Critical exponent for semi-linear wave equations with double damping terms in exterior domains

Abstract

In this paper, we consider wave equations with double damping terms expressed by $u_{t}$ and $-\Delta u_{t}$ and a power type of nonlinearity $\vert u\vert ^{p}$. We are concerned with mixed problems for these equations in exterior domains of a bounded obstacle. A main purpose is to determine a so-called critical exponent of the power p of the nonlinearity $\vert u\vert ^{p}$. In particular, in the two dimensional case, our results are optimal, and the critical exponent is given by the Fujita one. This shows a parabolic aspect (as $t \rightarrow \infty$) of our equations considered in exterior domains, and one can see that the usual frictional damping $u_{t}$ is more dominant than the strong one $-\Delta u_{t}$ as $t \rightarrow \infty$ even in the nonlinear problem case.