Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain

TL;DR

This paper investigates the existence and nonexistence of solutions to a biharmonic heat equation with singular potential and weighted nonlinearity in an exterior domain, revealing two distinct critical regimes.

Contribution

It extends previous work by analyzing a Hardy-Rellich potential and weighted nonlinearity, identifying two critical exponents governing solution behavior.

Findings

Identified a Fujita-type critical exponent for existence versus nonexistence.

Discovered a second critical exponent related to the decay of the source term.

Extended prior results to include singular Hardy-Rellich potentials and weighted nonlinearities.

Abstract

We study the existence and nonexistence of weak solutions to an inhomogeneous semilinear biharmonic heat equation in an exterior domain, involving a singular Hardy--Rellich potential, a weighted nonlinearity $|x|^{\sigma}|u|^{p}$, and a positive source term $f(x)$. We identify two distinct critical regimes governing the behavior of solutions. More precisely, we first determine a Fujita-type critical exponent that separates nonexistence from existence. We then show that, in the supercritical range, a second critical exponent arises in terms of the decay exponent of the source, in the sense of Lee and Ni. Our results extend the recent work \cite{Tobakhanov} by considering a singular Hardy--Rellich potential and a weighted nonlinearity, leading to a different critical behavior.