Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematics, 1967. / [Typescript]. Includes bibliography.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1981. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 89-91).

Thesis (MScIng)--University of Stellenbosch, 2005. / Bibliography. Also available via the Internet.

In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular initial data. These estimates are obtained by understanding the time decay of norms of solutions. First, we derive regularity results for the heat equation by estimating the decay of Lebesgue norms. Then, we apply similar methods to the Fokker-Planck equation with suitable assumptions on the advection and diffusion. Finally, we conclude by extending our techniques to the porous media equation. The sharpness of our results is confirmed by examining known solutions of these equations. The main contribution of this thesis is the use of functional inequalities to express decay of norms as differential inequalities. These are then combined with ODE methods to deduce estimates for the norms of solutions and their derivatives.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 104-109). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 104-109). Also available on the Internet.

by Chan Chun-hing. / Bibliography: leaves 66-68 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

Building on the work of Longuet-Higgins in 1972 and Calderbank and Macpherson in 2009, we study the combinatorics of symmetric configurations of hyperplanes and points in projective space, called generalized Cox configurations. To do so, we use the formalism of morphisms between incidence systems. We notice that the combinatorics of Cox configurations are closely related to incidence systems associated to certain Coxeter groups. Furthermore, the incidence geometry of projective space P (V ), where V is a vector space, can be viewed as an incidence system of maximal parabolic subalgebras in a semisimple Lie algebra g, in the special case g = pgl (V ) the projective general linear Lie algebra of V . Using Lie theory, the Coxeter incidence system for the Coxeter group, whose Coxeter diagram is the underlying diagram of the Dynkin diagram of the g, can be embedded into the parabolic incidence system for g. This embedding gives a symmetric geometric configuration which we call a standard parabolic configuration of g. In order to construct a generalized Cox configuration, we project a standard parabolic configuration of type Dn into the parabolic incidence system of projective space using a process called parabolic projection, which maps a parabolic subalgebra of the Lie algebra to a parabolic subalgebra of a lower dimensional Lie algebra. As a consequence of this construction, we obtain Cox configurations and their analogues in higher dimensional projective spaces. We conjecture that the generalized Cox configurations we construct using parabolic projection are nondegenerate and, furthermore, any non-degenerate Cox configuration is obtained in this way. This conjecture yields a formula for the dimension of the space of non-degenerate generalized Cox configurations of a fixed type, which enables us to develop a recursive construction for them. This construction is closely related to Longuet-Higgins’ recursive construction of (generalized) Clifford configurations but our examples are more general and involve the extra parameters.

ZelenyaK在一九六八年證明了所有二階擬線性拋物方程的有界全域解都會趨向一個穩態解，而其證明中的一個重要部分就是證明所有這類方程都存在一個數土結構，這是高階方程不定會有的。在這篇論文中，我們會證明Zelenyak 定理，以及找出一個四階、六階方程存在變分結構的充分必要條件。 / Zelenyak proved in 1968 that every bounded global solution of a second order quasilinear parabolic equation converges to a stationary solution. An important part in the proof is that every such equation has a variational structure. For higher order parabolic equations, this is not the case. In this thesis, we prove Zelenyak's theorem and find a necessary and sufficient condition for a fourth or sixth order equation to be variational. / Detailed summary in vernacular field only. / Chan, Hon To Hardy. / "October 2012." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leave 66). / Abstracts also in Chinese. / Introduction --- p.1 / Chapter 1 --- Convergence of Global Solutions of Second Order Parabolic Equations --- p.5 / Chapter 1.1 --- Main result --- p.5 / Chapter 1.2 --- Four auxiliary lemmas --- p.6 / Chapter 1.3 --- Proof of main result --- p.15 / Chapter 1.4 --- An extension to fourth order equations --- p.21 / Chapter 1.4.1 --- An example --- p.25 / Chapter 2 --- The Multiplier Problem for the Fourth Order Equa-tion --- p.28 / Chapter 2.1 --- Introduction --- p.28 / Chapter 2.2 --- Main results --- p.31 / Chapter 2.2.1 --- A necessary and sufficient condition for a variational structure --- p.31 / Chapter 2.2.2 --- An algorithm to check the existence of a variational structure --- p.32 / Chapter 2.3 --- Proof of main results --- p.33 / Chapter 2.4 --- Examples --- p.48 / Chapter 3 --- The Multiplier Problem for the Sixth Order Equa-tion --- p.52 / Chapter 3.1 --- Introduction --- p.52 / Chapter 3.2 --- Main results --- p.55 / Chapter 3.2.1 --- A necessary and sufficient condition for a variational structure --- p.55 / Chapter 3.2.2 --- An algorithm to check the existence of a variational structure --- p.56 / Chapter 3.3 --- Proof of main results --- p.59 / Bibliography --- p.66

First, a regularity property for global solutions of some superlinear parabolic problems is established. We obtain some new a priori estimates on the global classical solutions. Applying this property to the blow-up problem, we obtain a general criterion for the occurrence of blow-up. When applied to the study of global weak solutions, we obtain some regularity results, which answers some open questions in this topic. / In this thesis, we obtain some new results on the formation of singularities for parabolic problems. We are interested in two typical singularities in parabolic evolution problems: blow-up and quenching. / Second, dichotomy properties for some porous medium equations and some semilinear parabolic equations are discussed. Some conditions on universal quenching are also obtained. When the space dimension is one, we establish a new, strong dichotomy property. Bifurcation analysis of some stationary solutions in high dimension is also investigated. / by Zheng Gaofeng. / "June 2005." / Adviser: Chou Kai-Seng. / Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0310. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 84-89). / 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. / Abstract in English and Chinese. / School code: 1307.