【作者】 董晓丽；

【导师】 吴勃英；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2015， 硕士

【摘要】 现实生活中,有很多物理、化学、天文领域中的现象以具有非局部边界条件的抛物问题的数学模型为基础,例如热弹性力学,在这种情况下,问题的解通常描述的是物质每单位体积的熵。本文考虑非局部抛物方程和方程组的初边值问题,运用上、下解技术,研究这两个问题解的整体存在和爆破的性质。在第二章中,回顾了本论文研究所需要的偏微分方程的基础知识,包括一些术语和基本概念、几个常用的不等式以及Sobolev空间的定义。这些基本概念将会为课题研究提供必不可少的理论基础。在第三章中,考虑具有非局部源和加权非线性非局部边界条件的非牛顿多方渗流方程的初边值问题。由于扩散项是非线性而且是双退化的,所以使我们的研究面临了新的挑战。非牛顿多方渗流方程同时包涵了牛顿渗流方程和非牛顿渗流方程的情况,所以它的研究方法更加复杂。通过考虑非局部反应项、扩散项中的非线性机制之间的相互作用,以及边界条件中加权函数和非线性指数的不同取值对解的性质的影响,本文得到了该问题的解在什么条件下整体存在以及在有限时刻爆破,并在已有爆破速率估计的基础上,进一步得到了当p=2时爆破解的爆破渐进模式和爆破集。在第四章中,研究具有非局部源和加权线性非局部边界条件的抛物方程组。本章首先给出了该问题上(下)解的定义,并建立该问题的比较原理。然后通过椭圆方程和常微分方程组技术,构造合适的上(下)解,研究了非局部边界条件及非线性扩散项和非局部反应项之间的相互作用对解爆破与否的影响,得到了该问题的解整体存在和在有限时刻爆破所需的条件。p-Laplace更多还原

【Abstract】 In real life, a lot of phenomena in physics, chemistry and astronomy are based on mathematical models of parabolic problems with nonlocal boundary conditions, such as thermoelasticity. In this situation, solutions of the problem usually represent the material entropy per volume. This paper considers the initial boundary value problems of a nonlocal parabolic equation and a system, and studys the global existence and blow-up properties of solutions of the two problems by using the technique of super-solutions and sub-solutions.In chapter two, we first recall some basic knowledge of partial differential equation which our research needs concerning some terminologies and concepts, some frequently-used inequalities and the definition of Sobolev space. These will lay a necessary theoretical foundation for our research.In chapter three, we discuss the initial boundary value problem of nonNewtonian polytropic filtration equation with nonlocal source and weighted nonlinear nonlocal boundary condition. Because the diffusion term is nonlinear and doubly degenerate, our research is faced with new challenges. The nonNewtonian polytropic filtration includes not only the Newtonian filtration equation but also the non-Newtonian filtration equation, so the method for it should be synthetic. By investigating the impacts of the interaction between the nonlocal reaction term and the nonlinear element in the diffusion term and the different values of the weight function and the nonlinear exponent in the boundary condition on the properties of solutions, we obtain the conditions in which the solution of this problem exists globally and blows up. Furthermore, on the foundation of the deserved blow-up rate estimates, we get the blow-up profile and set of the blow-up solutions when p =2.In chapter four, we study the p-Laplacian parabolic system with nonlocal sources and weighted linear nonlocal boundary conditions. First, we define the super-solution and sub-solution of this problem, and establish a modified comparison principle. Then by the methods of the elliptic equations and ordinary differential equations, we construct proper super- solutions and sub-solutions and observe the influence of interaction among the boundary condition, the nonlinear diffusion term and the nonlocal reaction term on blowing up or not of solutions. Moreover, we obtain when the solutions of this problem exist globally and the conditions in which the solutions blow up.更多还原