Evolved from the author's lectures at the University of Bonn's Institut für angewandte Mathematik, this book reviews recent progress toward understanding of the local structure of solutions of degenerate and singular parabolic partial differential equations.

A method is presented for obtaining explicit upper and lower pointwise bounds for the solution of rather general interior boundary value problems. The differential equations associated with these problems are of the elliptic type in certain sections while both linear and non-linear parabolic equations are the subject of investigation in other sections. The bounds which are obtained are in terms of the integrals of the squares of known functions and hence, in the linear case, improvement is possible using the Rayleigh-Ritz technique.

ABSTRACT: This work is focused on questions of solvability of initial-boundary value problems for degenerate parabolic partial differential equations, and on the investigation of differential and certain qualitative properties of solutions of such equations. The generalized solvability of initial-boundary value problems is proved for linear equations with a degeneracy of parabolicity at the boundaries by employing numerical solutions obtained from finite difference numerical schemes. Furthermore, the regularity of generalized solutions is studied.

We adapt the Levi's parametrix method to prove existence, estimates and qualitative properties of a global fundamental solution to ultraparabolic partial differential equations of Kolmogorov type. Existence and uniqueness results for the Cauchy problem are also proved.

In this paper methods are derived for obtaining aribitrarily close upper and lower bounds for the solutions of Dirichlet and exterior Neumann problems for general systems of second order elliptic partial differential equations definted on arbitrary domains.