On Uniqueness and Stability of Self-Similar Blow-Up Solutions of Nonlinear Heat Equations: Evolution Approach

Abstract

We present evolution arguments of studying uniqueness and asymptotic stability of blow-up self-similar solutions of second-order nonlinear parabolic equations from combustion and filtration theory. The analysis uses intersection comparison techniques based on the Sturm Theorem on zero set for linear parabolic equations. We show that both uniqueness and stability of similarity ODE profiles are directly related to the asymptotic structure of their domain of attraction relative to the corresponding parabolic evolution.