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 $\sqrt{2n(t_*-t)}$.Ph