On inhomogeneous heat equation with inverse square potential

TL;DR

This paper investigates the inhomogeneous heat equation with an inverse square potential, establishing decay estimates, local and global well-posedness results, and analyzing asymptotic behavior of solutions in Lebesgue spaces.

Contribution

It extends analysis of heat equations to include inverse square potentials and inhomogeneous nonlinearities, providing new decay estimates and well-posedness results.

Findings

Established fixed-time decay estimates for the associated semigroup.

Proved local and global well-posedness in critical Lebesgue spaces.

Analyzed asymptotic behavior using self-similar solutions.

Abstract

We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^{\alpha}u,\] where $\mathcal{L}_a=-\Delta + a |x|^{-2}.$ We establish some fixed-time decay estimate for $e^{-t\mathcal{L}_a}$ associated with inhomogeneous nonlinearity $|\cdot|^{-b}$ in Lebesgue spaces. We then develop local theory in $L^q-$ scaling critical and super-critical regime and small data global well-posedness in critical Lebegue spaces. We further study asymptotic behaviour of global solutions by using self-similar solutions, provided the initial data satisfies certain bounds. Our method of proof is inspired from the work of Slimene-Tayachi-Weissler (2017) where they considered the classical case, i.e. $a=0$.