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