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

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 $\mathrm{\mid }x{\mathrm{\mid }}^{\sigma }\mathrm{\mid }u{\mathrm{\mid }}^p$, 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.