论文标题:几类非线性抛物方程（组）解的若干性质
On Some Properties of Solutions to Several Types of Nonlinear Parabolic Systems
论文作者
论文导师 穆春来,论文学位 博士,论文专业 应用数学
论文单位 四川大学,点击次数 142,论文页数 92页File Size2216K
2006-04-05论文网 http://www.lw23.com/lunwen_20989087/
nonlinear parabolic equations(system); interfaces; local-ization property; uniform boundedness; absorption; localization reaction;global existence; blow-up; blow-up rate
偏微分方程理论的飞速发展以及它在实践中的广泛应用使得抛物方程(组)基础理论的研究显得日益重要。本文研究了四类非线性抛物方程(组)解的性质,例如解的支集的性质,解的一致有界性,解的整体存在性,有限blow-up性,blow-up速率等。 全文共分五章。第一章是本文概述,叙述了本文所研究问题的实际背景,目前的发展现状以及本文的主要内容。 第二章讨论了一类在非均匀介质中带有吸收项的退化抛物方程的Cauchy问题解的支集性质。在高维情形下,我们研究了解的局部性和渐近性,以及解在有限时刻熄灭(extinction)问题;在一维空间里,我们讨论了解的交接面在有限时刻消失问题。 第三章研究一类有一般结构的局部化源的反应扩散方程ut = ?u+f(u(x_0(t),t))齐次Dirichlet初边值问题整体解的一致有界性问题。分两种情形进行讨论,即在高维区域考虑固定源(x0(t)恒为区域中的一固定点)情形下整体解的一致有界性和在一维区域里研究移动源情形下整体解的一致有界性。 第四章讨论具有非线性局部化源的抛物方程组齐次Dirichlet初边值问题解的爆破性质,讨论解的整体存在性和blow-up性的条件,blow-up速率,渐近估计以及边界层等。 第五章研究非线性局部化源的另一类反应扩散方程组:ut =△u+
The rapid development of the theory of partial di?erential equationsand the prolific growth of its applications have made it importance for usto study the basic theory of parabolic equations (systems). The propertiesof solutions to the four types of nonlinear parabolic equations (systems)have been studied in this dissertation, such as, support properties, uniformboundedness, global existence, finite-time blowup, and blow-up rate, etc. The thesis consists of five chapters. The first chapter is devote to thesummary of the dissertation, which recounts the background, the devel-opment, and the present circumstance about related problems. Moreover,main results of the present paper are stated. In chapter 2, we study the support properties of solutions to theCauchy problem on the following equation:ρ(|x|)ut = div(um?1|Du|λ?1Du)?f(u). We deal with localization property and asymptotic behavior of solu-tions to this type of equation in several dimensional spaces. On the otherhand, we study the appearance of interfaces in one dimensional space. In chapter 3, we investigate the uniform boundedness of global so-lutions to the equation with general structure localization reaction:ut =?u + f(u(x0(t),t)). We consider the problem in two cases: fixed sourceand moving source. For the former, we show that all global solutions areuniformly bounded in dimension n; for the latter, we also prove the corre-sponding conclusion holds in dimension 1. In chapter 4, we consider the blow-up properties of the solution to thefollowing parabolic system with localization reaction : ut = ?u + up1(x0,t)
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