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

Abstract

We study the initial boundary value problem for semilinear hyperbolic equations and parabolic equations with convex function nonlinear terms. By introducing a family of potential wells we not only obtain the invariant sets and vacuum isolation of solutions, but also give a threshold result of global existence and nonexistence of solutions. Moreover we discuss the global existence of solution for problem with critical initial condition E(0)=d (or J(u0)=d), which fills some important gaps regarding this problem. Finally we prove the global nonexistence of solution and asymptotic behavior of solution for semilinear parabolic equations.