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 ut − Lu = ∑i=1k ℓi(t) fi(u) in Lp(X, μ). Here (X,μ) represents a σ-finite measure space, L is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in Lp(X,μ), and for each i = 1, …, k, the functions ℓi: [0, ∞) → [0, ∞) and fi: [0, ∞) → [0, ∞) 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).
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BibTeX
@misc{castillo2026global,
title = {{Global and nonglobal solutions for an abstract semilinear heat equation}},
author = {Ricardo Castillo and Ricardo Freire and Miguel Loayza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04550},
eprint = {2610.04550},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04550},
}
RIS
TY - UNPB
TI - Global and nonglobal solutions for an abstract semilinear heat equation
AU - Ricardo Castillo
AU - Ricardo Freire
AU - Miguel Loayza
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04550
UR - https://arxiv.org/abs/2610.04550
AB - 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 = ∑_i=1^k ℓ_i(t) f_i(u) in L^p(X, μ). Here (X,μ) represents a σ-finite measure space, L is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in L^p(X,μ), and for each i = 1, …, k, the functions ℓ_i: [0, ∞) → [0, ∞) and f_i: [0, ∞) → [0, ∞) 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).
ER -