Non-existence of weak solutions to nonlinear damped wave equations in exterior domains

Takayoshi Ogawa, Hiroshi Takeda

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)

Abstract

We consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain Ω, {(∂t2 u - Δ u + ∂t u = | u |p, t > 0, x ∈ Ω,; u (0, x) = u0 (x), ∂t u (0, x) = u1 (x), x ∈ Ω,; u = 0, t > 0, x ∈ ∂ Ω .) When 1 < p < 1 + frac(2, n) and the initial data (u0, u1) ∈ H01 (Ω) × L2 (Ω) having compact support, we prove the non-existence of non-negative global solutions of the above problem. We employ the Kaplan-Fujita [H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = Δ u + u1 + α, J. Sci. Univ. Tokyo. Sec. I. 13 (1966) 109-124; S. Kaplan, On the growth of solutions quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963) 305-330] method to avoid the difficulty of the reflection from the boundary.

Original languageEnglish
Pages (from-to)3696-3701
Number of pages6
JournalNonlinear Analysis, Theory, Methods and Applications
Volume70
Issue number10
DOIs
Publication statusPublished - 2009 May 15

Keywords

  • Critical exponent
  • Exterior domains
  • Finite time blow-up
  • Nonlinear damped wave equations

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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