Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations

TL;DR

This paper investigates a fourth-order nonlinear degenerate parabolic equation, proving the existence of weak solutions, their positivity, and analyzing long-term behavior including attractors under certain conditions.

Contribution

It introduces a regularization approach to establish solutions and demonstrates positivity and attractor existence for a class of degenerate parabolic equations.

Findings

Existence of at least one weak solution was established.

Weak solutions become strictly positive instantly under added viscosity and conditions.

Trajectory and strong global attractors were shown to exist.

Abstract

A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a-priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is also investigated and the existence of trajectory attractors and, under more restrictive conditions, of strong global attractors is shown.