Global and nonglobal solutions for an abstract semilinear heat equation

Abstract

In this paper, we focus on the existence of both global and non-global nonnegative solutions for the abstract semilinear heat equation $u_t - Lu = \sum_{i=1}^k \ell_i(t) f_i(u)$ in $L^p(X, \mu)$. Here $(X,\mu)$ represents a $\sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,\mu)$, and for each $i = 1, \ldots, k$, the functions $\ell_i: [0, \infty) \rightarrow [0, \infty)$ and $f_i: [0, \infty) \rightarrow [0, \infty)$ are continuous. We utilize our findings to apply to the Laplacian operator on sub-Riemannian manifolds and locally finite graphs, obtaining Fujita-type results. Our results extend the recent work of Lenz, Schmidt, and Zimmermann,"Blow-up of nonnegative solutions of an abstract semilinear heat equation"(Calc. Var. Partial Differential Equations, 2023).