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

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain $\cup_{0 \leq t \leq \infty} \Omega(t) \times \{t\} \subset \mathbb{R}^N \times \mathbb{R}$. We are interested in finite energy solution. We derive an exponential decay of the energy in the case Ω(t) is bounded in $\mathbb{R}^N$ and the estimate $\int_0^\infty E(t)dt \leq C(E(0), ||u(0)||) < \infty $ in the case Ω (t) is unbounded. Existence and uniqueness of finite energy solution are also proved.