A necessary and sufficient condition for the global existence of solutions to nonlinear reaction‐diffusion equations on the half‐spaces in ℝN

Abstract

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations where is the half‐space , is a nonnegative continuous function, and is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data if and only if for every . In fact, we introduce a very special curve in to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result.