论文标题:退化抛物方程的若干问题
Problems about the Nonlinear Degenerate Parabolic Equation
论文作者
论文导师 赵俊宁,论文学位 博士,论文专业 基础数学
论文单位 厦门大学,点击次数 1053,论文页数 68页File Size2162K
2007-04-01论文网 http://www.lw23.com/lunwen_77762/
Degenerate parabolic equation ;; Renormalized solutions ;;Asymptotic behavior;; Global existence
本文讨论了非线性退化抛物方程的几个问题，全文分为三部分． 第一章讨论下面非线性奇异抛物方程的初边值问题重整化解的存在性及惟一性这里f∈L~1(Q)，g∈(L~(p")(Q))~N，p"=p/(p-1)，a(u，▽u)对|▽u|满足p-1次增长条件和单调性条件．此类问题来源于化学反应扩散问题．一般地，这类方程不存在弱解，原因是a(u，▽u)不属于(L_(loc)~1)~N且由于g∈(L~(p")(Q))~M，H(u)(f+divg)的意义不明确．为了克服这些困难，我们在本章利用重整化解的理论讨论比通常弱解更弱的解的存在性，即重整化解的存在性．重整化解的概念是Lions和Di Perna在研究Boltzmann方程中提出的，后来被应用到一类非线性抛物方程．本章的贡献在于利用重整化解的理论和技巧克服了由H(u)(f+divg)项带来的困难，证明了其重整化解的存在与惟一性． 第二章是研究p-Laplace方程ut=div(|▽u|~(p-2)▽u)和带有的吸收项的p-Laplace方程ut=div(|▽u|~(p-2)▽u)-u~qCauchy问题和Dirichlet问题弱解u_p(x，t)当→∞时的渐近极限性质．对于Cauchy问题，Evans[21]对初值u_0(x)具有紧支集情形讨论了弱解的渐近极限，明确给出弱解u_p(x，t)当p→∞时的渐近极限；当初值u_0(x)不具有紧支集时，易青和赵俊宁教授[34]证明了存在{u_p)的子列{u_(pj)}和函数u_∞∈C(R~N)，使得对任意紧集G(?)Q_T有在G上一致成立．在本章，我们改进了上述结果，对Cauchy问题和Dirichelet问题证明了解序列{u_p}极限的唯—性，从而给出解序列的整体渐进性质，即：在R~N的任何紧集G上一致成立． 第三章首先对L_(loc)~1(R~N)初值和强非线性热源的p-Laplace方程的Cauchy问题讨论解的局部存在性．证明了当sup_x∈R~N(∫B_ρ(x)|u_0(y)|~hdy)~(1/h)<∞，其中当qp-1+p/N时，h>N/p(q-p+1)，则所论问题存在局部解． 本章还对具有特殊扩散系数的p-Laplace方程讨论了解的整体存在性及解的性质．得到结果如下：设10，0<αλ_(N,p)，12，则存在有限时间T~*依赖于N，p，λ，|Ω|，使得问题(*)的解u(x，t)有，
In this paper we study some nonlinear degenerate parabolic equations. In the first chapter,we discuss the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equationsb(u)_t-div(a(u,▽u))=H(u)(f+divg),where f∈L~1(Q), g∈(L~(p")(Q))~N, p" =p/(p-1), a(u,▽u) satisfies p - 1 powers increasing conditions for|▽u|. These problems are motivated by control problems arising in chemical reactions. Under these assumptions, this problem does not admit, in general, a weak solution, since the fields a(u,▽u) do not belong to (L_(loc)~1)~N and the meaning of the term H(u)(f + divg) is not clear. To overcome this difficulty, we use in this paper the framework of renormalized solutions. This notion was introduced by Lions and Di Perna for the study of Boltzmann equation. And many people applies this notion to evolution problems in fluid mechanics. In this paper, we first give a suitable formulation of the problem to overcome the difficulty that the term H(u)(f + divg) brings , then the existence and uniquess of weak solution are proved. In the second chapter, we discuss the asymptotic behavior of the solution to the p-Laplacian equationand the p-Laplacian equation with absorptionfor the Cauchy problem and the Dirichlet initial-boundary value problem as p→∞. For the Cauchy problem, when the initial value u_0(x) has compact support, the same problem has been studied by Evans et al.[21], where some refined results are obtained. For the case, when the initial value u_0(x) has no compact support, the following result was proved in [34], there exists a subsequence {u_(pj)} of {u_p} and a function u_∞∈C(R~N), such that for any compact set G (?) Q_T uniformly on G. In this chapter, we improve the above results and study Dirichlet problem. We proved the the asymptotic limit of the solution is uniquess and obtained the results:(?) up(x,t) = u_∞(x) uniformly on G. The third chapter is devoted firstly to the local existence of the solution to the Cauchy problem of the p-Laplacian equation with strongly nonlinear sources when the initial value u_0(x)∈L_(loc)~1(R~N),We proved whenas qp -1 +p/N,h>N/p(q- p+1), the solution to the Cauchy problem exists localy . In this chapter, we also discuss the global existence of the solution to the Dirichlet initial-boundary value problem of the p-Laplacian equation with particular coefficient The following results we obtained . Let 1 < p < N,λ> 0, 0 <α< N, u_0(x)∈L~∞(Ω), onΩ, u_0(x)≥0. Denoteλ_(N,p) = ((N-p)/p)~p. Theorem 1 Let u_0∈W~(1,p)(Ω),λ<λ_(N,p), for any 1 < p < N, then the problem (*) exists a global solution. Theorem 2 Letλ<λ_(N,p) ,1T~* Theorem 4 Let Letλ≤μ_(N,p),12, there exists a finite time T~*, depending only upon N，p，λ，|Ω|, such that the solution u(x, t) of the problem (*),
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