一类带有时变系数的分数阶扩散方程解的爆破性

数学物理学报 ›› 2024, Vol. 44 ›› Issue (5): 1230-1241.

一类带有时变系数的分数阶扩散方程解的爆破性

高晓茹^*(),李建军,徒君

辽宁工程技术大学理学院辽宁阜新 123000

收稿日期:2023-09-18 修回日期:2024-04-28 出版日期:2024-10-26 发布日期:2024-10-16
通讯作者: *高晓茹, E-mail: g12921606482023@163.com
基金资助:
国家自然科学基金(51704140)

Blow-Up of Solutions for a Class of Fractional Diffusion Equations with Time Dependent Coefficients

Gao Xiaoru^*(),Li Jianjun,Tu Jun

College of science, Liaoning Technical University, Liaoning Fuxin 123000

Received:2023-09-18 Revised:2024-04-28 Online:2024-10-26 Published:2024-10-16
Supported by:
NSFC(51704140)

摘要/Abstract

摘要：

该文主要研究了一类带有时间系数的分数阶扩散方程初边值问题解的爆破性. 采用位势阱理论, Nehari 流形, 凹凸性等方法, 结合各种微分不等式技巧, 分别证明了次临界初始能级, 负初始能级和任意正初始能级的情况下解的爆破性, 并且给出了爆破时间的上界估计. 特别地, 由于能量泛函和位势阱深与时变系数 $f(t)$ 有关, 所以在次临界初始能级的情况下, 爆破时间的上界会随着时变系数 $f(t)$ 的变化而变化.

关键词: 分数阶, 时变系数, 位势阱, 爆破

Abstract:

In this paper, the blow-up properties of solutions for a class of fractional diffusion equations with time dependent coefficients is studied. By means of the potential well theory, Nehari manifold, concave conex method, and various differential inequalities, the finite time blow-up of the solutions under subcritical initial energy level, negative initial energy level and any positive initial energy level is discussed.And the upper bound of blow-up time is obtained.In particular, due to the energy functional and the depth of the potential well are related to the time-dependent coefficient $f(t)$, in the case of sub-critical initial energy level, the upper bound of blow-up time will change with $f(t)$.

Key words: Fractional, Time-dependent coefficient, Potential well, Blow-up

中图分类号:

O175.2

高晓茹, 李建军, 徒君. 一类带有时变系数的分数阶扩散方程解的爆破性[J]. 数学物理学报, 2024, 44(5): 1230-1241.

Gao Xiaoru, Li Jianjun, Tu Jun. Blow-Up of Solutions for a Class of Fractional Diffusion Equations with Time Dependent Coefficients[J]. Acta mathematica scientia,Series A, 2024, 44(5): 1230-1241.

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