## Departmental PhD Thesis Exam – Wenbin Kong

Monday, April 26, 2010, 3-4 p.m., in BA 6183, 40 St. George St.

PhD Candidate:  Wenbin Kong

Thesis title: Singularity formation in nonlinear heat and
mean curvature flow equations
(http://www.math.toronto.edu/kongwenb/thesis.pdf)

Thesis Abstract:

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 a certain family
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 to the standard
$n$-dimensional sphere in the Sobolev norm with the index greater than
$\frac{n}{2}+1$, 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 $\sqrt{2n(t_*-t)}$.