Boundary Value Problems
Volume 2007 (2007), Article ID 42954, 51 pages
doi:10.1155/2007/42954
Research Article
Blow up of the Solutions of Nonlinear Wave Equation
Department of Differential Equations, University of Sofia, Sofia 1164, Bulgaria
Received 14 March 2007; Accepted 26 May 2007
Academic Editor: Peter Bates
Copyright © 2007 Svetlin Georgiev Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We construct for every fixed n≥2 the metric gs=h1(r)dt2−h2(r)dr2−k1(ω)dω12−⋯−kn−1(ω)dωn−12, where h1(r), h2(r), ki(ω), 1≤i≤n−1, are continuous functions, r=|x|, for which we consider the Cauchy problem
(utt−Δu)gs=f(u)+g(|x|), where x∈ℝn, n≥2;
u(1,x)=u∘(x)∈L2(ℝn), ut(1,x)=u1(x)∈H˙−1(ℝn), where f∈𝒞1(ℝ1), f(0)=0, a|u|≤f′(u)≤b|u|, g∈𝒞(ℝ+), g(r)≥0, r=|x|, a and b are positive constants.
When g(r)≡0, we prove that the above Cauchy problem has a nontrivial
solution u(t,r) in the form u(t,r)=v(t)ω(r) for which limt→0‖u‖L2([0,∞))=∞.
When g(r)≠0, we prove that the above Cauchy problem has a nontrivial solution u(t,r) in the form u(t,r)=v(t)ω(r) for which limt→0‖u‖L2([0,∞))=∞.