具有含时系数的反应扩散方程的爆破问题
大连交通大学, 理学院, 辽宁大连 116028
| 摘 要： | 偏微分方程是以建立数学模型的方式分析和解决现实生活中存在的实际问题，反应扩散方程作为偏微分方程的重要部分之一，反应扩散方程爆破时间的研究不仅具有理论意义，而且具有更加实际的物理意义，尤其是在物理、化学、医学、生物科学和传染病学领域中物质的扩散现象分析等.在现实生活中，热原函数不仅与温度有关，还与时间开关函数k(t)有关，本文考虑了具有齐次Neumann边界条件和含时系数的散度型反应扩散方程的爆破问题.难点在于找到含时系数k(t)的影响以及内吸收项与边界源之间的关系来确定解的爆破.通过构造合适的能量泛函并定义恰当的辅助函数的方法，由凸性论证方法，并结合微分不等式估计和Young等一系列不等式，在k(t)和f(u)适当的假设条件下，计算得到了反应扩散方程的爆破时间. |
| 关 键 词： | 反应扩散方程; 含时系数; 爆破时间 |
| DOI： | 10.57237/j.wjms.2023.02.001 |
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收稿日期: 2023-07-06
接受日期: 2023-08-31
在线出版日期: 2023-09-08
Blow up Problem of Reaction Diffusion Equation with Time-dependent Coefficients
School of Science, Dalian Jiaotong University, Dalian 116028, China
| Abstract: | Partial differential equations are used to analyze and solve practical problems in real life by establishing mathematical models. As an important part of partial differential equations, reaction diffusion equations not only have theoretical significance, but also have more practical physical significance, especially in the fields of physics, chemistry, medicine, bioscience, and infectious disease. In real life, The pyrogen function is not only related to temperature, but also to the time switching function. This paper considers the blowup problem of a divergence type reaction diffusion equation with homogeneous Neumann boundary conditions and time-dependent coefficients. The difficulty lies in finding the influence of time-dependent coefficients and the relationship between the internal absorption term and the boundary source to determine the blowup of the solution. By constructing appropriate energy functional and defining appropriate auxiliary functions, the method of convexity argumentation is used, And combined with differential inequality estimation and a series of inequalities such as Young, the blow up time of the reaction diffusion equation was calculated under appropriate assumptions. |
| Keywords: | Reaction Diffusion Equation; Time Dependent Coefficient; Burst Time |
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