Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

Abstract

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain $\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}$. We are interested in finite energy solution. We derive an exponential decay of the energy in the case $\Omega(t)$ is bounded in $\mathbb{R}^N$ and the estimate $\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)\lt \infty$ in the case $\Omega(t)$ is unbounded. Existence and uniqueness of finite energy solution are also proved.