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 $$u_{t}$$ut and $$-\\Delta u_{t}$$-Δut and a power type of nonlinearity $$\\vert u\\vert ^{p}$$|u|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 $$\\vert u\\vert ^{p}$$|u|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 $$t \\rightarrow \\infty $$t→∞) of our equations considered in exterior domains, and one can see that the usual frictional damping $$u_{t}$$ut is more dominant than the strong one $$-\\Delta u_{t}$$-Δut as $$t \\rightarrow \\infty $$t→∞ even in the nonlinear problem case.