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On potential wells and vacuum isolating of solutions for semilinear wave equations

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
192
Issue
1
Identifiers
DOI: 10.1016/s0022-0396(02)00020-7
Keywords
  • Potential Wells
  • Semilinear Wave Equations
  • Global Solutions
  • Existence
  • Vacuum Isolating
Disciplines
  • Mathematics

Abstract

Abstract In this paper, we study the initial boundary value problem of semilinear wave equations: utt−Δu=|u|p−1u,x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(x,t)=0,x∈∂Ω,t⩾0,where Ω⊂RN is a bounded domain, 1<p<∞ for N=1,2; 1<p⩽N+2N−2 for N⩾3. First, by using a new method, we introduce a family of potential wells which include the known potential well as a special case. Then by using it, we obtain some new existence theorems of global solutions, and prove that for any e∈(0,d) (d is the depth of the known potential well) all solutions with initial energy E(0) satisfying 0<E(0)⩽e can only lie either inside of some smaller ball or outside of some bigger ball of space H01(Ω).

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