On potential wells and applications to semilinear hyperbolic equations and parabolic equations

In this paper we generalize the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations and parabolic equations u(tt) - Delta u = f(u), x epsilon Omega, t > 0, u(x,0) = u(0)(x), u(t)(x,0) =u(1)(x), x epsilon Omega u(x,t) = 0, x epsilon partial derivative Omega m, t >= 0 and u(t) - Delta u = f(u), x epsilon Omega, t > 0, u(x,0) = u(0)(x), x epsilon Omega, u(x,t) = 0, x epsilon partial derivative Omega, t >= 0, not only give a threshold result of global existence and nonexistence of solutions, but also obtain the vacuum isolating of solutions. Finally we prove the global existence of solutions for above problem with critical initial conditions I (u(0))>= 0, E(0) = d or I (u(0)) >= 0, J(u(0)) = d. So Payne and Sattinger's results are generalized and improved in essential. (c) 2005 Elsevier Ltd. All rights reserved.