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

TL;DR

This paper introduces a novel method for deriving weighted $L^{1}$-estimates for global solutions of the semilinear heat equation with super-critical Fujita exponent, emphasizing explicit computations and asymptotic behavior.

Contribution

It develops a new approach based on direct computation of commutation relations, avoiding traditional parabolic comparison or compactness methods.

Findings

Derived weighted $L^{1}$-estimates for solutions

Provided explicit asymptotic profiles with self-similarity

Established large time behavior of small amplitude solutions

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.