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

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 + 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.