Degenerate diffusion - behaviour at the boundary and kernel estimates

Abstract

We study evolution equations of the form: \begin{equation*} \frac{\partial u}{\partial t}(t,x)=m(x)(\triangle u)(t,x)\qquad t\in\R_+,\,x\in\Omega, \end{equation*} where $\Omega$ is a bounded domain in $\R^N$ and the function $m:\Omega\rightarrow (0,\infty)$ is assumed to be measurable. Dirichlet boundary conditions are posed. We investigate under which conditions on $m$ and $\partial\Omega$ the operator $m\triangle$ generates a strongly continuous semigroup on $C_0(\Omega)$. In the second part of the thesis we obtain various estimates on the kernel of the semigroup generated by $m\triangle$ on weighted $L^p$-spaces.