## Abstract

We consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain Ω, {(∂_{t}^{2} u - Δ u + ∂_{t} u = | u |^{p}, t > 0, x ∈ Ω,; u (0, x) = u_{0} (x), ∂_{t} u (0, x) = u_{1} (x), x ∈ Ω,; u = 0, t > 0, x ∈ ∂ Ω .) When 1 < p < 1 + frac(2, n) and the initial data (u_{0}, u_{1}) ∈ H_{0}^{1} (Ω) × L^{2} (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 u_{t} = Δ u + u^{1 + 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 language | English |
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Pages (from-to) | 3696-3701 |

Number of pages | 6 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 70 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2009 May 15 |

## Keywords

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