Energy decay of solutions to the Cauchy problem for a nondissipative wave equation

Abstract

In this paper, we consider energy decay for nondissipative wave equation in unbounded domain (the usual energy is not decreasing). We use an approach introduced by Guesmia, which leads to decay estimates (known in the dissipative case) when the integral inequalities method by Komornik [Exact Controllability and Stabilization. The Multiplier Method. Masson–Wiley, Paris, 1994] cannot be applied due to the lack of dissipativity. First, we study the stability of a wave equation with a weak nonlinear dissipative term based on the equation: \documentclass[12pt]{minimal}\begin{document}$u^{\prime \prime }-\Delta _{x}u+\lambda ^{2}(x) u+\sigma (t) g(u^{\prime })+ \theta (t)h(\nabla _{x} u)=0$\end{document}u′′−Δxu+λ2(x)u+σ(t)g(u′)+θ(t)h(∇xu)=0 in \documentclass[12pt]{minimal}\begin{document}${\mathchoice{\bf \hbox{$\displaystyle \rm I \hspace{-1.5pt}\rm R$}}{\bf \hbox{$\textstyle {\rm I} \hspace{-1.5pt}\rm R$}}{\bf \hbox{$\scriptstyle \rm I\hspace{-1.0pt}\rm R$}}{\bf \hbox{$\scriptscriptstyle \rm I \hspace{-1.0pt}\rm R$}}}^{n}$\end{document}IRn. We consider the general case with a function h satisfying a smallness condition, and we obtain decay of solutions under weak growth assumptions on the feedback function and without any control of the sign of the derivative of the energy related to the above equation. In the second case, we consider the case h(∇u) = −∇Φ∇u. We prove some precise decay estimates of equivalent energy.