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arXiv · 2106.06107

On global existence for semilinear wave equations with spacedependent critical damping

Motohiro Sobajima
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Abstract

The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata--Todorova--Yordanov [J.\ Math.\ Soc.\ Japan (2013), 183--236] but this clarifies the precise independence of the location of the support of initial data. The blowup phenomena is verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.

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Motohiro Sobajima. 2021-06-11. On global existence for semilinear wave equations with spacedependent critical damping. https://arxiv.org/abs/2106.06107

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