GLOBAL WELL-POSEDNESS OF SEMILINEAR HYPERBOLIC EQUATIONS, PARABOLIC EQUATIONS AND SCHRODINGER EQUATIONS

Abstract

This article studies the existence and nonexistence of global solutions to the initial boundary value problems for semilinear wave and heat equation, and for Cauchy problem of nonlinear Schr¨odinger equation. This is done under three possible initial energy levels, except the NLS as it does not have comparison principle. The most important feature in this article is a new hypothesis on the nonlinear source terms which can include at least eight important and popular power-type nonlinearities as special cases. This article also finds some kinds of divisions for the initial data to guarantee the global existence or finite time blowup of the solution of the above three problems.