Existence and nonexistence of solutions for the heat equation with a superlinear source term

Abstract

We develop a classification theory for the existence and non-existence of local in time solutions for initial value problems for nonlinear heat equations. By focusing on some quasi-scaling property and its invariant integral, we reveal the explicit threshold integrability of initial data that classifies the existence and nonexistence of solutions. Typical nonlinear terms, for instance polynomial type, exponential type and their sums, products and compositions can be treated as applications.