Asymptotically self-similar global solutions of a general semilinear heat equation

Abstract

We consider the general nonlinear heat equation $\partial_t u = \Delta u +a|u|^{p_1-1}u+g(u), u(0)=\varphi,$ on $(0,\infty)\times I\!\!R^n ,$ where $a\in I\!\!R, p_1>1+(2/n)$ and g satisfies certain growth conditions. We prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation. We also prove that some of those global solutions are asymptotic for large time to self-similar solutions of the single power nonlinear heat equation, i.e. with $g\equiv 0.$