局部有限图上具有幂-对数非线性的热方程正解的寿命
The lifespan of positive solutions of heat equation with power-logarithmic nonlinearity on locally finite graph
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中文总结 AI 辅助
研究局部有限连通图上含幂-对数非线性热方程正解寿命，用Kaplan第一特征值方法和Hu-Wang离散原理建立渐近行为，扩展了相关结果，还表明特定条件下非负初始数据也有类似寿命估计。
中文摘要 AI 辅助
在局部有限连通图\(G=(V,E)\)上，利用Kaplan引入的第一特征值方法以及Hu-Wang发展的离散Phragmén-Lindelöf原理，我们首先建立了具有幂-对数非线性\(u^p|\log u|^q\)的半线性热方程正解寿命的渐近行为，前提是初始数据由一个正常数下界界定。这些结果将Hu-Wang的结果扩展到具有幂-对数源项的方程。此外，通过更直接的论证，我们表明对于非负初始数据\(u(x,0)\)，只要在某个顶点\(x_i\in V\)处\(u(x_i,0)\)足够大，类似的寿命估计仍然有效。
英文摘要
On a locally finite connected graph $G=(V,E)$, using the first eigenvalue method introduced by Kaplan \cite{MR160044} and the discrete Phragmén-Lindelöf principle developed by Hu-Wang \cite{cvhuyuanyang}, we first establish the asymptotic behaviour of the lifespan of positive solutions to a semilinear heat equation with the power-logarithmic nonlinearity $u^p|\log u|^q$, provided that the initial datum is bounded below by a positive constant. These results extend those of Hu-Wang \cite{cvhuyuanyang} to equations with a power-logarithmic source term. Moreover, by means of a more direct argument, we show that analogous lifespan estimates remain valid for nonnegative initial datum $u(x,0)$, provided that $u(x_i,0)$ is suitably large at some vertex $x_i\in V$.
