## Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain.(English)Zbl 1463.35336

Summary: In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation $\begin{cases}\partial_t^2u-\Delta u+\partial_tu=f(u)&\text{in }\Omega\times(0,T),\\u=0&\text{on }\partial\Omega\times(0,T),\\u(0)=u_0,\partial_tu(0)=u_1&\text{in }\Omega\end{cases}$ in an exterior domain $$\Omega$$ in $$\mathbb{R}^N$$ $$(N\geq 2)$$, where $$f:\mathbb{R}\to\mathbb{R}$$ is a smooth function which behaves like $$f(u)\sim|u|^p$$. From the view point of weighted energy estimates given by M. Sobajima and Y. Wakasugi [Commun. Contemp. Math. 21, No. 5, Article ID 1850035, 30 p. (2019; Zbl 1421.35203)], the existence of global-in-time solutions with small initial data in the sense of $$\langle{x}\rangle^{\lambda}u_0$$, $$\langle{x}\rangle^{\lambda}\nabla u_0$$, $$\langle{x}\rangle^{\lambda}u_1\in L^2(\Omega)$$ with $$\lambda\in (0,\frac{N}{2})$$ is shown under the condition $$p\geq 1+\frac{4}{N+2\lambda}$$. The lower and upper bounds for the lifespan of blowup solutions with small initial data $$(\varepsilon u_0,\varepsilon u_1)$$ are also given.

### MSC:

 35L20 Initial-boundary value problems for second-order hyperbolic equations 35B33 Critical exponents in context of PDEs

### Keywords:

lifespan; weighted energy estimate; blow-up

Zbl 1421.35203
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