具有加权函数的发散型反应扩散方程的爆破分析

Blow-Up Analysis of Divergent Reaction Diffusion Equations with Weighted Functions

摘要: 反应扩散方程以建立数学模型的方式解决物理、化学、生物学、传染病学和核科学领域中的实际问题，反应扩散方程可以恰当地描述很多自然现象，例如浓度和密度的扩散以及化学药剂的燃烧等众多自然现象。反应扩散方程是一类非线性方程，其非线性项可能来自边界条件、扩散项、反应项或者是三者所组合的不同形式的耦合关系，所有这些非线性项均可影响爆破解，另外还发现无论解的全局存在性还是爆破性都受到加权项的影响。本文考虑了一类具有加权函数和含时系数的发散型反应扩散方程的爆破问题，难点在于找到位置加权和时间加权项与边界源对于爆破解的影响，给出了非线性函数u(x,t)正解全局存在的条件，利用微分不等式技术，并构造恰当的辅助函数，分别得到了爆破时间的上界和下界。

Abstract: The reaction diffusion equation solves practical problems in the fields of physics, chemistry, biology, infectious disease, and nuclear science by establishing mathematical models. The reaction diffusion equation can appropriately describe many natural phenomena, such as diffusion of concentration and density, and combustion of chemical agents. The reaction diffusion equation is a type of nonlin-ear equation, whose nonlinear terms may come from boundary conditions, diffusion terms, reaction terms, or different forms of coupling between the three. All of these nonlinear terms can affect the explosive solution. In addition, it has been found that both the global existence and explosive prop-erty of the solution are affected by the weighted terms. This article considers the blow up problem of a class of divergent reaction diffusion equations with weighted functions and time-dependent co-efficients. The difficulty lies in finding the effects of position weighted and time weighted terms and boundary sources on the blow up solution. The conditions for the global existence of positive solu-tions to nonlinear functions u(x,t) are given. By using differential inequality techniques and con-structing appropriate auxiliary functions, the upper and lower bounds of the blow up time are ob-tained, respectively.