Decay rates for dissipative wave equations
This paper is concerned with decay rates for the energy of the solutions of the wave equation with a nonlinear internal damping \[ \begin{cases} u''-\Delta u+g(u')=0 \quad & \text{in }\Omega \times(0,\infty),\\ u=0\quad & \text{in }\Gamma\times (0,\infty),\\ u(0)=u^0,\;u'(0)=u^1 \quad & \text{in }\Omega, \end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\Gamma=\partial \Omega\), and \(g\) is an increasing function on \(\mathbb{R}\) such that \(g(0)=0\). The author succeeded in weakening the growth condition of \(g\) near the origin \(c|s|^r\leq |g(s)|\leq C|s|^{1/r}\), for \(|s|\leq 1\) in the existing results to the conditions: for some constant \(q>0\) \(|g(s)|\leq C|s|^q\), for \(|s|\leq 1\), and there exists an increasing and convex function \(\varphi\) defined on \([0,\infty)\) and twice differentiable outside \(s=0\) such that \(\varphi(s^2)\leq g(s)s\) on \([-1,1]\). As an example it is shown that a logarithmic decay \(E(t)\leq c_1/\log(c_2t)\) holds in the case \(g(s)=s^3e^{-1/s^2}\), for \(|s|\leq 1\).
scientific article; zbMATH DE number 1370121