On the Hardy-Hénon heat equation with an inverse square potential

Abstract

We study Cauchy problem for the Hardy-Hénon parabolic equation with an inverse square potential, namely,where , α > 1 and or μuα, . We establish sharp fixed time-time decay estimates for heat semigroups in weighted Lebesgue spaces. This may be of independent interest. As an application, we establish local well-posedness in scale subcritical and critical weighted Lebesgue spaces and small data global existence in critical weighted Lebesgue spaces. Further, under certain conditions on γ and α, we show that local solution cannot be extended to global one for certain initial data in the subcritical regime. Thus, finite time blow-up in the subcritical Lebesgue space norm is exhibited. We also demonstrate nonexistence of local positive weak solution (and hence failure of local well-posedness) in supercritical case for the Fujita exponent.