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

In this paper, we study the initial boundary value problem of semilinear wave equations: u(u) - Deltau = \u\(p-1)u, xis an element ofOmega, t > 0, u(x,0) = u(0)(x), u(t)(x,0) = u(1)(x), xis an element ofOmega, u(x,t) = 0, xis an element ofpartial derivativeOmega, tgreater than or equal to0, where Omegasubset ofR(N) is a bounded domain, 1<p<infinity for N = 1, 2; 1<pless than or equal toN+2/N-2 for Ngreater than or equal to3. 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 eis an element of(0, d) (d is the depth of the known potential well) all solutions with initial energy E(0) satisfying 0<E(0)less than or equal toe can only lie either inside of some smaller ball or outside of some bigger ball of space H-0(1)(Omega). (C) 2002 Published by Elsevier Science (USA).