Global existence, uniqueness, and asymptotic behavior of solution for the Euler-Bernoulli viscoelastic equation

1. Introduction

This work is concerned with the global existence, uniqueness, and asymptotic behavior of solution for the Euler-Bernoulli viscoelastic equation where $\Omega$ is a bounded domain of $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$, and $\nu$ is the unit outer normal on $\partial\Omega$. Here $g_{1}$ and $g_{2}$ are positive functions satisfying some conditions to be specified later, and $$\begin{equation*} g_{i}\ast\chi(t)=\int_{0}^{t}g_{i}(t-\tau)\chi(\tau)d\tau,\quad i=1,2. \end{equation*}$$ The Euler-Bernoulli equation describes the deflection $u(x,t)$ of a beam (when $n=1)$ or a plate (when $n=2$), where $\Delta^{2}u:=\Delta(\Delta u)=\sum_{j=1}^{n}\left(\sum_{i=1}^{n}u_{x_{i}}u_{x_{i}}\right)_{x_{j}x_{i}},$ and $h$ and $f$ represent the friction damping and the source respectively.
Lange and Menzala [1] considered where $x\in\mathbb{R}^{n}$, $t\geq0$, $a(t)=m(\|\nabla v(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{n}})$ and the real-valued function $m:[0,+\infty)\rightarrow [1,+\infty)$ will be assumed to be of class $C^{1}$ satisfying the condition $m(s)\geq 1+s$ for all $s\geq0$. They remarked that the imaginary part of the solutions of Schr\"{o}dinger's equation $$\begin{equation*} iw_{t}=\Delta w+im\left(\|\nabla(Imw)\|_{L^{2}(\mathbb{R}^{n}}^{2}\right)Rew=0, \end{equation*}$$ are precisely the solutions for (3). Then, using Fourier transform, the existence of global classical solutions and algebraic decay rate were proved for initial data whose regularity depends on the spacial dimension $n$. Messaoudi [2] studied the equation where $a,b>0$, $p,m>2$. He established an existence result for (4) and showed that the solution continued to exist globally if $m\geq p$. If we take the viscoelastic materials into consideration, the model (2) becomes where $g$ is so-called viscoelastic kernel. The term $\int_{0}^{t}g(t-s)\Delta^{2}u(x,s)ds$ describes the hereditary properties of the viscoelastic materials [3]. It expresses the fact that the stress at any instant $t$ depends on the past history of strains which the material has undergone from time $0$ up to $t$. Tatar [4] obtained the property of the energy decay of the model (5) for $h=f=0$ and from this, we know that the term $\int_{0}^{t}g(t-s)\Delta^{2}u(x,s)ds$, similar to the friction damping, can cause the inhibition of the energy. Messaoudi and Mukiawa [5] studied the fourth-order viscoelastic plate equation $$\begin{equation*} u_{tt}(x,t)+\Delta^{2}u(x,t)-\int_{0}^{t}g(t-s)\Delta^{2}u(x,s)ds=0, \end{equation*}$$ in the bounded domain $\Omega=(0,\pi )\times (-l,l)\subset\mathbb{R}^{2}$ with nontraditional boundary conditions. The authors established the well-posedness of the solution and a decay result.
Rivera et al. [6] investigated the plate model: $$\begin{equation*} u_{tt}+\Delta^{2}u-\sigma\Delta u_{tt}+\int_{0}^{t}g(t-s)\Delta^{2} u(s)ds=0, \end{equation*}$$ in the bounded domain $\Omega\subset \mathbb{R}^{2}$ with mixed boundary condition and suitable geometrical hypotheses on $\partial\Omega$. They established that the energy decays to zero with the same rate of the kernel $g$ such as exponential and polynomial decay. To do so in the second case they made assumptions on $g, g'$ and $g''$ which means that $g\simeq(1+t)^{-p}$ for $p>2$. Then they obtained the same decay rate for the energy. However, their approach can not be applied to prove similar results for $1< p\leq2$.
Cavalcanti et al. [7] investigated the global existence, uniqueness and stabilization of energy of $$\begin{equation*} u_{tt}+\Delta^{2}u-\int_{0}^{t}g(t-s)\Delta^{2}u(s)ds+a(t)u_{t}=0 \end{equation*}$$ where $$\begin{equation*} a(t)=M\left(\int_{\Omega}|\nabla u(x,t)|^{2}dx\right) \ \ \mbox {with} \ \ \ M\in C^{1}([0,+\infty)). \end{equation*}$$

By taking a bounded or unbounded open set $\Omega$ where $M(s)>m_{0}>0$ for all $s\geq0$, the authors showed in [7] that the energy goes to zero exponentially, provided that $g$ goes to zero at the same form.

The aim of this work is to study the global existence of regular and weak solutions of problem (1) for the bounded domain, then for $\xi:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ a increasing $C^{2}$ function such that the solution features the asymptotic behavior $$\begin{equation*} E(t)\leq E(0)e^{-\kappa \xi(t)},\quad \forall t\geq0, \end{equation*}$$ where $E(t)$ is defined in (38) and $\kappa$ is a positive constant independent of the initial energy $E(0)$.

2. Preliminaries and main results

We begin by introducing some notation that will be used throughout this work. For functions $u(x,t)$, $v(x,t)$ defined on $\Omega$, we introduce $$(u,v)=\int_{\Omega}u(x)v(x)dx \ \ \mbox {and} \ \ \|u\|_{2}=\left(\int_{\Omega}|u(x)|^{2}dx\right)^{\frac{1}{2}}.$$ Define $$X=\left\{u\in H^{2}_{0}(\Omega);\Delta^{2}u\in L^{2}(\Omega)\right\}$$ Then, $X$ is a Hilbert space endowed with the natural inner product $$(u,v)_{X}=(u,v)_{H^{2}_{0}}+(\Delta^{2}u,\Delta^{2}v).$$ Now let us precise the hypotheses on $g_{1}$ and $g_{2}$.
(H1) $g_{1}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is a bounded function satisfying $$\begin{equation*} g_{1}(t)\in C^{2}(\mathbb{R}^{+})\cap L^{1}(\mathbb{R}^{+}),\quad g_{1}(0)>0. \end{equation*}$$ (H2) There exist positive constants $\alpha_{1},\alpha_{2}$ and $\alpha_{3}$ such that $$\begin{equation*} -\alpha_{1}g_{1}(t)\leq g_{1}'(t)\leq-\alpha_{2}g_{1}(t),\quad \forall t\geq0, \end{equation*}$$ (H3) $$\begin{equation*} 0\leq g_{1}''(t)\leq\alpha_{3}g_{1}(t),\quad \forall t\geq0, \end{equation*}$$ (H4) $g_{2}:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is a bounded function satisfying $$\begin{equation*} g_{2}(t)\in C^{1}(\mathbb{R}^{+})\cap L^{1}(\mathbb{R}^{+}),\quad g_{2}(0)>0. \end{equation*}$$ (H5) There exist positive constants $\eta_{1}$ and $\eta_{2}$ such that $$\begin{equation*} -\eta_{1}g_{2}(t)\leq g_{2}'(t)\leq-\eta_{2}g_{2}(t),\quad \forall t\geq0, \end{equation*}$$ (H6) $$\begin{equation*} 1-\int_{0}^{t}\left(g_{1}(s)+\lambda_{1}^{-1} g_{2}(s)\right)ds=l>0, \end{equation*}$$ where $\lambda_{1}>0$ is the first eigenvalue of the spectral Dirichlet problem $$\begin{equation*} \Delta^{2}u=\lambda_{1}u\quad \ \mbox {in} \ \ \Omega,\quad u=\frac{\partial u}{\partial \nu}=0 \ \ \mbox {in} \ \ \partial\Omega, \end{equation*}$$ $$\begin{equation*} \|\nabla u\|_{2}\leq \frac{1}{\sqrt{\lambda_{1}}}\|\Delta u\|_{2}. \end{equation*}$$

Lemma 1. For $\phi,\psi\in C^{1}([0,+\infty[,\mathbb{R})$ we have $$\begin{equation*} 2\int_{0}^{t}\int_{\Omega}\phi(t-s)\psi\psi'dxds=-\frac{d}{dt}\left((\phi\square\psi)(t) -\int_{0}^{t}\phi(s)ds\|\psi\|^{2}_{2}\right)+(\phi'\square\psi)(t) -\phi(t)\|\phi\|^{2}_{2}, \end{equation*}$$ where $$\begin{equation*} (\phi\square\psi)(t)=\int_{0}^{t}\phi(t-s)\|\psi(t)-\psi(s)\|_{2}^{2}ds. \end{equation*}$$

Theorem 2. Assume that $(H1)-(H6)$ hold, and that $\left\{u_{0},u_{1}\right\}$ belong to $H^{2}_{0}(\Omega)\times L^{2}(\Omega)$. Then, Problem (1) admits a unique weak solution $u$ in the class $$u\in C^{0}([0,\infty);H^{2}_{0}(\Omega))\cap C^{1}([0,\infty);L^{2}(\Omega)).$$ Moreover, for $\xi:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ a increasing $C^{2}$ function satisfying ($\ref{eq6}$) and, if $\|g_{1}\|_{L^{1}(0,\infty)}$ is sufficiently small, we have for $\kappa>0$ $$\begin{equation*} E(t)\leq E(0)e^{-\kappa \xi(t)},\quad \forall t\geq0. \end{equation*}$$

3. Existence of Solutions

In this section we first prove the existence and uniqueness of regular solutions to Problem (1). Then, we extend the same result to weak solutions using density arguments.

3.1. Regular solutions

Let $(w_{j})$ be a Galerkin basis in $X$, and let $V_{m}$ be the subspace generated by the first $m$ vectors $w_{1},...,w_{m}$. We search for a function $$\begin{equation*} u_{m}(t)=\sum_{i=1}^{m}k_{im}(t)w_{i}(x),\ m=1,2,.... \end{equation*}$$ satisfying the approximate Cauchy problem By standard methods in differential equations, we can prove the existence of solutions to the problem $(5)-(6)$ on $[0,t_{m})$ with $0< t_{m}< T$. In order to extend the solution of $(7)-(8)$ to the whole $[0,T]$, we need the following priori estimate.
Estimate 1. Taking $v=2u'_{m}(t)$ in $(7)$, we have Using Lemma 1, we obtain and Inserting Equations (10) and (11) into Equation (9) and integrating over $[0,t]\subset[0, T]$, we obtain By using the fact that $$\begin{eqnarray*} &&(g_{1}\square\Delta u_{m})(t)+(g_{2}\square\nabla u_{m})(t)-\int_{0}^{t}(g'_{1}\square\Delta u_{m})(s)ds-\int_{0}^{t}(g'_{2}\square\nabla u_{m})(s)ds\nonumber\\ &&+\int_{0}^{t}\int_{\Omega}g_{1}(s)|\Delta u_{m}(s)|^{2}dxds+\int_{0}^{t}\int_{\Omega}g_{2}(s)|\nabla u_{m}(s)|^{2}dxds\geq0, \end{eqnarray*}$$ and $$\begin{eqnarray*} \left(1-\int_{0}^{t}g_{1}(s)ds\right)\|\Delta u_{m}(t)\|^{2}_{2}-\left(\int_{0}^{t}g_{2}(s)ds\right)\|\nabla u_{m}(t)\|^{2}_{2}&\geq&\left(1-\int_{0}^{t}\left[g_{1}(s)+\lambda_{1}^{-1} g_{2}(s)\right]ds\right)\|\Delta u_{m}(t)\|^{2}_{2}\nonumber\\ &\geq& l\|\Delta u_{m}(t)\|^{2}_{2}, \end{eqnarray*}$$ Equation (12) yields Taking the convergence of Equation (8) into consideration, we arrive at where $L_{1}=\|u_{1}\|^{2}_{2}+\|\Delta u_{0}\|^{2}_{2}$.
Estimate 2. Firstly, we obtain an estimate for $u''_{m}(0)$ in the $L^{2}$ norm. indeed, setting $v=u''_{m}(0)$ and $t=0$ in Equation (7), we obtain From Equations (8), (14) and (15), it follows that where $L_{2}$ is a positive constant independent of $m\in\mathbb{N}$. Differentiating Equation (7) with respect to $t$, and setting $v=u''_{m}(t)$, we obtain By (H5), H\"{o}lder's inequality and Young's inequality give From Equation (14) we obtain and From (H3), we deduce Inserting Equations (18)-(21) in Equation (17), we get where $$C_{3}=\left[\frac{\eta_{1}^{2}\|g_{2}\|_{L^{1}}}{2}+\frac{[g_{2}(0)]^{2}L_{1}}{2l} +\frac{|g'_{1}(0)|L_{1}}{2l}\right]T+\left[\frac{\eta_{1}^{2}\|g_{2}\|_{L^{1}(0,\infty)}\|g_{2}\|_{L^{\infty} (0,\infty)}}{2}+\frac{\alpha_{1}^{2}\|g_{1}\|_{L^{1}(0,\infty)}\|g_{1}\|_{L^{\infty} (0,\infty)}}{2}\right]\frac{L_{1}T}{l}$$ and $$C_{4}=\frac{|g'_{1}(0)|}{2}+\frac{1}{2}.$$ Using H\"{o}lder's inequality, we know that, for any $\delta>0$, where $$C_{5}=\left[\frac{[g_{1}(0)]^{2}}{4\delta}+\frac{\alpha_{1}^{2}}{4\delta}\|g_{1}\|_{L^{1}(0,\infty)}\|g_{1}\|_{L^{\infty} (0,\infty)}T\right]\frac{L_{1}}{l}.$$ Combining Equation (22) and Equation (23), we get Fixing $\delta>0$, sufficiently small, so that $\frac{1}{2}-2\delta>0$ in Equation (24), and taking into account Equations (8) and (16), we get from Gronwall's Lemma the second estimate, where $L_{3}$ is a positive constant independent of $m\in\mathbb{N}$ and $t\in[0,T]$.
Estimate 3. Let $m_{1}\geq m_{2}$ be two natural numbers, and consider $z_{m}=u_{m_{1}}-u_{m_{2}}$. Then, applying the same way as in the estimate 1 and observing that $\left\{u_{0m}\right\}$ and $\left\{u_{1m}\right\}$ are Cauchy sequence in $X$ and $H^{2}_{0}(\Omega)$, respectively, we deduce for all $t\in[0,T]$.
Therefore, from Equations (24), (25) and (26), we deduce that there exist a subsequence $\left\{u_{\mu}\right\}$ of $\left\{u_{m}\right\}$ and $u$ such that The above convergences (27)-(29) are enough to pass to the limit in Equation (7), to obtain $$\begin{equation*}\label{W} \begin{array}{ll} u''+\Delta^{2}u -\int_{0}^{t}g_{1}(t-s)\Delta^{2}u(s)ds+\int_{0}^{t}g_{2}(t-s)\Delta u(s)ds+u'=0\ \ \mbox {in} \ \ \ L^{\infty}(0,\infty;L^{2}(\Omega)),\\ u(0)=u_{0},\quad u'(0)=u_{1}. \end{array}. \end{equation*}$$ Next, we want to show the uniqueness of solution of (7)-(8). Let $u^{(1)}$, $u^{(2)}$ be two solutions of (7)-(8). Then $z=u^{(1)}-u^{(2)}$ satisfies $$\begin{equation*} z(x,0)=z'(x,0)=0,\quad x\in\Omega, \end{equation*}$$ $$\begin{equation*} z=0,\quad \frac{\partial z}{\partial\nu}=0,\quad x\in\partial\Omega,\ t>0. \end{equation*}$$ Setting $v=2z'(t)$ in (30), then as in deriving (14), we see that Therefore, we have the uniqueness.

3.2. Weak solutions

Let $(u_{0},u_{1})\in H^{2}_{0}(\Omega)\times L^{2}(\Omega)$. Then, since $X\times H^{2}_{0}(\Omega)$ is dense in $H^{2}_{0}(\Omega)\times L^{2}(\Omega)$ there exists $(u_{0\mu},u_{1\mu})\subset X\times H^{2}_{0}(\Omega)$ such that Then, for each $\mu\in\mathbb{N}$, there exists a unique regular solution $u_{\mu}$ of Problem (1) in the class In view of Equation (33) and using an analogous argument to that in Estimate 1 and Estimate 3, we find a sequence $\left\{u_{\mu}\right\}$ of solutions to Problem (1) such that The convergences (33)-(36) are sufficient to pass to the limit in order to obtain a weak solution of Problem (1), which satisfies $$\begin{array}{ll} u''+\Delta^{2}u -\int_{0}^{t}g_{1}(t-s)\Delta^{2}u(s)ds+\int_{0}^{t}g_{2}(t-s)\Delta u(s)ds+u'=0\ \ \mbox {in} \ \ \ L^{2}(0,\infty;H^{-2}(\Omega)),\\ u(0)=u_{0},\quad u'(0)=u_{1}. \end{array}$$ . The uniqueness of weak solutions requires a regularization procedure and can be obtained using the standard method of Visik-Ladyzhenskaya, c.f. Lions and Magenes [8, Chap. 3, Sec. 8.2.2].

4. Asymptotic Behaviour

In this section, we discuss the asymptotic behavior of the above-mentioned weak solutions. Let us define the energy associated to Problem (1) as To demonstrate our decay result, the lemmas below are essential.

Lemma 3. For any $t>0$ $$\begin{equation*} 0\leq E(t)\leq \frac{1}{2}\Big[\|u_{t}(t)\|_{2}^{2}+\|\Delta u(t)\|^{2}_{2}+(g_{1}\square\Delta u)(t)+(g_{2}\square\nabla u)(t)\Big]. \end{equation*}$$

Proof. Using the fact that $\|\nabla u(t)\|^{2}_{2}\leq\lambda_{1}^{-1}\|\Delta u(t)\|^{2}_{2}$, we have $$\begin{eqnarray*} &&\left(1-\int_{0}^{t}g_{1}(\tau)d\tau\right)\|\Delta u(t)\|^{2}_{2}-\left(\int_{0}^{t}g_{2}(\tau)d\tau\right)\|\nabla u(t)\|^{2}_{2}\nonumber\\ &&\quad \geq\left(1-\int_{0}^{t}\left[g_{1}(s)+\lambda_{1}^{-1}g_{2}(s)\right]ds\right)\|\Delta u(t)\|^{2}_{2} \end{eqnarray*}$$ and according to $(H6)$ we have $E(t)\geq0$,
and $$\begin{eqnarray*} E(t)&=&\frac{1}{2}\|u_{t}(t)\|_{2}^{2}+\frac{1}{2}\|\Delta u(t)\|^{2}_{2}+\frac{1}{2}(g_{1}\square\Delta u)(t)+\frac{1}{2}(g_{2}\square\nabla u)(t)\nonumber\\ &&-\frac{1}{2}\left\{\left(\int_{0}^{t}g_{1}(s)ds\right)\|\Delta u(t)\|^{2}_{2}+\left(\int_{0}^{t}g_{2}(s)ds\right)\|\nabla u(t)\|^{2}_{2}\right\}\nonumber\\ &\leq&\frac{1}{2}\Big[\|u_{t}(t)\|_{2}^{2}+\|\Delta u(t)\|^{2}_{2}+(g_{1}\square\Delta u)(t)+(g_{2}\square\nabla u)(t)\Big]. \end{eqnarray*}$$

Lemma 4. The energy $E(t)$ satisfies

Proof. Multiplying the first equation in (1) by $u_{t}$ and integrating over $\Omega$, we obtain $$\begin{equation*} \frac{d}{dt}\left[\frac{1}{2}\|u_{t}(t)\|_{2}^{2}+\frac{1}{2}\|\Delta u(t)\|^{2}_{2}\right]+\|u_{t}(t)\|^{2}_{2} = \int_{0}^{t}g_{1}(t-\tau)\Delta u(\tau)\cdot\Delta u_{t}(t)dxd\tau +\int_{0}^{t}g_{2}(t-\tau)\nabla u(\tau)\cdot\nabla u_{t}(t)dxd\tau. \end{equation*}$$ Exploiting (10)-(11) and by $(H1)-(H5)$, we deduce

From assumptions $(H2)$ and since $\int_{0}^{t}g_{1}'(\tau)d\tau=g_{1}(t)-g_{1}(0)$, we obtain Combining Equation (40) and Equation (41), we conclude that $$\begin{eqnarray*} E'(t)&\leq&-\|u_{t}(t)\|^{2}_{2}-\frac{1}{2}\alpha_{2}(g_{1}\square\Delta u)(t)-\frac{1}{2}\eta_{2}(g_{2}\square\nabla u)(t)\nonumber\\ &&-\frac{1}{2}\left[g_{1}(0)-\alpha_{1}\|g_{1}\|_{L^{1}(0,\infty)}\right]\|\Delta u(t)\|^{2}_{2}\leq0. \end{eqnarray*}$$ Multiplying Equation (39) by $e^{\kappa\xi(t)}$ $(\kappa>0)$ and utilizing Lemma 3, we have Using the fact that $\xi'$ is decreasing we arrive at Choosing $\|g_{1}\|_{L^{1}(0,\infty)}$ sufficiently small so that $$g_{1}(0)-\alpha_{1}\|g_{1}\|_{L^{1}(0,\infty)}=L>0,$$ and choosing $\kappa$ sufficiently small in order to have $$2-\kappa\xi'(0)>0,\quad \alpha_{2}-\kappa\xi'(0)>0,\quad \eta_{2}-\kappa\xi'(0)>,\quad L-\kappa\xi'(0)>0.$$ from Equation (43) we arrive at Integrating the above inequality over $(0,t)$, it follows that

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.