Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary

The self-similar expanding curve for the curvature flow equation

“…By applying a comparison principle for nonlinear equations, we are able to derive the existence and uniqueness of the self-similar expanding curve. Furthermore, the stability of this self-similar expanding curve can be proved by the same argument as the one given by Kohsaka in [18] (see also [12]). …”

“…Then the behavior of the interface can be studied. In other methods, one seeks for a parametrization [19], graph representation [1,2,15,18,20,21], or auxiliary function [7] of the interface and then studies some alternative equations coming from (1.1).…”

“…They proved that such graphs also converge to some surface which moves at a constant speed. In [18], Kohsaka considered one-dimensional graphs evolving by a general class of quasilinear equations on a domain with one free end. In [20,21], Ninomiya and Taniguchi considered one-dimensional graph representations of interfaces on the whole real line for F (κ, ν) = κ + c, where c is a nonzero constant.…”

“…In [18], the author obtained the existence, uniqueness, and asymptotic stability of the self-similar expanding curve when θ 1 < θ 2 . For other related problems, we refer the readers to the papers of Caffarelli-Vazquez [5], Galaktionov-Hulshof-Vazquez [10], Hilhorst-Hulshof [14], and Vazquez [24].…”

On a two-point free boundary problem

“…We note that, by a reflection, this case is the symmetric case of our two-point free boundary problem with β 1 = β 2 = 0 and −α 2 = α 1 = π/4. For the case with α 1 , α 2 ∈ (0, π/2), we refer the readers to [11].…”