A general decay result of a viscoelastic equation with past history and boundary feedback
The authors investigate a viscoelastic initial-boundary value problem \[ \begin{aligned} &u_{tt}(t)-\Delta u(t)+\int_0^{+\infty} g(s)\Delta u(x,t-s)\,ds=0\;\text{in}\;\Omega\times \mathbb{R}^+, \\ &\frac{\partial u}{\partial \nu}(t)-\int_0^{+\infty} g(s)\frac{\partial u}{\partial \nu}(t-s)\,ds+h(u_t(t))=0 \;\text{on}\;\Gamma_1\times \mathbb{R}^+, \\ &u(t)=0,\;\text{on}\; \Gamma_0\times \mathbb{R}^+ , \\ &u(-t)=u_0(t),\;u_t(0)=u_1,\;\text{in}\;\Omega\times \mathbb{R}^+, \end{aligned} \] where \(\Omega\) is bounded in \(\mathbb{R}^N,(N\geq 1)\) with a smooth boundary \(\partial\Omega=\Gamma_0\cap\Gamma_1,\;\Gamma_0\cup\Gamma_1=\emptyset,\;|\Gamma_0|>0\) and \(g,\;h\) are functions satisfying suitable assumptions. The main results has the form \(E(t)\leq k_4W_1^{-1}(k_2t+k_3)\;\text{ for all } t\in \mathbb{R}^+,\) where \[ E(t)=\frac 12 \|u_t(t)\|^2_2+\frac {1-\ell}2\|\nabla u(t)\|^2_2+\frac 12(g\diamond \nabla u)(t) \] is the modified energy, \(\;(g\diamond \nabla u)(t)=\int_0^{+\infty}g(s)\|\nabla u(t)-\nabla u(t-s)\|^2_2\,ds\;\), and \(W_1\) is the specific function.
scientific article; zbMATH DE number 6478440
scientific article