Global and nonglobal solutions for an abstract semilinear heat equation

In this paper, we focus on the existence of both global and non-global nonnegative solutions for the abstract semilinear heat equation $u_t-Lu={\sum }_{i=1}^k{\mathrm{\ell }}_i(t)f_i(u)$ 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 $i=1,\dots ,k$, the functions ${\mathrm{\ell }}_i:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ and $f_i:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ 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).