Role of critical exponents in blowup theorems

Abstract

In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation ut = Δu + uP in RN with p > 1 and nonnegative initial values. Fujita showed that if 1 < p < 1 + 2/N, then the initial value problem had no nontrivial global solutions while if p > 1 + 2/N, there were nontrivial global solutions. This paper discusses similar results for other geometries and other equations including a nonlinear wave equation and a nonlinear Schrodinger equation.