Double criticality for a Hardy–Rellich biharmonic heat equation in an exterior domain

Abstract

We study the existence and nonexistence of weak solutions to an inhomogeneous semilinear biharmonic heat equation in an exterior domain involving a Hardy–Rellich potential, a weighted nonlinearity of the form | 𝑥 | 𝜎 ⁢ | 𝑢 | 𝑝 , and a source term 𝑓 ⁡ ( 𝑥 ) . We identify two distinct critical regimes governing the existence and nonexistence of solutions. First, we determine a Fujita-type critical exponent separating the nonexistence and existence regimes. Next, in the supercritical range, we identify a critical exponent associated with the asymptotic behavior of the source term at infinity, in the sense of Lee and Ni. Our results extend the recent work [29] by incorporating a Hardy–Rellich potential and a weighted nonlinearity, which give rise to different critical phenomena.