AI 中文总结
本文研究抽象半线性热方程全局与非全局非负解的存在性，并应用于子黎曼流形和局部有限图，推广了 Fujita 型结果及近期相关研究。
AI 中文摘要
本文研究抽象半线性热方程 $u_t - Lu = \sum_{i=1}^k \ell_i(t) f_i(u)$ 在 $L^p(X, \mu)$ 中全局非负解与非全局非负解的存在性。其中 $(X,\mu)$ 表示一个 $\sigma$-有限测度空间，$L$ 是 $L^p(X,\mu)$ 中由有界线性算子构成的子马尔可夫强连续半群的无穷小生成元，且对每个 $i = 1, \ldots, k$，函数 $\ell_i: [0, \infty) \rightarrow [0, \infty)$ 和 $f_i: [0, \infty) \rightarrow [0, \infty)$ 均为连续函数。我们利用所得结果应用于子黎曼流形和局部有限图上的拉普拉斯算子，获得了 Fujita 型结果。我们的结果推广了 Lenz、Schmidt 和 Zimmermann 近期的工作“抽象半线性热方程非负解的爆破”（Calc. Var. Partial Differential Equations, 2023）。
英文摘要
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 $L^p(X, μ)$. 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 $i = 1, \ldots, k$, 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).
CommentsSubmitted for publication on November 11, 2024. 28 pages
