Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches

Abstract

This paper is devoted to some aspects of well-posedness of the Cauchy problem for a quasilinear degenerate fourth-order parabolic thin film equation u_{t} = -\nabla \cdot(|u|^{n} \nabla\D u) in \ren \times \re_+, \quad u(x,0)=u_0(x) in \ren, where $n>0$ is a fixed exponent, with bounded smooth compactly supported initial data. Dealing with the CP (for, at least, $n \in (0, \frac 32)$) requires introducing classes of infinitely changing sign solutions that are oscillatory close to finite interfaces. The main goal of the paper is to detect proper solutions of the CP for the degenerate TFE--4 by uniformly parabolic analytic $\e$-regularizations at least for values of the parameter $n$ sufficiently close to 0.