrm{\prime }}(t)\le -mE(t)+c(b\circ \mathrm{\Delta }u)(t)+c(b\circ \mathrm{\Delta }u{)}^{\frac{1}{1+{ϵ}_0}}(t)(54)L\_1\text{'}(t)\; \backslash leq\; -mE(t)\; +\; c(b\; \backslash circ\; \backslash Delta\; u)(t)\; +\; c(b\; \backslash circ\; \backslash Delta\; u)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}(t)\; \backslash quad\; (54)$$

Multiplying $\xi(t)$ to (54) and using Remarks 4.1 and 4.2, and for all $t \geq t_0$ we get

$$\xi (t)L_1^{\mathrm{\prime }}(t)\le -m\xi (t)E(t)+c(-E^{\mathrm{\prime }}(t){)}^{\frac{1}{1+{ϵ}_0}}\mathrm{.}\backslash xi(t)L\_1\text{'}(t)\; \backslash leq\; -m\backslash xi(t)E(t)\; +\; c(-E\text{'}(t))^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}.$$

Multiplying the above inequality by $\xi^{\epsilon_0}(t)E^{\epsilon_0}(t)$ and using the Young's inequality and for any $\epsilon_1 > 0$ and $t \geq t_0$ , we obtain

Denote $L_2(t) = \xi^{1+\epsilon_0}(t)E^{\epsilon_0}(t)L_1(t) + cE(t)$ and choosing $\epsilon_1 < \frac{m}{c}$ and using the properties of $\xi$ and $E$ to get

$$L_2^{\mathrm{\prime }}(t)\le -c{\xi }^{1+{ϵ}_0}(t)E^{1+{ϵ}_0}(t),\mathrm{\forall }t\ge t_0,(55)L\_2\text{'}(t)\; \backslash leq\; -c\backslash xi^\{1+\backslash epsilon\_0\}(t)E^\{1+\backslash epsilon\_0\}(t),\; \backslash quad\; \backslash forall\; t\; \backslash geq\; t\_0,\; \backslash quad\; (55)$$

where $c_1 = m - \epsilon_1 c$ . Since, $L_2 \sim E$ and from (55), it is easy to see that

$$E^{\mathrm{\prime }}(t)\le -c{\xi }^{1+{ϵ}_0}(t)E^{1+{ϵ}_0}(t),\mathrm{\forall }t\ge t_0,E\text{'}(t)\; \backslash leq\; -c\backslash xi^\{1+\backslash epsilon\_0\}(t)E^\{1+\backslash epsilon\_0\}(t),\; \backslash quad\; \backslash forall\; t\; \backslash geq\; t\_0,$$

Integrating the above inequality from $(t_0, t)$ , we obtain (52).Case 2: $G$ is nonlinear. Using (49),

Denote $L_1(t) = L(t) + cE(t)$ , then using Lemma 4.3 and Remark 4.1 the above inequality becomes

the last line in the above inequality follows from the fact that, there exists $t_1 > t_0$ such that $\frac{1}{t-t_0} < 1$ when ever $t > t_1$ . Hence we have

$${\bar{B}}^{-1}\left(\frac{\delta M(t)}{(t-t_0)\xi (t)}\right)\le {\bar{B}}^{-1}\left(\frac{\delta M(t)}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}\xi (t)}\right)\mathrm{\forall }t>t_1\mathrm{.}\backslash bar\{B\}^\{-1\}\; \backslash left(\; \backslash frac\{\backslash delta\; M(t)\}\{(t\; -\; t\_0)\backslash xi(t)\}\; \backslash right)\; \backslash leq\; \backslash bar\{B\}^\{-1\}\; \backslash left(\; \backslash frac\{\backslash delta\; M(t)\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; \backslash xi(t)\}\; \backslash right)\; \backslash quad\; \backslash forall\; t\; >\; t\_1.$$

Denote

$$\mathcal{K}(t)={\left[{\left({\bar{B}}^{-1}\right)}^{\frac{1}{1+{ϵ}_0}}\right]}^{-1}(t),\alpha (t)=\frac{\delta M(t)}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}\xi (t)}\mathrm{.}(56)\backslash mathcal\{K\}(t)\; =\; \backslash left[\; \backslash left(\; \backslash bar\{B\}^\{-1\}\; \backslash right)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; \backslash right]^\{-1\}\; (t),\; \backslash quad\; \backslash alpha(t)\; =\; \backslash frac\{\backslash delta\; M(t)\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; \backslash xi(t)\}.\; \backslash quad\; (56)$$

From the definition of $\mathcal{K}$ , it is clear that $\mathcal{K}'(t) > 0$ and $\mathcal{K}''(t) > 0$ on $(0, r]$ where $r = \min{r_1, r_2}$ . Using these notations, we obtain that for all $t \geq t_1$ ,

$$L_1^{\mathrm{\prime }}(t)\le -mE(t)+c(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}{\mathcal{K}}^{-1}(\alpha (t))\mathrm{.}(57)L\text{'}\_1(t)\; \backslash leq\; -mE(t)\; +\; c(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; \backslash mathcal\{K\}^\{-1\}(\backslash alpha(t)).\; \backslash quad\; (57)$$

Now, we define the functional $L_2$ as follows

$$L_2(t):={\mathcal{K}}^{\mathrm{\prime }}\left(\frac{{ϵ}_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(t)}{E(0)}\right)L_1(t),L\_2(t)\; :=\; \backslash mathcal\{K\}\text{'}\; \backslash left(\; \backslash frac\{\backslash epsilon\_1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash right)\; L\_1(t),$$

for some $\epsilon_1 \in (0, r)$ . Using the fact that $\mathcal{K}'(t) > 0$ , $\mathcal{K}''(t) > 0$ and $E'(t) \leq 0$ on $(0, r]$ , and using (57) we obtain

and

$$L_2\sim E\mathrm{.}(59)L\_2\; \backslash sim\; E.\; \backslash quad\; (59)$$

Let $\mathcal{K}^*$ denote the convex conjugate of $\mathcal{K}$ in the sense of Young (see [24]), then we have

$${\mathcal{K}}^{\ast }(\tau )=\tau ({\mathcal{K}}^{\mathrm{\prime }}{)}^{-1}(\tau )-\mathcal{K}([{\mathcal{K}}^{\mathrm{\prime }}{]}^{-1}(\tau )),\text{ if }\tau \in (0,{\mathcal{K}}^{\mathrm{\prime }}(r)),\backslash mathcal\{K\}^*(\backslash tau)\; =\; \backslash tau(\backslash mathcal\{K\}\text{'})^\{-1\}(\backslash tau)\; -\; \backslash mathcal\{K\}([\backslash mathcal\{K\}\text{'}]^\{-1\}(\backslash tau)),\; \backslash text\{\; if\; \}\; \backslash tau\; \backslash in\; (0,\; \backslash mathcal\{K\}\text{'}(r)),$$

here $\mathcal{K}^*$ satisfies the generalized Young Inequality:

$$ab\le {\mathcal{K}}^{\ast }(a)+\mathcal{K}(b),\text{ if }a\in (0,{\mathcal{K}}^{\mathrm{\prime }}(r)),b\in (0,r]\mathrm{.}ab\; \backslash leq\; \backslash mathcal\{K\}^*(a)\; +\; \backslash mathcal\{K\}(b),\; \backslash text\{\; if\; \}\; a\; \backslash in\; (0,\; \backslash mathcal\{K\}\text{'}(r)),\; b\; \backslash in\; (0,\; r].$$

So, by assuming $a = \mathcal{K}'\left(\frac{\epsilon_1}{(t-t_0)^{\frac{1}{1+\epsilon_0}}} \frac{E(t)}{E(0)}\right)$ and $b = \mathcal{K}^{-1}(\alpha(t))$ and using (58) we obtain

Multiplying the above inequality by $\xi(t)$ and using (43) and (56) for all $t \geq t_1$ we get

By setting $L_3 := \xi L_2 + cE$ (notice that $L_3 \sim E$ ), we get for all $t \geq t_1$

$$L_3^{\mathrm{\prime }}(t)\le -(mE(0)-c{ϵ}_1)\xi (t)\frac{E(t)}{E(0)}{\mathcal{K}}^{\mathrm{\prime }}\left(\frac{{ϵ}_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(t)}{E(0)}\right)L\text{'}\_3(t)\; \backslash leq\; -(mE(0)\; -\; c\backslash epsilon\_1)\backslash xi(t)\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash mathcal\{K\}\text{'}\backslash left(\backslash frac\{\backslash epsilon\_1\}\{(t-t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(t)\}\{E(0)\}\backslash right)$$

by choosing $\epsilon_1$ such that $mE(0) - c\epsilon_1 > 0$ , we obtain

$$L_3^{\mathrm{\prime }}(t)\le -c\xi (t)\frac{E(t)}{E(0)}{\mathcal{K}}^{\mathrm{\prime }}\left(\frac{{ϵ}_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(t)}{E(0)}\right)(60)L\text{'}\_3(t)\; \backslash leq\; -c\backslash xi(t)\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash mathcal\{K\}\text{'}\backslash left(\backslash frac\{\backslash epsilon\_1\}\{(t-t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(t)\}\{E(0)\}\backslash right)\; \backslash quad\; (60)$$Integrating (60) from $t_1$ to $t$ to get,

$${\int }_{t_1}^{t}c\xi (\tau )\frac{E(\tau )}{E(0)}{\mathcal{K}}^{\mathrm{\prime }}\left(\frac{{ϵ}_1}{(\tau -t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(\tau )}{E(0)}\right)d\tau \le -{\int }_{t_1}^t{L}_3^{\mathrm{\prime }}(t)dt\le L_3(t_1)\backslash int\_\{t\_1\}^t\; c\backslash xi(\backslash tau)\; \backslash frac\{E(\backslash tau)\}\{E(0)\}\; \backslash mathcal\{K\}\text{'}\; \backslash left(\; \backslash frac\{\backslash epsilon\_1\}\{(\backslash tau\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(\backslash tau)\}\{E(0)\}\; \backslash right)\; d\backslash tau\; \backslash leq\; -\; \backslash int\_\{t\_1\}^t\; L\text{'}\_3(t)\; dt\; \backslash leq\; L\_3(t\_1)$$

Using the properties of $\mathcal{K}$ , $E$ and $\forall t \geq t_1$ ,

$$\begin{array}{r}{\displaystyle c\frac{E(t)}{E(0)}{\mathcal{K}}^{\mathrm{\prime }}\left(\frac{{ϵ}_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(t)}{E(0)}\right){\int }_{t_1}^t\xi (\tau )d\tau }\\ {\displaystyle \le c\xi (\tau )\frac{E(\tau )}{E(0)}{\mathcal{K}}^{\mathrm{\prime }}\left(\frac{{ϵ}_1}{(\tau -t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(\tau )}{E(0)}\right)d\tau \le L_3(t_1)\mathrm{.}}\end{array}\backslash begin\{aligned\}\; c\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash mathcal\{K\}\text{'}\; \backslash left(\; \backslash frac\{\backslash epsilon\_1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash right)\; \backslash int\_\{t\_1\}^t\; \backslash xi(\backslash tau)\; d\backslash tau\; \backslash \backslash \; \backslash leq\; c\backslash xi(\backslash tau)\; \backslash frac\{E(\backslash tau)\}\{E(0)\}\; \backslash mathcal\{K\}\text{'}\; \backslash left(\; \backslash frac\{\backslash epsilon\_1\}\{(\backslash tau\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(\backslash tau)\}\{E(0)\}\; \backslash right)\; d\backslash tau\; \backslash leq\; L\_3(t\_1).\; \backslash end\{aligned\}$$

Hence,

$$c\left(\frac{1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(t)}{E(0)}\right){\mathcal{K}}^{\mathrm{\prime }}\left(\frac{{ϵ}_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(t)}{E(0)}\right){\int }_{t_1}^t\xi (\tau )d\tau \le \frac{c_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\mathrm{.}c\; \backslash left(\; \backslash frac\{1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash right)\; \backslash mathcal\{K\}\text{'}\; \backslash left(\; \backslash frac\{\backslash epsilon\_1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash right)\; \backslash int\_\{t\_1\}^t\; \backslash xi(\backslash tau)\; d\backslash tau\; \backslash leq\; \backslash frac\{c\_1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}.$$

Setting $\mathcal{K}_1(\tau) = \tau \mathcal{K}'(\epsilon_1 \tau)$ , the above inequality reduces to

$$c{\mathcal{K}}_1\left(\frac{{ϵ}_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}}\frac{E(t)}{E(0)}\right){\int }_{t_1}^t\xi (\tau )d\tau \le \frac{c_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}},(61)c\; \backslash mathcal\{K\}\_1\; \backslash left(\; \backslash frac\{\backslash epsilon\_1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\}\; \backslash frac\{E(t)\}\{E(0)\}\; \backslash right)\; \backslash int\_\{t\_1\}^t\; \backslash xi(\backslash tau)\; d\backslash tau\; \backslash leq\; \backslash frac\{c\_1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\},\; \backslash quad\; (61)$$

After rearranging the terms in (61), we conclude that

$$E(t)\le c(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}{\mathcal{K}}_1^{-1}\left(\frac{c_1}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}{\int }_{t_1}^t\xi (\tau )d\tau }\right),\mathrm{\forall }t\ge t_1\mathrm{.}E(t)\; \backslash leq\; c(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; \backslash mathcal\{K\}\_1^\{-1\}\; \backslash left(\; \backslash frac\{c\_1\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; \backslash int\_\{t\_1\}^t\; \backslash xi(\backslash tau)\; d\backslash tau\}\; \backslash right),\; \backslash quad\; \backslash forall\; t\; \backslash geq\; t\_1.$$

Hence, the theorem is proved. $\square$

Remark 4.3. If $H(0) = 0$ and $H$ is strictly convex on $(0, r]$ , then

$$H(\theta s)\le \theta H(s),\theta \in [0,1],\text{and }s\in (0,r]\mathrm{.}(62)H(\backslash theta\; s)\; \backslash leq\; \backslash theta\; H(s),\; \backslash quad\; \backslash theta\; \backslash in\; [0,\; 1],\; \backslash quad\; \backslash text\{and\; \}\; s\; \backslash in\; (0,\; r].\; \backslash quad\; (62)$$

Lemma 4.7. Assume that there exists $C$ such that $\tau h(\tau) \leq C$ . Then for some constant $c > 0$ , we have the following estimate:

$${\bar{H}}^{-1}(G(t))\le c(t-t_1{)}^{\frac{1}{1+ϵ}}{\bar{H}}^{-1}{\left(\frac{M(t)}{(t-t_1{)}^{\frac{1}{1+ϵ}}}\right)}^{\frac{1}{1+ϵ}}\mathrm{.}(63)\backslash bar\{H\}^\{-1\}(G(t))\; \backslash leq\; c(t\; -\; t\_1)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; \backslash bar\{H\}^\{-1\}\; \backslash left(\; \backslash frac\{M(t)\}\{(t\; -\; t\_1)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\}\; \backslash right)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}.\; \backslash quad\; (63)$$

Proof. Since, $\lim_{t \rightarrow \infty} \frac{1}{t - t_1} = 0$ , $\exists t_2$ such that $\frac{1}{t - t_1} < 1$ whenever $t > t_2$ .

Using the strict convexity and strictly increasing properties of $\bar{H}$ , with $\theta = \frac{1}{(t - t_1)^{\frac{1}{1+\epsilon}}} < 1$ and using (62), we get

$${\bar{H}}^{-1}(G(t))\le (t-t_1{)}^{\frac{1}{1+ϵ}}{\bar{H}}^{-1}\left(\frac{M(t)}{(t-t_1{)}^{\frac{1}{1+ϵ}}}\right)\mathrm{.}(64)\backslash bar\{H\}^\{-1\}(G(t))\; \backslash leq\; (t\; -\; t\_1)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; \backslash bar\{H\}^\{-1\}\; \backslash left(\; \backslash frac\{M(t)\}\{(t\; -\; t\_1)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\}\; \backslash right).\; \backslash quad\; (64)$$Since, $\tau h(\tau) \leq C$ it is easy to see that $M(t) \leq C$ and also $\frac{M(t)}{(t-t_1)^{\frac{1}{1+\epsilon}}} \leq C$ .
Therefore

Therefore (63) follows from (64) and (65). Hence, the lemma is proved. $\square$

Theorem 4.2. Let $(u_0, u_1) \in H_0^2(\Omega) \times H_0^2(\Omega)$ and $h_1, B$ are nonlinear. Assume that the hypothesis of Lemma 4.7 and the hypothesis (H1) – (H4) holds. Then for all $t \geq t_1$ , we have

$$E(t)\le c(t-t_0{)}^{\frac{1}{1+ϵ}}{W}_2^{-1}\left(\frac{c}{(t-t_0{)}^{\frac{1}{1+ϵ}}{\int }_{t_0}^t\xi (s)ds}\right),(66)E(t)\; \backslash leq\; c(t-t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; W\_2^\{-1\}\backslash left(\backslash frac\{c\}\{(t-t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; \backslash int\_\{t\_0\}^t\; \backslash xi(s)\; ds\}\backslash right),\; \backslash quad\; (66)$$

where $W_2(t) = tW'(\epsilon_1 t)$ and $W(t) = \left((\bar{B}^{-1})^{\frac{1}{1+\epsilon}} + (\bar{H}^{-1})^{\frac{1}{1+\epsilon}}\right)^{-1}(t)$ .

Proof. Using (49) and Lemma (4.2), observe that

Denote $L_1(t) = L(t) + cE(t)$ , then using Remark 4.1, Lemma's 4.3 and 4.7 and for all $t \geq t_1$ the above inequality becomes

Denote

$$W(t)={(({\bar{B}}^{-1}{)}^{\frac{1}{1+ϵ}}+({\bar{H}}^{-1}{)}^{\frac{1}{1+ϵ}})}^{-1}(t),(67)W(t)\; =\; \backslash left(\; (\backslash bar\{B\}^\{-1\})^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; +\; (\backslash bar\{H\}^\{-1\})^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\; \backslash right)^\{-1\}(t),\; \backslash quad\; (67)$$

and

$$\beta (t)=\max \{\frac{\delta M(t)}{(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}\xi (t)},\frac{M(t)}{(t-t_1{)}^{\frac{1}{1+ϵ}}}\}\mathrm{.}(68)\backslash beta(t)\; =\; \backslash max\; \backslash left\backslash \{\; \backslash frac\{\backslash delta\; M(t)\}\{(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; \backslash xi(t)\},\; \backslash frac\{M(t)\}\{(t\; -\; t\_1)^\{\backslash frac\{1\}\{1+\backslash epsilon\}\}\}\; \backslash right\backslash \}.\; \backslash quad\; (68)$$

From the definition of $W$ , it is clear that $W'(t) > 0$ and $W''(t) > 0$ on $(0, r]$ , where $r = \min{r_1, r_2}$ . Using these notations, we obtain for all $t \geq t_1$ ,

$$L_1^{\mathrm{\prime }}(t)\le -mE(t)+c(t-t_0{)}^{\frac{1}{1+{ϵ}_0}}{W}^{-1}(\beta (t))\mathrm{.}(69)L\text{'}\_1(t)\; \backslash leq\; -mE(t)\; +\; c(t\; -\; t\_0)^\{\backslash frac\{1\}\{1+\backslash epsilon\_0\}\}\; W^\{-1\}(\backslash beta(t)).\; \backslash quad\; (69)$$

To conclude (66), we follow the similar lines as shown in the proof of Theorem 4.1. This completes the proof of this theorem. $\square$

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