Abstract
We consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain Ω , { â t 2 u â Î u + â t u = | u | p , t > 0 , x â Ω , u ( 0 , x ) = u 0 ( x ) , â t u ( 0 , x ) = u 1 ( x ) , x â Ω , u = 0 , t > 0 , x â â Ω . When 1 < p < 1 + 2 n and the initial data ( u 0 , u 1 ) â H 0 1 ( Ω ) Ă L 2 ( Ω ) having compact support, we prove the non-existence of non-negative global solutions of the above problem. We employ the KaplanâFujita [H. Fujita, On the blowing up of solutions of the Cauchy problem for u t = Î u + u 1 + α , J. Sci. Univ. Tokyo. Sec. I. 13 (1966) 109â124; S. Kaplan, On the growth of solutions quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963) 305â330] method to avoid the difficulty of the reflection from the boundary.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have