Degenerate diffusion behaviour at the boundary and kernel estimates 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.