The 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 $u_t = \Delta u + u^p $ in $R^N $ 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 Schrödinger equation.