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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

Citations:

Zbl 1421.35203
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