Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity

Abstract

In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.

Keywords

Viscoelasticity global existence decay estimates convexity logarithmic nonlinearity

1. Introduction

In this work, we are interested to study the global existence and the general decay behaviour of the following plate problem: $\begin{matrix} (1) & \{\begin{matrix} | u_{t} |^{ρ} u_{t t} + Δ^{2} u + Δ^{2} u_{t t} + u - \int_{0}^{t} b (t - s) Δ^{2} u (s) d s \\ + h (u_{t}) = k u ln | u |, & (x, t) \in Ω \times (0, \infty), \\ u (x, t) = \frac{\partial u}{\partial ν} (x, t) = 0, & (x, t) \in \partial Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \end{matrix} \end{matrix}$ where $Ω \subset R^{n}$ is a bounded domain with smooth boundary, ν is the unit outward normal to $\partial Ω$ and k, ρ are positive constants. We assume that ρ satisfies $0 < ρ ⩽ \frac{2}{n - 2}$ if $n ⩾ 3$ . We consider the usual scalar product and norms to the Lebesgue spaces $L^{2} (Ω)$ and $H_{0}^{2} (Ω)$ . 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. Our goal here is to capture the behaviour of suspension bridges, that must have a suitable number of degrees of freedom and contain some nonlinearity. As would be expected, nonlinearities enable the detection of some obscure events (see [19] for more details). The distribution’s form is altered by a logarithmic expression; it minimises sample skewness and, in some situations, data skewness. Since a suspension bridge’s behaviour is nonlinear, nonlinearity can be used to explain why torsional oscillation occurs. If we use logarithmic nonlinearity, the oscillation’s amplitude will also decrease.

Let us discuss some works related to the plate problems. In [12], the authors considered the following problem $\begin{matrix} \{\begin{matrix} u_{t t} - Δ u + α (t) h (u_{t}) = 0, & (x, t) \in Ω \times (0, \infty), \\ u = 0, & (x, t) \in \partial Ω \times (0, \infty), \end{matrix} \end{matrix}$ where the function h is having a polynomial growth near the origin. They have provided few energy decay results. In [6] author consider the problem: $\begin{matrix} u_{t t} - σ Δ u_{t t} + Δ^{2} u - \int_{0}^{+ \infty} k (s) Δ^{2} u (t - s) d s = 0, \end{matrix}$ and show that the stability of this problem holds for a much larger class of kernals. A.M. Al-Mahdi [5] consider the problem with viscoelastic damping localized on a portion of the boundary and nonlinear damping in the domain. For a larger class of relaxation functions, they develop general and optimal decay rate results. In [30], Lasiecka and Tataru considered the wave equation with nonlinear boundary conditions and nonlinear boundary velocity feedback and established uniform decay rates. In [32], Liu considered the following problem $\begin{matrix} \{\begin{matrix} | u_{t} |^{ρ} u_{t t} - Δ u - Δ u_{t t} - \int_{0}^{t} g (t - s) Δ u (s) d s + α (t) h (u_{t}) \\ = b | u |^{ρ - 2} u, & (x, t) \in Ω \times (0, \infty), \\ u (x, t) = 0, & (x, t) \in \partial Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), & x \in Ω, \end{matrix} \end{matrix}$ and proved a general decay result that depends on the behavior of g, α and h without imposing any restrictive growth assumption on the damping term at origin. The authors in [36] consider the swelling elastic system with frictional and viscoelastic damping terms and established explicit and optimal energy decay rates. For more results in the direction of the plate problems, see [7,9,16,18,26–29,33–35,37,40] and the references there in.

Now, let us review few works with a logarithmic term that are related to the problem (1). Cazenave and Haraux [15] studied the following problem: $\begin{matrix} (2) & u_{t t} - Δ u - K u ln | u |^{2}, (x, t) \in R^{3} \times R^{+}, \end{matrix}$ and established the existence and uniqueness of the solution for the Cauchy problem. Gorka [20] 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 (2). The authors in [11] considered the one dimensional Cauchy problem for equation (2) and they establish the existence of classical solutions and also they study the weak solutions. Birula and Mycielski [13,14] considered $\begin{matrix} \{\begin{matrix} u_{t t} - u_{x x} + u = ϵ u ln | u |^{2}, & (x, t) \in (a, b) \times (0, T), \\ u (a, t) = u (b, t) = 0, & t \in (0, T), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in [a, b], \end{matrix} \end{matrix}$ 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 $p \to 1$ (see [21,39]). The authors in [3] considered equation (1) without damping term and they establish the existence of solution and proved the energy decay rates. The authors in [1] considered the viscoelastic problem with logarithmic nonlinearities and variable exponent: $\begin{matrix} u_{t t} - Δ u + u + \int_{0}^{t} b (t - s) Δ u (s) d s + | u_{t} |^{γ (\cdot) - 2} u_{t} = u ln | u |^{α}, (x, t) \in Ω \times R^{+}, \end{matrix}$ and they establish global existence result using the well-depth method. Moreover, under a wide class of relaxation functions they prove explicit and general decay results. Gongwei Liu [31] considered the differential equation $\begin{matrix} u_{t t} + Δ^{2} u + | u_{t} |^{m - 2} u_{t} = u | u |^{p - 2} log | u |^{k}, (x, t) \in Ω \times R^{+}, \end{matrix}$ with the boundary conditions given in (1). They have established the local existence result by the fixed point techniques. At sub-critical initial energy, the global existence and decay estimate of the solution is obtained, and they additionally prove that the solution with negative initial energy blows up in finite time. Moreover, in linear damping (i.e., $m = 2$ ) case, at the arbitrarily high initial energy they find out the blow-up in finite time of the solution.

Let us discuss few works related to the past history to the plate equations. We start with Dafermos’ pioneer work [17] in which he consider the following one-dimensional viscoelastic problem: $\begin{matrix} \{\begin{matrix} ρ u_{t t} = c u_{x x} - \int_{- \infty}^{t} g (t - τ) u_{x x} d τ, & x \in [0, 1], t \in [0, \infty), \\ u (0, t) = u (1, t) = 0, & t \in (- \infty, \infty), \end{matrix} \end{matrix}$ and proved that the solution goes to zero as t approaches infinity for smooth monotonic decreasing relaxation functions. Moreover, he also established an existence result. The existence results and the stability estimates of viscoelastic plate equation has been attracted by many people. Guesmia [23] considered the following problem: $\begin{matrix} u_{t t} + A u - \int_{0}^{+ \infty} h (s) B u (t - s) d s = 0, t > 0, \end{matrix}$ with a class of infinite history kernels satisfying $\int_{0}^{+ \infty} \frac{h (s)}{H^{- 1} (- h^{'} (s))} d s + {sup}_{s \in R_{+}} \frac{h (s)}{H^{- 1} (- h^{'} (s))} < + \infty$ , where H is strictly increasing convex function with $H (0), H^{'} (0) = 0$ and ${lim}_{t \to + \infty} H^{'} (t) = + \infty$ . He proved the more general decay result using the properties of convex function and Young’s inequality. In [25] authors consider the viscoelastic plate equation with nonlinear damping, logarithmic nonlinearity and past history and proved a general decay rate results of the solution by using the convex properties, logarithmic inequalities and generalised Young’s inequality. For more work related to the past history problems, refer to [8,24,38](and the references therein).

Through out the paper, we use c to denote a generic positive constant and we consider the following hypotheses:

Let $b : (0, \infty) \to (0, \infty)$ is a $C^{1}$ -nonincreasing function satisfying $b (0) > 0$ and $\begin{matrix} (3) & 1 - \int_{0}^{\infty} b (τ) d τ = l > 0 . \end{matrix}$

Assume that there exist a nonincreasing positive differentiable function ξ such that $ξ (0) > 0$ and $ξ : R^{+} \to R^{+}$ . Further assume that there exists a $C^{1}$ function $B : (0, \infty) \to (0, \infty)$ which is linear or strictly convex $C^{2}$ function and strictly increasing on $(0, r_{1}], r_{1} ⩽ b (0)$ , $B (0) = B^{'} (0) = 0$ and B satisfies $\begin{matrix} (4) & b^{'} (t) ⩽ - ξ (t) B (b (t)), \forall t ⩾ 0 . \end{matrix}$

Let $h : R \to R$ is a nondecreasing continuous function such that there exist a $h_{1} \in C^{1} (R^{+})$ with $h_{1} (0) = 0$ which is strictly increasing function and $h_{1}$ satisfies $\begin{array}{l} h_{1} (| s |) ⩽ | h (s) | ⩽ h_{1}^{- 1} (| s |), \forall | s | ⩽ ϵ, \\ c_{1} | s | ⩽ | h (s) | ⩽ c_{2} | s |, \forall | s | ⩾ ϵ, \end{array}$ where $c_{1}$ , $c_{2}$ , ϵ are positive constants. Moreover, define H to be a strictly convex $C^{2}$ function in $(0, r_{2}]$ for some $r_{2} ⩾ 0$ such that $H (s) = \sqrt{s} h_{1} (\sqrt{s})$ when $h_{1}$ is nonlinear.

(Lasiecka I. and Tataru D. [30] was the first ones who used the hypothesis (H3).)

The constant k in (1) is such that $\begin{matrix} (5) & 0 < k < k_{0} = \frac{2 π l e^{3}}{c_{p}}, \end{matrix}$ where $c_{p} > 0$ is the smallest number satisfying $\begin{matrix} (6) & ‖ \nabla u ‖_{2}^{2} ⩽ c_{p} ‖ Δ u ‖_{2}^{2}, u \in H_{0}^{2} (Ω), where ‖ . . ‖_{2} = ‖ . . ‖_{L^{2} (Ω)} . \end{matrix}$

Remark 1.1.
Hypothesis (H3) implies that $τ h (τ) > 0$ , $\forall τ > 0$ .
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 Remark 2.3 in [2] for details). Similarly, we denote the extension of H to be $\bar{H}$ .
In this article, we are engaged with the global existence and stability of the plate problem (1) with kernel 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 general decay rate results.
2. Local existence

In this section, we prove the local existence result for the problem (1). We use the Faedo–Galerkin approach to find a sequence of approximate solution to (1). Using a few compactness results, we can find a subsequence to the approximate solution whose limit will be a local solution to (1).

The energy functional associated to problem (1) is given by: $\begin{matrix} (7) & \begin{matrix} E (t) & = \frac{1}{ρ + 2} ‖ u_{t} ‖_{ρ + 2}^{ρ + 2} + \frac{1}{2} [(1 - \int_{0}^{t} b (s) d s) ‖ Δ u ‖_{2}^{2} + ‖ Δ u_{t} ‖_{2}^{2} - k \int_{Ω} u^{2} ln | u | d x \\ + ‖ u ‖_{2}^{2} + b \circ Δ u] + \frac{k}{4} ‖ u ‖_{2}^{2}, \end{matrix} \end{matrix}$ where the product ∘ is defined by $\begin{matrix} (b \circ Δ u) (t) = \int_{0}^{t} b (t - s) {‖ Δ u (s) - Δ u (t) ‖}_{2}^{2} d s . \end{matrix}$ Direct differentiation of (7) with respect to t and using (1) we observe that $\begin{matrix} (8) & \begin{matrix} E^{'} (t) & = \frac{1}{2} (b^{'} \circ Δ u) (t) - \frac{1}{2} b (t) ‖ Δ u ‖_{2}^{2} - \int_{Ω} u_{t} h (u_{t}) d x \\ ⩽ \frac{1}{2} (b^{'} \circ Δ u) (t) - \int_{Ω} u_{t} h (u_{t}) d x \\ ⩽ 0 . \end{matrix} \end{matrix}$ Before we state the main result in this section, we state few results which are helpful for the analysis later on.

Lemma 2.1 (Logarithmic Sobolev inequality).

Let $u \in H_{0}^{1} (Ω)$ and $a > 0$ be any number. Then $\begin{matrix} (9) & \int_{Ω} u^{2} ln | u | d x ⩽ \frac{1}{2} ‖ u ‖_{2}^{2} ln ‖ u ‖_{2}^{2} + \frac{a^{2}}{2 π} ‖ ▽ u ‖_{2}^{2} - (1 + ln a) ‖ u ‖_{2}^{2} . \end{matrix}$

Using the similar arguments employed in [22], one can establish Lemma 2.1. So, we omit the details here.

Corollary 2.1.
Let $u \in H_{0}^{2} (Ω)$ and $a > 0$ be any number. Then $\begin{matrix} (10) & \int_{Ω} u^{2} ln | u | d x ⩽ \frac{1}{2} ‖ u ‖_{2}^{2} ln ‖ u ‖_{2}^{2} + \frac{c_{p} a^{2}}{2 π} ‖ Δ u ‖_{2}^{2} - (1 + ln a) ‖ u ‖_{2}^{2} . \end{matrix}$

The proof of the above corollary directly follows from inequality (6).
Lemma 2.2 (cf. lemma 2.4, [3]).

Let $ϵ_{0} \in (0, 1)$ , then there exists $d_{ϵ_{0}} > 0$ such that $\begin{matrix} (11) & s | ln s | ⩽ s^{2} + d_{ϵ_{0}} s^{1 - ϵ_{0}}, \forall s > 0 . \end{matrix}$

Definition 2.1.
A function $\begin{matrix} u \in L^{2} ([0, T], H_{0}^{2} (Ω)) \end{matrix}$ is called a weak solution of (1) on $[0, T]$ if, for any $t \in [0, T]$ and $\forall w \in H_{0}^{2} (Ω)$ , u satisfies $\begin{matrix} (12) & \{\begin{matrix} \int_{Ω} | u_{t} |^{ρ} u_{t t} (x, t) w (x) d x + \int_{Ω} Δ u (x, t) Δ w (x) d x + \int_{Ω} Δ u_{t t} (x, t) Δ w (x) d x \\ + \int_{Ω} u (x, t) w (x) d x - \int_{Ω} Δ w (x) \int_{0}^{t} b (t - s) Δ u (s) d s \\ + \int_{Ω} h (u_{t} (x, t)) w (x) d x = k \int_{Ω} u (x, t) w (x) ln (| u (x, t) |) d x, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x) . \end{matrix} \end{matrix}$

We now state and prove the main theorem ( $i . e$ ., local existence result) in this section. The majority of the proof for the local existence portion is a small adaptation of the relevant research in [3] and [4]. Because of the presence of damping term, we need to prove few more estimates in order to prove the local existence results. Theorem 2.1.
Assume that the hypothesis (H1)–(H4) hold. Let $(u_{0}, u_{1}) \in H_{0}^{2} (Ω) \times H_{0}^{2} (Ω)$ . Then the problem ( 1 ) has weak solution on $[0, T]$ .
Proof.
As mentioned earlier, we use the Faedo–Galerkin approach for the existence of a solution to (1). Denote ${(w_{j})}_{j = 1}^{\infty}$ be an orthogonal basis of the separable space $H_{0}^{2} (Ω)$ . Denote $V_{m} = span (w_{1}, w_{2}, \dots, w_{m})$ and assume that the projections of the initial data on $V_{m}$ is given by $\begin{matrix} (13) & u_{0}^{m} (x) = \sum_{j = 1}^{m} α_{j} w_{j} (x), u_{1}^{m} (x) = \sum_{j = 1}^{m} β_{j} w_{j} (x) . \end{matrix}$ We seek for an approximation solution $\begin{matrix} (14) & u^{m} (x) = \sum_{j = 1}^{m} g_{j}^{m} (t) w_{j} (x), \end{matrix}$ of the approximate problem in $V_{m}$ : $\begin{array}{rcl} (15) & \{\begin{matrix} \int_{Ω} [| u_{t}^{m} |^{ρ} u_{t t}^{m} w + Δ u^{m} Δ w + Δ u_{t t}^{m} Δ w + u^{m} w + h (u_{t}^{m}) w \\ - \int_{0}^{t} b (t - s) Δ u^{m} (s) Δ w d s] d x = k \int_{Ω} w u^{m} ln | u^{m} | d x, \forall w \in V_{m}, \\ u^{m} (0) : = u_{0}^{m} = \sum_{j = 1}^{m} (u_{0}, w_{j}) w_{j}, \\ u_{t}^{m} (0) : = u_{1}^{m} = \sum_{j = 1}^{m} (u_{1}, w_{j}) w_{j} . \end{matrix} \end{array}$ Equation (15) gives a system of ordinary differential equations (ODE’s) for the unknown functions $g_{j}^{m} (t)$ . Using the standard existence theory for ODE’s, one can obtain functions $\begin{matrix} g_{j} : [0, t_{m}) \to R, j = 1, 2, \dots, m, \end{matrix}$ which satisfy (15) in a maximal interval $[0, t_{m}), t_{m} \in (0, T]$ . Later, we prove that $t_{m}$ can be extended to T and the local solution is uniformly bounded which is independent of m and t.

We divide the proof into three steps. In Step 1, we prove $(u^{m})$ , $(u_{t}^{m})$ are uniformly bounded. Later, we prove that $(u_{t t}^{m})$ is bounded in Step 2 and in the last step, we pass the limit as $m \to \infty$ and conclude.

Step 1: In this step, our aim is to prove $\begin{matrix} (16) & \{\begin{matrix} (u^{m}) is uniformly bounded in L^{\infty} (0, T; H_{0}^{2} (Ω)), \\ (u_{t}^{m}) is uniformly bounded in L^{\infty} (0, T; L^{ρ + 2} (Ω)) \cap L^{\infty} (0, T; H_{0}^{2} (Ω)) . \end{matrix} \end{matrix}$ To do this, substitute $w = u_{t}^{m}$ in (15) and using integration by parts we obtain $\begin{matrix} (17) & \frac{d}{d t} E^{m} (t) ⩽ \frac{1}{2} (b^{'} o Δ u^{m}) - \int_{Ω} u_{t}^{m} h (u_{t}^{m}) d x ⩽ 0, \end{matrix}$ where $\begin{array}{rcl} E^{m} (t) & = & \frac{1}{ρ + 2} {‖ u_{t}^{m} ‖}_{ρ + 2}^{ρ + 2} + \frac{1}{2} ((1 - \int_{0}^{t} b (s) d s) {‖ Δ u^{m} ‖}_{2}^{2} + {‖ Δ u_{t}^{m} ‖}_{2}^{2} \\ (18) & - k \int_{Ω} {| u^{m} |}^{2} ln | u^{m} | d x) + \frac{k + 2}{4} {‖ u^{m} ‖}_{2}^{2} + \frac{1}{2} (b \circ Δ u^{m}) . \end{array}$ From (17), we have $E^{m} (t) ⩽ E^{m} (0)$ , $\forall t ⩾ 0$ . We now choose a such that $\begin{matrix} (19) & e^{- 3 / 2} < a < \sqrt{\frac{2 π l}{k c_{p}}}, \end{matrix}$ and following the similar arguments presented in the proof of theorem 3.2 in [3], we arrive at the following estimate: $\begin{matrix} (20) & sup_{t \in (0, t_{m})} [(b \circ Δ u^{m}) (t) + {‖ u_{t}^{m} ‖}_{ρ + 2}^{ρ + 2} + {‖ Δ u_{t}^{m} ‖}_{2}^{2} + {‖ Δ u^{m} ‖}_{2}^{2} + {‖ u^{m} ‖}_{2}^{2}] ⩽ c . \end{matrix}$ Hence, from (20), we can extend $t_{m}$ to T and also (16) follows.

Step 2: Here we prove that $(u_{t t}^{m})$ is bounded in $L^{2} (0, T; H_{0}^{2} (Ω))$ . This step is not straight-forward as presented in [3] because of the presence of the damping term. We substitute $w = u_{t t}^{m}$ in (12). Using (1), we see that $\begin{array}{rcl} \int_{Ω} {| u_{t}^{m} |}^{ρ} {| u_{t t}^{m} |}^{2} d x + {‖ Δ u_{t t}^{m} ‖}_{2}^{2} & = & - \int_{Ω} (Δ u^{m} Δ u_{t t}^{m} + u^{m} u_{t t}^{m}) d x + \int_{Ω} \int_{0}^{t} b (t - s) Δ u^{m} (s) Δ u_{t t}^{m} (t) d s d x \\ - \int_{Ω} h (u_{t}^{m}) u_{t t}^{m} d x + k \int_{Ω} u_{t t}^{m} u^{m} ln | u^{m} | d x . \end{array}$ By using the Cauchy–Schwarz’ inequality, Young’s inequality, for $δ > 0$ , we obtain $\begin{matrix} \int_{Ω} {| u_{t}^{m} |}^{ρ} {| u_{t t}^{m} |}^{2} d x + {‖ Δ u_{t t}^{m} ‖}_{2}^{2} & ⩽ δ {‖ Δ u_{t t}^{m} ‖}_{2}^{2} + \frac{1}{4 δ} {‖ Δ u^{m} (t) ‖}_{2}^{2} + δ {‖ u_{t t}^{m} ‖}_{2}^{2} + \frac{1}{4 δ} {‖ u^{m} (t) ‖}_{2}^{2} + δ {‖ Δ u_{t t}^{m} ‖}^{2} \\ + \frac{1}{4 δ} {(\int_{0}^{t} b (t - s) ‖ Δ u^{m} (s) ‖ d s)}^{2} + \frac{1}{4 δ} \int_{Ω} h^{2} (u_{t}^{m}) d x + δ {‖ u_{t t}^{m} ‖}_{2}^{2} \\ + k \int_{Ω} u_{t t}^{m} u^{m} ln | u^{m} | d x . \end{matrix}$ Using the similar calculations as in [3] we arrive at the following estimate, $\begin{matrix} (21) & \begin{matrix} \int_{Ω} {| u_{t}^{m} |}^{ρ} {| u_{t t}^{m} |}^{2} d x + (1 - c δ) {‖ Δ u_{t t}^{m} ‖}_{2}^{2} \\ ⩽ \frac{c}{4 δ} ({‖ Δ u^{m} ‖}_{2}^{4} + {‖ u^{m} ‖}_{2}) \\ + \frac{1}{4 δ} [{‖ Δ u^{m} (t) ‖}^{2} + {(\int_{0}^{t} b (t - s) ‖ Δ u^{m} (s) ‖ d s)}^{2} + \int_{Ω} h^{2} (u_{t}^{m}) d x + {‖ u^{m} ‖}^{2}], \end{matrix} \end{matrix}$ where δ is chosen such that $(1 - c δ) > 0$ . We now prove that $h (u_{t}^{m})$ is bounded in $L^{2} (0, T; L^{2} (Ω))$ . For this purpose, we consider two cases:

Case 1. When $h_{1}$ is linear on $[0, ϵ]$ , we can directly prove that $\begin{matrix} \int_{0}^{t} \int_{Ω} h^{2} (u_{t}^{m}) d x d t ⩽ C . \end{matrix}$

Case 2. When $h_{1}$ is nonlinear, consider $| s | ⩽ ϵ$ , then $H (h^{2} (s)) = | h (s) | h_{1} (| h (s) |) ⩽ s h (s)$ , this implies that $\begin{matrix} H^{- 1} (s h (s)) ⩾ h^{2} (s), \forall | s | ⩽ ϵ, \end{matrix}$ then following the similar lines from Theorem 3.2 in [4], we get $\int_{0}^{t} \int_{Ω} h^{2} (u_{t}^{m}) d x d t ⩽ C_{T}$ . Therefore $h (u_{t}^{m})$ is bounded in $L^{2} (0, T; L^{2} (Ω))$ .

Integrating (21) from $(0, T)$ , using the hypothesis (H1) and (20), we obtain $\begin{array}{l} \int_{0}^{T} \int_{Ω} {| u_{t}^{m} |}^{ρ} {| u_{t t}^{m} |}^{2} d x d t + (1 - c δ) \int_{0}^{T} {‖ Δ u_{t t}^{m} ‖}_{2}^{2} d t \\ (22) & ⩽ \frac{c}{δ} \int_{0}^{T} [(b \circ Δ u^{m}) (t) + {‖ Δ u^{m} ‖}_{2}^{2} + {‖ Δ u^{m} ‖}_{2}^{4} + {‖ u^{m} ‖}_{2}] d t + c_{δ} \int_{0}^{T} \int_{Ω} h^{2} (u_{t}^{m}) d x d t . \end{array}$ From (22) and using the fact that $h (u_{t}^{m})$ is bounded in $L^{2} (0, T; L^{2} (Ω))$ , it is easy to observe that for δ small enough, $\begin{matrix} (23) & (u_{t t}^{m}) is bounded in L^{2} (0, T; H_{0}^{2} (Ω)) . \end{matrix}$

Step 3: From (16) and (23), there exist a subsequence of ( $u^{m}$ ) (denoted by ( $u^{m}$ ) itself), such that $\begin{matrix} (24) & \{\begin{matrix} u^{m} \overset{}{\to} u in L^{\infty} (0, T; H_{0}^{2} (Ω)), \\ u^{m} ⇀ u in L^{2} (0, T; H_{0}^{2} (Ω)), \\ u_{t}^{m} \overset{}{\to} u_{t} in L^{\infty} (0, T; L^{ρ + 2} (Ω)) \cap L^{\infty} (0, T; H_{0}^{2} (Ω)), \\ u_{t}^{m} ⇀ u_{t} in L^{2} (0, T; L^{ρ + 2} (Ω)) \cap L^{2} (0, T; H_{0}^{2} (Ω)), \\ u_{t t}^{m} ⇀ u_{t t} in L^{2} (0, T; H_{0}^{2} (Ω)), \end{matrix} \end{matrix}$ where $\overset{*}{\to}$ represent the weak ∗ convergence and ⇀ represent weak convergence. For the analysis of nonlinear terms (to pass the limit), we again refer to [3] (for the terms $u^{m} ln (u^{m})$ , $| u_{t}^{m} |^{ρ} u_{t}^{m}$ ) and [4] (for the term $h (u_{t}^{m})$ ). Thus, if u be the limit for $u^{m}$ , and for all $w \in H_{0}^{2} (Ω)$ and $a . e . t \in (0, T)$ , we obtain $\begin{array}{l} \int_{Ω} | u_{t} |^{ρ} u_{t t} w d x + \int_{Ω} Δ u Δ w d x + \int_{Ω} Δ u_{t t} Δ w d x + \int_{Ω} u w d x \\ + \int_{Ω} h (u_{t}) w d x - \int_{Ω} (\int_{0}^{t} g (t - s) Δ u (s) d s) Δ w d x = k \int_{Ω} w u ln | u | d x d s . \end{array}$ This complete the proof of this theorem. □

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: $\begin{array}{rcl} J (t) & : = & J (u (t)) \\ = & \frac{1}{2} [‖ Δ u_{t} ‖_{2}^{2} + (1 - \int_{0}^{t} b (s) d s) ‖ Δ u ‖_{2}^{2} + ‖ u ‖_{2}^{2} + (b \circ Δ u) - \int_{Ω} u^{2} ln | u |^{k} d x] \\ (25) & + \frac{k}{4} ‖ u ‖_{2}^{2}, \end{array}$ and $\begin{matrix} (26) & I (t) : = I (u (t)) = ‖ Δ u_{t} ‖_{2}^{2} + (1 - \int_{0}^{t} b (s) d s) ‖ Δ u ‖_{2}^{2} + ‖ u ‖_{2}^{2} + (b \circ Δ u) - \int_{Ω} u^{2} ln | u |^{k} d x . \end{matrix}$ Note: From (7), (25) and (26), it is clear that $\begin{matrix} (27) & J (t) = \frac{1}{2} I (t) + \frac{k}{4} ‖ u ‖_{2}^{2}, and E (t) = \frac{1}{ρ + 2} ‖ u_{t} ‖_{ρ + 2}^{ρ + 2} + J (t) . \end{matrix}$ Notation: Define $\overline{ρ} = e^{\frac{2 Q_{0} - k}{k}}$ , $d = \frac{1}{2} Q_{0} {\overline{ρ}}^{2} - \frac{k}{4} {\overline{ρ}}^{2} ln {\overline{ρ}}^{2}$ and $Q_{0} = \frac{k + 2}{2} + k (1 + ln a)$ , where $0 < a < \sqrt{\frac{2 π c_{p} l}{k}}$ .

Lemma 3.1.
Let $(u_{0}, u_{1}) \in H_{0}^{2} (Ω) \times H_{0}^{2} (Ω)$ and assume that the hypothesis (H1) holds. Further assume that $‖ u_{0} ‖ < \overline{ρ}$ and $0 < E (0) < d$ . Then $I (u) > 0$ $\forall t \in [0, T)$ .
Proof.
We divide the proof of this lemma into two steps. In step (1), we prove that $‖ u ‖_{2} < \overline{ρ}$ , $\forall t \in [0, T),$ and in step (2), we prove $I (t) > 0$ , $t \in [0, T)$ .

Step 1. From (7) and (25), and using Logarithmic Sobolev inequality it is easy to see that $\begin{matrix} E (t) ⩾ J (t) ⩾ \frac{1}{2} (l - \frac{c_{p} k a^{2}}{2 π}) ‖ Δ u ‖^{2} + \frac{1}{2} (\frac{k + 2}{2} + k (1 + ln a) - \frac{k}{2} ln ‖ u ‖_{2}^{2}) ‖ u ‖_{2}^{2} . \end{matrix}$ Using (19), we obtain $\begin{matrix} (28) & E (t) ⩾ Q (\tilde{ρ}) = \frac{1}{2} Q_{0} {\tilde{ρ}}^{2} - \frac{k}{4} {\tilde{ρ}}^{2} ln {\tilde{ρ}}^{2}, \end{matrix}$ where $\tilde{ρ} = ‖ u ‖_{2}$ . Observe that (28) implies Q is increasing on $(o, \overline{ρ})$ and decreasing on $(\overline{ρ}, \infty)$ . Also, note that $Q (\tilde{ρ}) \to - \infty$ as $\tilde{ρ} \to + \infty$ . Let $\begin{matrix} max_{0 < \tilde{ρ} < + \infty} Q (\tilde{ρ}) = \frac{1}{2} Q_{0} {\overline{ρ}}^{2} - \frac{α}{4} {\overline{ρ}}^{2} ln {\overline{ρ}}^{2} = Q (\overline{ρ}) = d . \end{matrix}$ Suppose that $‖ u ‖_{2} < \overline{ρ}$ does not hold in $[0, T)$ , then there exist $t_{0} \in (0, T)$ and $‖ u (x, t_{0}) ‖_{2} = \overline{ρ}$ . Using (28), we get $E (t_{0}) ⩾ Q (‖ u (x, t_{0}) ‖_{2}) = Q (\overline{ρ}) = d$ , which is contradiction to the fact $E (t) ⩽ E (0) < d$ $\forall t > 0$ . Hence, $‖ u ‖_{2} < \overline{ρ},$ for all $t \in [0, T)$ .

Step 2. Using the definition of $I (t)$ and (19), we notice that for $t \in [0, T)$ , $\begin{matrix} I (t) & ⩾ (l - \frac{c_{p} k a^{2}}{2 π}) ‖ Δ u ‖_{2}^{2} + (1 + k (1 + ln a) - \frac{k}{2} ln ‖ u ‖_{2}^{2}) ‖ u ‖_{2}^{2} \\ ⩾ (l - \frac{c_{p} k a^{2}}{2 π}) ‖ Δ u ‖_{2}^{2} + ‖ u ‖_{2}^{2} \\ ⩾ 0 . \end{matrix}$ This complete the proof of this lemma. □
Remark 3.1.
Under the assumptions of Lemma (3.1), for $t \in [0, T)$ , we have $\begin{matrix} ‖ u_{t} ‖_{ρ + 2}^{ρ + 2} ⩽ (ρ + 2) E (t) ⩽ (ρ + 2) E (0), \end{matrix}$ and $\begin{matrix} ‖ Δ u_{t} ‖_{2}^{2} ⩽ 2 E (t) ⩽ 2 E (0) . \end{matrix}$ 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 solution of problem (1). We recall few results from [3] and [2] which are useful to prove the stability results.

Lemma 4.1 (Cf. [3]).

Assume that b satisfies the hypothesis (H1). Then, for $u \in H_{0}^{2} (Ω)$ , we have $\begin{matrix} \int_{Ω} {(\int_{0}^{t} b (t - s) (u (t) - u (s)) d s)}^{2} d x ⩽ c (b \circ Δ u) (t), \end{matrix}$ and $\begin{matrix} \int_{Ω} {(\int_{0}^{t} b^{'} (t - s) (u (t) - u (s)) d s)}^{2} d x ⩽ - c (b^{'} \circ Δ u) (t), \end{matrix}$ for some $c > 0$ .

Lemma 4.2 (Cf. [2]).

Assume that h satisfies the hypothesis (H3). Then, the solution of ( 1 ) satisfies the following estimates: $\begin{array}{l} (29) & \int_{Ω} h^{2} (u_{t}) d x ⩽ c \int_{Ω} u_{t} h (u_{t}) d x ⩽ - c E^{'} (t), if h_{1} is linear, \\ (30) & \int_{Ω} h^{2} (u_{t}) d x ⩽ c H^{- 1} (G (t)) - c E^{'} (t), if h_{1} is nonlinear, \end{array}$ where $G (t) : = \frac{1}{| Ω_{1} |} \int_{Ω_{1}} u_{t} h (u_{t}) d x ⩽ - c E^{'} (t)$ and $Ω_{1} = {x \in Ω : | u_{t} | ⩽ ϵ}$ for some $c, ϵ > 0$ .

Lemma 4.3 (Cf. [2]).

Assume that b satisfies the hypothesis (H1) and (H2). Then the solution of ( 1 ) satisfies the following estimate: $\begin{matrix} (31) & \int_{0}^{t} b (s) \int_{Ω} {| Δ u (t) - Δ u (t - s) |}^{2} d x d s ⩽ \frac{t}{δ} {\bar{B}}^{- 1} (\frac{δ M (t)}{t ξ (t)}), \end{matrix}$ where $δ \in (0, 1)$ , $\bar{B}$ is an extension of B and $\begin{matrix} (32) & M (t) : = - \int_{0}^{t} b^{'} (s) \int_{Ω} {| Δ u (t) - Δ u (t - s) |}^{2} d x d s ⩽ - c E^{'} (t) . \end{matrix}$

Lemma 4.4.
Under the hypothesis (H1)–(H4), the functional $Ψ_{1} (t)$ defined as $\begin{matrix} Ψ_{1} (t) : = \frac{1}{ρ + 1} \int_{Ω} | u_{t} |^{ρ} u_{t} u d x + \int_{Ω} Δ u Δ u_{t} d x, \end{matrix}$ satisfies the estimate: $\begin{array}{rcl} Ψ_{1}^{'} (t) & ⩽ & \frac{1}{ρ + 1} ‖ u_{t} ‖_{ρ + 2}^{ρ + 2} + ‖ Δ u_{t} ‖_{2}^{2} - \frac{l}{4} ‖ Δ u ‖_{2}^{2} - ‖ u ‖_{2}^{2} + c \int_{Ω} h^{2} (u_{t}) d x \\ (33) & + c (b \circ Δ u) (t) + k \int_{Ω} u^{2} ln | u | d x . \end{array}$
Proof.
Differentiating $Ψ_{1}$ with respect to t and using (1), we get $\begin{array}{rcl} Ψ_{1}^{'} (t) & = & \frac{1}{ρ + 1} ‖ u_{t} ‖_{ρ + 2}^{ρ + 2} + k \int_{Ω} u^{2} ln | u | d x - ‖ Δ u ‖_{2}^{2} - ‖ u ‖_{2}^{2} + ‖ Δ u_{t} ‖_{2}^{2} \\ (34) & + \int_{Ω} Δ u (t) \int_{0}^{t} b (t - s) Δ u (s) d s d x - \int_{Ω} u h (u_{t}) d x . \end{array}$ At first, we estimate the sixth term in the right hand side of (34). Using the hypothesis (H1) and Lemma 4.1, we have $\begin{matrix} \int_{Ω} Δ u (t) \int_{0}^{t} b (t - s) Δ u (s) d s d x \\ = \int_{Ω} Δ u (t) \int_{0}^{t} b (t - s) (Δ u (s) - Δ u (t)) d s d x + \int_{Ω} {(Δ u (t))}^{2} \int_{0}^{t} b (t - s) d s d x \\ ⩽ \frac{η}{2} \int_{Ω} | Δ u |^{2} d x + \frac{1}{2 η} \int_{Ω} {(\int_{0}^{t} b (t - s) | Δ u (t) - Δ u (s) | d s)}^{2} d x + (1 - l) \int_{Ω} | Δ u |^{2} d x \\ = (1 - l + \frac{η}{2}) \int_{Ω} | Δ u |^{2} d x + \frac{c}{η} (b \circ Δ u) (t) \\ = (1 - \frac{l}{2}) \int_{Ω} | Δ u |^{2} d x + \frac{c}{η} (b \circ Δ u) (t) . \end{matrix}$ The last line in the above inequality is obtained by choosing $η = l$ . Now, we estimate the last term in (34). $\begin{matrix} \int_{Ω} | u h (u_{t}) | d x & ⩽ δ c ‖ \nabla u ‖_{2}^{2} + \frac{c}{δ} \int_{Ω} h^{2} (u_{t}) d x \\ ⩽ δ c ‖ Δ u ‖_{2}^{2} + \frac{c}{δ} \int_{Ω} h^{2} (u_{t}) d x . \end{matrix}$ Now choose $δ = \frac{l}{4 c}$ to get (33). Hence proved. □
Lemma 4.5.
Under the hypothesis (H1)–(H4), the functional $Ψ_{2} (t)$ defined as $\begin{matrix} Ψ_{2} (t) : = - \int_{Ω} (Δ^{2} u_{t} + \frac{1}{ρ + 1} | u_{t} |^{ρ} u_{t}) \int_{0}^{t} b (t - s) (u (t) - u (s)) d s d x, \end{matrix}$ satisfies the estimate: $\begin{matrix} (35) & \begin{matrix} Ψ_{2}^{'} (t) & ⩽ - \frac{1}{ρ + 1} (\int_{0}^{t} b (s) d s) ‖ u_{t} ‖_{ρ + 2}^{ρ + 2} + (\frac{δ}{2} + δ_{1} + 2 δ_{1} {(1 - l)}^{2}) ‖ Δ u ‖_{2}^{2} \\ + (δ + c δ_{2} {(E (0))}^{ρ} - \int_{0}^{t} b (s) d s) ‖ Δ u_{t} ‖_{2}^{2} - \frac{c}{δ} (b^{'} \circ Δ u) (t) \\ + (c + \frac{c}{δ} + c δ_{1} + \frac{c}{δ_{1}}) (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c \int_{Ω} h^{2} (u_{t}) d x, \end{matrix} \end{matrix}$ for some $ϵ_{0} \in (0, 1)$ .
Proof.
Differentiating $Ψ_{2}$ with respect to t and using (1), we get $\begin{matrix} (36) & \begin{matrix} Ψ_{2}^{'} (t) & = - \int_{Ω} u ln | u |^{k} (\int_{0}^{t} b (t - s) (u (t) - u (s)) d s) d x \\ + \int_{Ω} Δ u (t) (\int_{0}^{t} b (t - s) (Δ u (t) - Δ u (s)) d s) d x \\ + \int_{Ω} u (\int_{0}^{t} b (t - s) (u (t) - u (s)) d s) d x \\ - \int_{Ω} (\int_{0}^{t} b (t - s) Δ u (s) d s) (\int_{0}^{t} b (t - s) (Δ u (t) - Δ u (s)) d s) d x \\ + \int_{Ω} h (u_{t}) (\int_{0}^{t} b (t - s) (u (t) - u (s)) d s) d x \\ - \int_{Ω} \frac{1}{ρ + 1} | u_{t} |^{ρ} u_{t} (\int_{0}^{t} b^{'} (t - s) (u (t) - u (s)) d s) d x \\ - \int_{Ω} \frac{1}{ρ + 1} | u_{t} |^{ρ} u_{t}^{2} (\int_{0}^{t} b (s) d s) d x \\ - \int_{Ω} Δ u_{t} (\int_{0}^{t} b^{'} (t - s) (Δ u (t) - Δ u (s)) d s) d x \\ - (\int_{0}^{t} b (s) d s) (\int_{Ω} | Δ u_{t} |^{2} d x) . \end{matrix} \end{matrix}$ In order to estimate the all the nine terms in the right hand side of (36), 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 [4]. For the sake of completeness, we will write the final estimates.
$- \int_{Ω} u ln | u |^{k} (\int_{0}^{t} b (t - s) (u (t) - u (s)) d s) d x ⩽ \frac{δ}{4} ‖ Δ u ‖_{2}^{2} + \frac{c}{δ} (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t)$ ,

$\int_{Ω} Δ u (t) (\int_{0}^{t} b (t - s) (Δ u (t) - Δ u (s)) d s) d x ⩽ δ_{1} ‖ Δ u ‖_{2}^{2} + \frac{c}{δ_{1}} (b \circ Δ u) (t)$ ,

$\int_{Ω} u (\int_{0}^{t} b (t - s) (u (t) - u (s)) d s) d x ⩽ \frac{δ}{4} ‖ Δ u ‖_{2}^{2} + \frac{c}{δ} (b \circ Δ u) (t)$ ,

$\int_{Ω} (\int_{0}^{t} b (t - s) Δ u (s) d s) (\int_{0}^{t} b (t - s) (Δ u (t) - Δ u (s)) d s) d x ⩽ (c δ_{1} + \frac{c}{δ_{1}}) (b \circ Δ u) (t) + 2 δ_{1} {(1 - l)}^{2} ‖ Δ u ‖_{2}^{2}$ ,

$\int_{Ω} h (u_{t}) (\int_{0}^{t} b (t - s) (u (t) - u (s)) d s) d x ⩽ c \int_{Ω} h^{2} (u_{t}) d x + (1 - l) c^{2} (b \circ Δ u) (t)$ .

$\int_{Ω} \frac{1}{ρ + 1} | u_{t} |^{ρ} u_{t} (\int_{0}^{t} b^{'} (t - s) (u (t) - u (s)) d s) d x ⩽ c δ_{2} {(E (0))}^{ρ} ‖ Δ u_{t} ‖_{2}^{2} - \frac{c}{δ_{2}} (b^{'} \circ Δ u) (t)$ ,

$\int_{Ω} \frac{1}{ρ + 1} | u_{t} |^{ρ} u_{t}^{2} (\int_{0}^{t} b (s) d s) d x = - \frac{1}{ρ + 1} (\int_{0}^{t} b (s) d s) ‖ u_{t} ‖_{ρ + 2}^{ρ + 2}$ ,

$\int_{Ω} Δ u_{t} (\int_{0}^{t} b^{'} (t - s) (Δ u (t) - Δ u (s)) d s) d x ⩽ δ ‖ Δ u_{t} ‖_{2}^{2} - \frac{c}{δ} (b^{'} \circ Δ u) (t)$ ,

$(\int_{0}^{t} b (s) d s) (\int_{Ω} | Δ u_{t} |^{2} d x) = - (\int_{0}^{t} b (s) d s) ‖ Δ u_{t} ‖_{2}^{2}$ .
Combining all the above estimates we arrive at (35). □
Lemma 4.6.
Assume that the hypothesis of Lemma 3.1 and the hypothesis of (H1)–(H4) holds. Then there exists $N, ϵ > 0$ such that the functional $\begin{matrix} L (t) = N E (t) + ϵ Ψ_{1} (t) + Ψ_{2} (t) \end{matrix}$ satisfies, $\begin{matrix} (37) & L \sim E, \end{matrix}$ and for all $t ⩾ 0$ , there exists $m > 0$ and $ϵ_{0} \in (0, 1)$ such that $\begin{matrix} (38) & L^{'} (t) ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c \int_{Ω} h^{2} (u_{t}) d x . \end{matrix}$
Proof.
In order to prove (37), we use the Sobolev embedding $H_{0}^{1} (Ω) ↪ L^{ρ + 2} (Ω)$ , $\int_{Ω} | u_{t} |^{2 (ρ + 1)} d x ⩽ c ‖ Δ u_{t} ‖^{2}$ and Remark 3.1. From (38), we observe that $\begin{matrix} | L (t) - N E (t) | \\ ⩽ ϵ | Ψ_{1} | + | Ψ_{2} | \\ ⩽ \frac{ϵ}{ρ + 1} \int_{Ω} | u_{t} |^{ρ + 1} | u | d x \\ + | ϵ \int_{Ω} Δ u Δ u_{t} d x | + | \frac{1}{ρ + 1} \int_{Ω} | u_{t} |^{ρ} u_{t} \int_{0}^{t} b (t - s) (u (t) - u (s)) d s d x | \\ + | \int_{Ω} {(Δ u_{t})}^{2} \int_{0}^{t} b (t - s) (u (t) - u (s)) d s d x | \\ ⩽ \frac{ϵ}{ρ + 2} ‖ u_{t} ‖_{ρ + 2}^{ρ + 2} + \frac{ϵ}{(ρ + 1) (ρ + 2)} ‖ u ‖_{ρ + 2}^{ρ + 2} + \frac{ϵ}{2} ‖ Δ u ‖^{2} + \frac{ϵ}{2} ‖ Δ u_{t} ‖_{2}^{2} + \frac{1}{2 (ρ + 1)} ‖ u_{t} ‖_{2 (ρ + 1)}^{2 (ρ + 1)} \\ + \frac{1 - l}{2 (ρ + 1)} c_{p} (b \circ Δ u) (t) + \frac{1}{2} ‖ Δ u_{t} ‖_{2}^{2} + \frac{1 - l}{2} (b \circ Δ u) (t) \\ ⩽ c (1 + ϵ) E (t) . \end{matrix}$ Therefore, by choosing N large enough we obtain (37). We use the standard arguments presented in [3] to conclude the inequality (38). □
Remark 4.1.
Since, $\begin{matrix} E (t) ⩾ J (t) ⩾ \frac{1}{2} (b \circ Δ u) (t), \end{matrix}$ this implies $(b \circ Δ u) (t) ⩽ 2 E (t) ⩽ 2 E (0)$ .

Hence, $\begin{matrix} (39) & (b \circ Δ u) (t) ⩽ {(b \circ Δ u)}^{\frac{ϵ_{0}}{1 + ϵ_{0}}} (t) {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) ⩽ c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) . \end{matrix}$
Remark 4.2.
If b is linear then we have $\begin{matrix} (40) & \begin{matrix} ξ (t) {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) & = {[ξ^{ϵ_{0}} (t) ξ (t) (b \circ Δ u) (t)]}^{\frac{1}{1 + ϵ_{0}}} \\ ⩽ {[ξ^{ϵ_{0}} (0) ξ (t) (b \circ Δ u) (t)]}^{\frac{1}{1 + ϵ_{0}}} \\ ⩽ c {(ξ (t) (b \circ Δ u) (t))}^{\frac{1}{1 + ϵ_{0}}} \\ ⩽ c {(- E^{'} (t))}^{\frac{1}{1 + ϵ_{0}}} . \end{matrix} \end{matrix}$

Now, we are ready to prove the decay estimates.
Theorem 4.1.
Let $(u_{0}, u_{1}) \in H_{0}^{2} (Ω) \times H_{0}^{2} (Ω)$ and $h_{1}$ is linear. Under the assumptions of Lemma 4.6 , there exist strictly positive constants c, $ϵ_{1}$ such that the solution to ( 1 ) satisfies, $\begin{matrix} (41) & E (t) ⩽ c {(1 + \int_{t_{0}}^{t} ξ^{1 + ϵ_{0}} (s) d s)}^{- \frac{1}{ϵ_{0}}}, \forall t ⩾ t_{0}, if B is linear, \end{matrix}$ and $\begin{matrix} (42) & E (t) ⩽ c t^{\frac{1}{1 + ϵ_{0}}} K_{1}^{- 1} (\frac{c}{t^{\frac{1}{1 + ϵ_{0}}} \int_{t_{1}}^{t} ξ (s) d s}), \forall t ⩾ t_{1} = max {t_{0}, 1}, if B is nonlinear, \end{matrix}$ where $K_{1} (t) = t K^{'} (ϵ_{1} t)$ and $K (t) = {({({\bar{B}}^{- 1})}^{\frac{1}{1 + ϵ}})}^{- 1} (t)$ .
Proof.
We will divide the proof of this theorem into two cases.

Case 1: If B is linear, using (38), we get $\begin{matrix} L^{'} (t) & ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c \int_{Ω} h^{2} (u_{t}) d x \\ ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c (- E^{'} (t)) . \end{matrix}$ Denote $L_{1} (t) = L (t) + c E (t)$ , then the above inequality becomes $\begin{matrix} (43) & L_{1}^{'} (t) ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) . \end{matrix}$ Multiplying $ξ (t)$ to (43) and using Remarks 4.1 and 4.2, and for all $t ⩾ t_{0}$ we get $\begin{matrix} ξ (t) L_{1}^{'} (t) ⩽ - m ξ (t) E (t) + c {(- E^{'} (t))}^{\frac{1}{1 + ϵ_{0}}} . \end{matrix}$ Multiplying the above inequality by $ξ^{ϵ_{0}} (t) E^{ϵ_{0}} (t)$ and using the Young’s inequality and for any $ϵ_{1} > 0$ and $t ⩾ t_{0}$ , we obtain $\begin{matrix} ξ^{1 + ϵ_{0}} (t) E^{ϵ_{0}} (t) L_{1}^{'} (t) & ⩽ - m ξ^{1 + ϵ_{0}} (t) E^{1 + ϵ_{0}} (t) + c {(ξ E)}^{ϵ_{0}} (t) {(- E^{'} (t))}^{\frac{1}{1 + ϵ_{0}}} \\ ⩽ - m ξ^{1 + ϵ_{0}} (t) E^{1 + ϵ_{0}} (t) + c (ϵ_{1} ξ^{1 + ϵ_{0}} (t) E^{1 + ϵ_{0}} (t) - c_{ϵ_{1}} E^{'} (t)) \\ ⩽ - (m - ϵ_{1} c) ξ^{1 + ϵ_{0}} (t) E^{1 + ϵ_{0}} (t) - c E^{'} (t) . \end{matrix}$ Denote $L_{2} (t) = ξ^{1 + ϵ_{0}} (t) E^{ϵ_{0}} (t) L_{1} (t) + c E (t)$ and choosing $ϵ_{1} < \frac{m}{c}$ and using the properties of ξ and E to get $\begin{matrix} (44) & L_{2}^{'} (t) ⩽ - c ξ^{1 + ϵ_{0}} (t) E^{1 + ϵ_{0}} (t), \forall t ⩾ t_{0}, \end{matrix}$ where $c_{1} = m - ϵ_{1} c$ . Since, $L_{2} \sim E$ and from (44), it is easy to see that $\begin{matrix} E^{'} (t) ⩽ - c ξ^{1 + ϵ_{0}} (t) E^{1 + ϵ_{0}} (t), \forall t ⩾ t_{0} . \end{matrix}$ Integrating the above inequality from $(t_{0}, t)$ , we obtain (41).

Case 2: If B is nonlinear, using (38) we observe that, $\begin{matrix} L^{'} (t) & ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c \int_{Ω} h^{2} (u_{t}) d x \\ ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c (- E^{'} (t)) . \end{matrix}$ Denote $L_{1} (t) = L (t) + c E (t)$ , then using Lemma 4.3 and Remark 4.1 the above inequality becomes $\begin{matrix} L_{1}^{'} (t) & ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) \\ ⩽ - m E (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) \\ ⩽ - m E (t) + c t^{\frac{1}{1 + ϵ_{0}}} {[{\bar{B}}^{- 1} (\frac{δ M (t)}{t ξ (t)})]}^{\frac{1}{1 + ϵ_{0}}} \\ ⩽ - m E (t) + c t^{\frac{1}{1 + ϵ_{0}}} {[{\bar{B}}^{- 1} (\frac{δ M (t)}{t^{\frac{1}{1 + ϵ_{0}}} ξ (t)})]}^{\frac{1}{1 + ϵ_{0}}}, \end{matrix}$ the last line in the above inequality follows from the fact that, $\frac{1}{t} < 1$ when ever $t > t_{1}$ . Hence we have $\begin{matrix} {\bar{B}}^{- 1} (\frac{δ M (t)}{t ξ (t)}) ⩽ {\bar{B}}^{- 1} (\frac{δ M (t)}{t^{\frac{1}{1 + ϵ_{0}}} ξ (t)}) \forall t > t_{1} . \end{matrix}$ Denote $\begin{matrix} (45) & K (t) = {[{({\bar{B}}^{- 1})}^{\frac{1}{1 + ϵ_{0}}}]}^{- 1} (t), α (t) = \frac{δ M (t)}{t^{\frac{1}{1 + ϵ_{0}}} ξ (t)} . \end{matrix}$ From the definition of $K$ , it is clear that $K^{'} (t) > 0$ and $K^{″} (t) > 0$ on $(0, r]$ where $r = min {r_{1}, r_{2}}$ . Using these notations, we obtain that for all $t ⩾ t_{1}$ , $\begin{matrix} (46) & L_{1}^{'} (t) ⩽ - m E (t) + c t^{\frac{1}{1 + ϵ_{0}}} K^{- 1} (α (t)) . \end{matrix}$ Now, we define the functional $L_{2}$ as follows $\begin{matrix} L_{2} (t) : = K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) L_{1} (t), \end{matrix}$ for some $ϵ_{1} \in (0, r)$ . Using the fact that $K^{'} (t) > 0$ , $K^{″} (t) > 0$ and $E^{'} (t) ⩽ 0$ on $(0, r]$ , and using (46) we obtain $\begin{matrix} (47) & L_{2}^{'} (t) ⩽ - m E (t) K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) + c t^{\frac{1}{1 + ϵ_{0}}} K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) K^{- 1} (α (t)), \forall t ⩾ t_{1}, \end{matrix}$ and $\begin{matrix} (48) & L_{2} \sim E . \end{matrix}$ Let $K^{}$ denote the convex conjugate of $K$ in the sense of Young (see [10]), then we have $\begin{matrix} K^{} (τ) = τ {(K^{'})}^{- 1} (τ) - K ({[K^{'}]}^{- 1} (τ)), if τ \in (0, K^{'} (r)), \end{matrix}$ here $K^{}$ satisfies the generalized Young Inequality: $\begin{matrix} a b ⩽ K^{} (a) + K (b), if a \in (0, K^{'} (r)), b \in (0, r] . \end{matrix}$ So, by assuming $a = K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)})$ and $b = K^{- 1} (α (t))$ and using (47) we obtain $\begin{matrix} L_{2}^{'} (t) & ⩽ - m E (t) K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) + c t^{\frac{1}{1 + ϵ_{0}}} K^{} [K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)})] + c t^{\frac{1}{1 + ϵ_{0}}} α (t) \\ ⩽ - m E (t) K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) + c ϵ_{1} \frac{E (t)}{E (0)} K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) + c t^{\frac{1}{1 + ϵ_{0}}} α (t) . \end{matrix}$ Multiplying the above inequality by $ξ (t)$ and using (32) and (45) for all $t ⩾ t_{1}$ we get $\begin{matrix} ξ (t) L_{2}^{'} (t) ⩽ - m ξ (t) E (t) K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) + c ϵ_{1} ξ (t) \frac{E (t)}{E (0)} K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) - c E^{'} (t) . \end{matrix}$ By setting $L_{3} : = ξ L_{2} + c E$ (notice that $L_{3} \sim E$ ), we get for all $t ⩾ t_{1}$ $\begin{matrix} L_{3}^{'} (t) ⩽ - (m E (0) - c ϵ_{1}) ξ (t) \frac{E (t)}{E (0)} K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}), \end{matrix}$ choose $ϵ_{1}$ such that $m E (0) - c ϵ_{1} > 0$ , we obtain $\begin{matrix} (49) & L_{3}^{'} (t) ⩽ - c ξ (t) \frac{E (t)}{E (0)} K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) . \end{matrix}$ Integrating (49) from $t_{1}$ to t to get, $\begin{matrix} \int_{t_{1}}^{t} c ξ (τ) \frac{E (τ)}{E (0)} K^{'} (\frac{ϵ_{1}}{τ^{\frac{1}{1 + ϵ_{0}}}} \frac{E (τ)}{E (0)}) d τ ⩽ - \int_{t_{1}}^{t} L_{3}^{'} (t) d t ⩽ L_{3} (t_{1}) . \end{matrix}$ Using the properties of $K$ , E and $\forall t ⩾ t_{1}$ , $\begin{array}{rcl} c \frac{E (t)}{E (0)} K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) \int_{t_{1}}^{t} ξ (τ) d τ ⩽ c \int_{t_{1}}^{t} ξ (τ) \frac{E (τ)}{E (0)} K^{'} (\frac{ϵ_{1}}{τ^{\frac{1}{1 + ϵ_{0}}}} \frac{E (τ)}{E (0)}) d τ ⩽ L_{3} (t_{1}) . \end{array}$ Hence, $\begin{matrix} c (\frac{1}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) K^{'} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) \int_{t_{1}}^{t} ξ (τ) d τ ⩽ \frac{c_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} . \end{matrix}$ Setting $K_{1} (τ) = τ K^{'} (ϵ_{1} τ)$ , the above inequality reduces to $\begin{matrix} (50) & c K_{1} (\frac{ϵ_{1}}{t^{\frac{1}{1 + ϵ_{0}}}} \frac{E (t)}{E (0)}) \int_{t_{1}}^{t} ξ (τ) d τ ⩽ \frac{c_{1}}{t^{\frac{1}{1 + ϵ_{0}}}}, \end{matrix}$ after rearranging the terms in (50), we conclude that $\begin{matrix} E (t) ⩽ c t^{\frac{1}{1 + ϵ_{0}}} K_{1}^{- 1} (\frac{c_{1}}{t^{\frac{1}{1 + ϵ_{0}}} \int_{t_{1}}^{t} ξ (τ) d τ}), \forall t ⩾ t_{1} . \end{matrix}$ Hence, the theorem is proved. □
Theorem 4.2.
Let* $(u_{0}, u_{1}) \in H_{0}^{2} (Ω) \times H_{0}^{2} (Ω)$ and $h_{1}$ , B are nonlinear. Under the assumptions of Lemma 4.6 , there exist strictly positive constants c, $ϵ_{1}$ such that the solution to ( 1 ) satisfies, for all $t ⩾ t_{1} = max {t_{0}, 1}$ , $\begin{matrix} (51) & E (t) ⩽ c t^{\frac{1}{1 + ϵ_{0}}} W_{2}^{- 1} (\frac{c}{t^{\frac{1}{1 + ϵ_{0}}} \int_{t_{1}}^{t} ξ (s) d s}), \end{matrix}$ where $W_{2} (t) = t W^{'} (ϵ_{1} t)$ and $W (t) = {({({\bar{B}}^{- 1})}^{\frac{1}{1 + ϵ_{0}}} + {\bar{H}}^{- 1})}^{- 1} (t)$ .
Proof.
Using (38) and Lemma 4.2, observe that $\begin{matrix} L^{'} (t) & ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c \int_{Ω} h^{2} (u_{t}) d x \\ ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c H^{- 1} (G (t)) + c (- E^{'} (t)) . \end{matrix}$ Denote $L_{1} (t) = L (t) + c E (t)$ , then using Remark 4.1 and Lemma 4.3 for all $t ⩾ t_{1}$ the above inequality becomes $\begin{matrix} L_{1}^{'} (t) & ⩽ - m E (t) + c (b \circ Δ u) (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c H^{- 1} (G (t)) \\ ⩽ - m E (t) + c {(b \circ Δ u)}^{\frac{1}{1 + ϵ_{0}}} (t) + c H^{- 1} (G (t)) \\ ⩽ - m E (t) + c t^{\frac{1}{1 + ϵ_{0}}} {[{\bar{B}}^{- 1} (\frac{δ M (t)}{t ξ (t)})]}^{\frac{1}{1 + ϵ_{0}}} + c t^{\frac{1}{1 + ϵ_{0}}} {\bar{H}}^{- 1} (\frac{G (t)}{t^{\frac{1}{1 + ϵ_{0}}}}) \\ ⩽ - m E (t) + c t^{\frac{1}{1 + ϵ_{0}}} {[{\bar{B}}^{- 1} (\frac{δ M (t)}{t^{\frac{1}{1 + ϵ_{0}}} ξ (t)})]}^{\frac{1}{1 + ϵ_{0}}} + c t^{\frac{1}{1 + ϵ_{0}}} {\bar{H}}^{- 1} (\frac{G (t)}{t^{\frac{1}{1 + ϵ_{0}}}}), \end{matrix}$ Denote $\begin{matrix} W (t) = {({({\bar{B}}^{- 1})}^{\frac{1}{1 + ϵ_{0}}} + {\bar{H}}^{- 1})}^{- 1} (t), β (t) = max {\frac{δ M (t)}{t^{\frac{1}{1 + ϵ_{0}}} ξ (t)}, \frac{G (t)}{t^{\frac{1}{1 + ϵ_{0}}}}} . \end{matrix}$ 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 $t ⩾ t_{1}$ , $\begin{matrix} (52) & L_{1}^{'} (t) ⩽ - m E (t) + c t^{\frac{1}{1 + ϵ_{0}}} W^{- 1} (β (t)) . \end{matrix}$ Following the similar steps from Case 2, Theorem 4.1, inequality (51) can be established. □

To illustrate the above theorems, let us consider few examples:
Example 4.1 (If $h_{1}$ is linear and B is linear).

Let $B (t) = t$ , $ξ (t) = 1$ , $ϵ_{0} \in (0, 1)$ and $h_{1} (t) = t$ then inequality (41) gives $\begin{matrix} E (t) ⩽ c {(1 + t)}^{- \frac{1}{ϵ_{0}}} . \end{matrix}$

Example 4.2 (If $h_{1}$ is linear and B is nonlinear).

Let $B (t) = t^{p}$ with $1 < p < \frac{3}{2}$ , $ξ (t) = 1$ and $ϵ_{0} \in (0, 1)$ then we obtain $K (t) = t^{p (1 + ϵ_{0})}$ and $K_{1} (t) = p (1 + ϵ_{0}) t^{p (1 + ϵ_{0})}$ . Using (42) we get $\begin{matrix} E (t) ⩽ c t^{\frac{- (2 + ϵ_{0} - p (1 + ϵ_{0}))}{p {(1 + ϵ_{0})}^{2}}} . \end{matrix}$

Remark 4.3.
In the above mentioned example (i.e., $B (t) = t^{p}$ ), we have restricted the values of p in order to get the decay result. It is worth mentioning that if $p > \frac{3}{2}$ the energy estimate methods discussed in this section will not work and we need to rely on some other techniques.
Example 4.3 (If $h_{1}$ is nonlinear and B is nonlinear).

Let $ξ (t) = 1$ , $B (t) = t^{\frac{4}{3}}$ . Assume $h_{1} (t) = t^{3}$ , so that $H (t) = t^{2}$ . For $ϵ_{0} = \frac{1}{2}$ we get $\begin{matrix} W (t) = t^{2} and W_{2} (t) = c t^{- \frac{5}{4}} . \end{matrix}$ Therefore, applying (51) we obtain, $\begin{matrix} E (t) ⩽ c t^{- \frac{7}{12}} . \end{matrix}$

Footnotes

Acknowledgements

The Authors are very grateful to the anonymous referees for his/her comments and suggestions that greatly helped to improve this manuscript.

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Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity

Abstract

Keywords

1. Introduction

Lemma 2.1 (Logarithmic Sobolev inequality).

Lemma 4.1 (Cf. [3]).

Lemma 4.2 (Cf. [2]).

Lemma 4.3 (Cf. [2]).

Example 4.2 (If h 1 is linear and B is nonlinear).

Footnotes

Acknowledgements

References

Example 4.2 (If $h_{1}$ is linear and B is nonlinear).