Existence of strong global attractors for hyperbolic equation with linear memory
The authors consider the following nonlinear integro-differential equation which describes a homogeneous and isotropic viscoelastic solid: \[ \begin{cases} u_{tt}+ \alpha u_t - K(0)\Delta u -\int^\infty_0 {K^ \prime}(s) \Delta u(t-s)ds + g(u)= f,&\text{in }\Omega\times \mathbb{R}^+,\\ u (x,t) = 0, &x \in \Gamma,\;\;t\in{\mathbb{R}}, \\ u (x,t) = u_0 (x,t), &x\in \Omega,\;\;t\leq 0, \end{cases}\tag{1} \] where \(\Omega\) is an open and bounded subset of \({\mathbb {R}}^n \) with smooth boundary \( \Gamma.\) \( \alpha > 0\) is a given constant; and \(K^\prime (s) \leq 0 \) for all \(s\in {\mathbb{R}^+},\) and \(K(0), K(\infty)\) are positive. \(g\in C^2 ({\mathbb {R}}),\) and \(f\in H^1_ 0 (\Omega). \) Under suitable conditions, and using semigroup theory, the authors prove the existence of a global attractor for problem (1).
scientific article; zbMATH DE number 2117761