一类具有对数非线性源项的分数阶 $p$-Laplace 扩散方程解的存在性和爆破

摘要/Abstract

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摘要：

该文研究了一类具有对数非线性源项的分数阶 $p$ -Laplace 扩散方程的初边值问题. 文中利用 Galerkin 近似、势阱理论和 Nehari 流形的方法证明了方程在亚临界状态和临界状态下解的全局存在性, 然后通过构造辅助函数、应用微分不等式给出了解在有限时间内爆破的一些充分条件.

关键词: 分数阶 $p$ -Laplace 方程, Galerkin 近似, 全局解, 微分不等式, 爆破

Abstract:

The paper study the initial-boundary value problem for a class of fractional $p$ -Laplace diffusion equation with logarithmic nonlinearity. Using the Galerkin approximation, potential well theory and Nehari manifold methods, the global existence of solutions in subcritical and critical states is proven. Then, by constructing auxiliary functions and applying differential inequality techniques, the existence of blow-up solutions in finite time is established.

Key words: fractional $p$ -Laplacian, Galerkin approximation, the solutions for global existence, differential inequality, blow-up

中图分类号:

O175.23

李建军,李阳晨. 一类具有对数非线性源项的分数阶 $p$ -Laplace 扩散方程解的存在性和爆破[J]. 数学物理学报, 2025, 45(2): 465-478.

Jianjun Li,Yangchen Li. The Existence and Blow-Up of Solutions for a Class of Fractional $p$ -Laplace Diffusion Equation with Logarithmic Nonlinearity[J]. Acta mathematica scientia,Series A, 2025, 45(2): 465-478.

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