The heat equation for the Dirichlet fractional Laplacian with Hardy's potentials: properties of minimal solutions and blow-up

TL;DR

This paper investigates the heat equation involving the Dirichlet fractional Laplacian with Hardy potentials, focusing on minimal solutions' properties and demonstrating blow-up phenomena in supercritical cases.

Contribution

It provides new insights into the behavior of minimal solutions and establishes blow-up results for the fractional heat equation with Hardy potentials.

Findings

Minimal solutions exhibit specific local and global properties.

Nonnegative solutions blow up instantaneously in the supercritical regime.

The study advances understanding of fractional Laplacian equations with singular potentials.

Abstract

Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.