一类带局部化源项的非局部扩散问题解的性质

发布时间：2018-06-21 09:45

本文选题：非局部扩散 + 局部化源项　；参考：《东南大学》2015年硕士论文

【摘要】：非线性偏微分方程(组)解的性质一直以来都是非线性分析和偏微分方程这两个研究领域讨论的一个重要内容.生物学、化学和物理学等应用学科中的很多数学模型也都与这些方程紧密相关.随着科学技术的日新月异和数学研究方法的日臻完善,非线性偏微分方程(组)的形式越来越多样.近年来,非局部扩散方程以及推广而来的一系列非局部扩散问题,引起了很多科学工作者的兴趣和关注.这些非局部扩散问题被广泛地用来描述扩散的进程,其中u(x,t)可以用来表示某个物种在t时刻在点x处的密度,J(x-y)代表从点y移动到点x的概率分布,卷积(J+u)(x,t)=∫RN J(x-y)u(y,t)dy表示物种从其它点到达点x的速度.本论文研究如下带有局部化源项的非局部扩散方程组解的性质,包括解的存在性与唯一性,解的整体存在与有限时刻爆破,解的两个分量u和v发生同时爆破和不同时爆破的条件以及解的爆破速率的估计等问题.在本文中,我们首先运用压缩映像原理证明了解的存在性和唯一性；然后通过建立新的比较原理,并利用上下解方法导出了解在有限时刻发生爆破的条件；接着利用一些常用的不等式和分析的技巧并借鉴文献[1]中的方法,我们得到：u和v在有限时刻T同时爆破的充分条件和必要条件,并进一步讨论了u和u的爆破模式与爆破点集的刻画；最后借助于比较原理和常微分方程不等式等方法和技巧,导出了u和v在有限时刻T不同时爆破的充分条件和必要条件以及爆破速率的估计和爆破点集的刻画.
[Abstract]:The properties of solutions of nonlinear partial differential equations have always been an important part of nonlinear analysis and partial differential equations. Many mathematical models in applied disciplines such as biology, chemistry and physics are also closely related to these equations. With the rapid development of science and technology and the improvement of mathematical research methods, the forms of nonlinear partial differential equations are becoming more and more diverse. In recent years, nonlocal diffusion equations and a series of generalized nonlocal diffusion problems have attracted the interest and attention of many scientists. These nonlocal diffusion problems are widely used to describe the diffusion process, in which the density of a species at point x can be used to represent the probability distribution of moving from point y to point x. Convolution J ~ (U) ~ (X) T ~ (1) = ~ (1) RN ~ (J) ~ (x) -Y ~ (+) ~ = the velocity of species reaching point x In this paper, we study the properties of solutions of nonlocal diffusion equations with localized source terms, including the existence and uniqueness of solutions, the global existence of solutions and the finite time blow-up. The conditions for simultaneous and non-simultaneous blasting of the two components u and v of the solution and the estimation of the blow-up rate of the solution are discussed. In this paper, we first prove the existence and uniqueness of the solution by using the contraction mapping principle, and then establish a new comparison principle and derive the conditions for the solution to burst at finite time by using the method of upper and lower solutions. Then, by using some commonly used techniques of inequality and analysis and using the method in [1], we obtain the sufficient and necessary conditions for the simultaneous blow-up of T at finite time. Furthermore, the characterization of the blow-up mode and the set of blasting points for u and u are discussed, and finally, by means of comparison principle and ordinary differential equation inequality and other methods and techniques, In this paper, the sufficient conditions and necessary conditions for the simultaneous blasting of u and v at finite time T are derived, as well as the estimation of the blasting rate and the characterization of the set of blasting points.
【学位授予单位】：东南大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29

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