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Singularity Formation in Nonlinear Heat and Mean Curvature Flow Equations

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation
(also known as reaction-diffusion equation) and mean curvature flow equation.

For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks
like a sphere of radius $\sqrt{2n(t_*-t)}$.

Identiferoai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OTU.1807/26198
Date15 February 2011
CreatorsKong, Wenbin
ContributorsSigal, Israel Michael
Source SetsLibrary and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada
Languageen_ca
Detected LanguageEnglish
TypeThesis

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