Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Time-dependent Damping

Abstract

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{@{}ll} u_{tt} - \Delta u + b(t)u_t=f(u)\,, & (t,x) \in {\bf R}^+ \times {\bf R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x)\,, & x \in {\bf R}^N\,. \end{array}\right. {(*)} $$ hen $b(t)=(t+1)^{-\beta}$ with $0\le \beta <1$, the damping is effective and the solution $u$ to ($*$) behaves as that to the corresponding parabolic problem. When $f(u)=O(|u|^{\rho})$ as $u \to 0$ with $1<\rho < \frac{N+2}{[N-2]_+}$(the Sobolev exponent), our main aim is to show the time-global existence of solutions for small data in the supercritical exponent $\rho>\rho_F(N):=1+2/N$. We also obtain some blow-up results on the solution within a finite time, so that the smallness of the data is essential to get global existence in the supercritical exponent case.