An approach without using Hardy inequality for the linear heat equation with singular potential

TL;DR

This paper introduces a Hardy inequality-free method using Fourier transform and $PM^{k}$-spaces to establish well-posedness for the linear heat equation with singular potentials, including inverse square singularities, and analyzes long-term solution behavior.

Contribution

It presents a novel approach avoiding Hardy inequalities to prove well-posedness for heat equations with singular potentials in Fourier-based spaces, providing explicit conditions for global solutions.

Findings

Well-posedness established without Hardy inequalities.

Explicit smallness condition for inverse square potential.

Multiple asymptotic behaviors for solutions.

Abstract

The aim of this paper is to employ a strategy known from fluid dynamics in order to provide results for the linear heat equation $u_{t}-\Delta u-V(x)u=0$ in $\mathbb{R}^{n}$ with singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely $PM^{k}$-spaces. For arbitrary data $u_{0}\in PM^{k}$, the approach allows to compute an explicit smallness condition on $V$ for global existence in the case of $V$ with finitely many inverse square singularities. As a consequence, well-posedness of solutions is obtained for the case of the monopolar potential $V(x)=\frac{\lambda}{\left|x\right|^{2}}$ with $\left|\lambda\right|<\lambda_{\ast}=\frac{(n-2)^{2}}{4}$. This threshold value is the same one obtained for the global well-posedness of $L^{2}$-solutions by means of Hardy inequalities and energy estimates. Since there is no any inclusion relation between $L^{2}$ and $PM^{k}$, our results indicate that $\lambda_{\ast}$ is intrinsic of the PDE and independent of a particular approach. We also analyze the long time behavior of solutions and show there are infinitely many possible asymptotics characterized by the cells of a disjoint partition of the initial data class $PM^{k}$.