Global existence versus blow-up for a Hardy-Hénon parabolic equation on arbitrary domains

Abstract

We are concentrating on the nonlinear parabolic problem described by the equation $u_t-\Delta u=h\left(t\right)|\cdot {|}^{\gamma }{u}^p$ in $\Omega \times (0,T)$ subject to zero Dirichlet conditions on the boundary ∂Ω, where Ω is a general domain that may be either bounded or unbounded. Here, $h\in C(0,\infty )$, $\gamma >-2$, $p>1$, and we consider only nonnegative initial data. We have derived new conditions for global existence and blow up in finite time in terms of the behavior of the heat semigroup. Our results are particularly relevant when $\gamma =0$, as they align with Meier's findings in Meier (1990) [29]. When $\gamma \ne 0$, our results provide new Fujita exponents.