Classical solutions for a thin–film equation

Abstract

The main part of the thesis provides existence, uniqueness and regularity for the 1-d thin-film equation with linear mobility. The equation is viewed as a classical free boundary problem. The focus is laid on the blow up situation near the free boundary. The strategy is based on a priori energy type estimates which provide minimal conditions on the initial data such that a unique global solution exists. As a result, smoothness of the solution is obtained as well as the large time behavior of the free boundary. The second part of the thesis is concerned with Schauder estimates for a related degenerate parabolic linear operator of fourth order. The last part of the thesis provides an optimal lower bound for solutions to 1-d thin-film equations whenever the initial data are almost flat.