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Finally, we get various estimates on the rupture set of the solution to the Thin Film type equations. / In the first part of the thesis, we focus on the semilinear equations with supercritical growth, and give upper bounds on the Hausdorff dimension of the singular sets for borderline solution. As a result, we can prove that the positive borderline solution must blow up in finite time. / Secondly, for the semilinear equations with critical growth, we apply a fundamental e-regularity property to illustrate the concentration phenomenon for the positive borderline solution when time goes to infinity. More precisely, we show that the lost energy can be counted exactly by the standard bubbles. / Du, Shizhong. / Adviser: Kai-Seng Chou. / Source: Dissertation Abstracts International, Volume: 70-09, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 113-119). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Shi-Zhong Du. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 56-58). / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- A Nondegenerate Cahn-Hilliard Type Equation --- p.11 / Chapter 2.1 --- A priori estimation --- p.13 / Chapter 2.2 --- Long time existence --- p.19 / Chapter 3 --- The Thin Film Type Equation --- p.31 / Chapter 3.1 --- Positivity for n>4 --- p.31 / Chapter 3.2 --- Improved entropy estimates --- p.36 / Chapter 4 --- Finite Speed of Propagation --- p.43 / Chapter 4.1 --- Finite speed of propagation --- p.44 / Chapter 4.2 --- The regularity of free boundary --- p.54

[Typescript] Includes bibliography.

v, 125 leaves ; 26 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.1976) from the Dept. of Pure Mathematics, University of Adelaide

In this dissertation we study multiscale numerical methods for nonlinear parabolic equations, turbulent diffusion problems, and high contrast parabolic equations. We focus on designing and analysis of multiscale methods which can capture the effects of the small scale locally. At first, we study numerical homogenization of nonlinear parabolic equations in periodic cases. We examine the convergence of the numerical homogenization procedure formulated within the framework of the multiscale finite element method. The goal of the second problem is to develop efficient multiscale numerical techniques for solving turbulent diffusion equations governed by celluar flows. The solution near the separatrices can be approximated by the solution of a system of one dimensional heat equations on the graph. We study numerical implementation for this asymptotic approach, and spectral methods and finite difference scheme on exponential grids are used in solving coupled heat equations. The third problem we study is linear parabolic equations in strongly channelized media. We concentrate on showing that the solution depends on the steady state solution smoothly. As for the first problem, we obtain quantitive estimates for the convergence of the correctors and some parts of truncation error. These explicit estimates show us the sources of the resonance errors. We perform numerical implementations for the asymptotic approach in the second problem. We find that finite difference scheme with exponential grids are easy to implement and give us more accurate solutions while spectral methods have difficulties finding the constant states without major reformulation. Under some assumption, we justify rigorously the formal asymptotic expansion using a special coordinate system and asymptotic analysis with respect to high contrast for the third problem.

[Typescript] / Includes bibliography. / 170 leaves : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematics, 1967

Thesis (Ph.D. 1976) from the Department of Pure Mathematics, University of Adelaide.