Global existence, asymptotic behavior, and uniform attractor for a nonautonomousequation (Q2868883)

In the paper, the authors consider the following nonautonomous linear viscoelastic system NEWLINE\[NEWLINEu_{tt}(x,t)-\Delta u(x,t) + \int_{0}^{t}g(t-s)\;\Delta u(x,s)\;ds+\mu_1 u_t(x,t)+\mu_2 u_t(x,t-\tau)=f(x,t)NEWLINE\]NEWLINE for \((x,t) \in \Omega \times \mathbb{R}^{+}\) with homogeneous Dirichlet boundary condition, initial conditions \(u(x,0)=u_{0}(x)\), \(u_t(x,0)=u_1(x)\) in \(\Omega\) (\(\subset \mathbb{R}^n\) bounded and regular domain), and \(u_t(x,t-\tau)=f_{0}(x,t-\tau)\). \(\mu_1,\mu_2\) are positive constants with \(\mu_1 \leq \mu_2\) and \(\tau>0\) is the time delay. The following conditions are imposed on the relaxation function \(g:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) is a \(C^{1}\) function satisfying \(g(0)>0\), NEWLINE\[NEWLINE1-\int_{0}^{+\infty}g(s)\;ds>0,NEWLINE\]NEWLINE and there exists a positive nonincreasing differentiable function \(\eta(t)\) such that NEWLINE\[NEWLINE-g^{'}(t)\geq\eta (t)g(t)\text{ and }\int_{0}^{+\infty}\eta(t)\;dt=+\infty.NEWLINE\]NEWLINE The data satisfy \(u_0 \in H_{0}^{1}(\Omega)\), \(u_1 \in L^{2}(\Omega)\), \(f_0 \in L^{2}(\Omega \times (0,1))\) and for each \(T>0\), \(f \in L^{2}(0,T;L^{2}(\Omega))\). Under these assumptions, there exists a unique weak solution on \((0,T)\). Using the energy method and suitable Lyapunov functionals, it is shown that the energy of the system \(\mathcal{E}(t)\) tends to zero as \(t \rightarrow +\infty\), if \(\mu_2 \leq \mu_1\), \(f \in L^{2}(\mathbb{R}^{+},L^{2}(\Omega))\) and \(\|f\|_2^{2}/\eta(t) \rightarrow 0\) as \(t \rightarrow \infty\). If, for certain positive constants \(K_0\), \(K_1\), NEWLINE\[NEWLINE\|f\|_{2}^2 \leq K_1\;e^{-K_0\;\int_{0}^{t}\eta(s)\;ds},NEWLINE\]NEWLINE then \(E(t)\) has a similar decay. In the last section of the paper, the authors prove the existence of a uniform attractor.