Boundary Value Problems
Volume 2007 (2007), Article ID 42954, 51 pages
doi:10.1155/2007/42954

Blow up of the solutions of nonlinear wave equation

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,∞))=∞.