Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation

Abstract

We present a new method to obtain weighted $L^1$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in ${\mathbb{R}}^n$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.

arXiv comment30 pages