# On a One-Phase Stefan Problem

created by scianna on 02 Sep 2012

[BibTeX]

Preprint

Inserted: 2 sep 2012
Last Updated: 2 sep 2012

Year: 2012
Notes:

Work in progress

Abstract:

We are interested in the following one-phase Stefan problem: $\partial_t(u+\chi)=\triangle u+f\chi$, $u(0)=g$, $\chi\in \mathcal{H}(u(t))$, where $u$ is defined on $[0, +\infty)\times\mathbb{R}^n$, both $g$ and $f$ are non-negative time-independent functions and $\mathcal{H}:\mathbb{R}\rightarrow\mathcal{P}([0, 1])$ is the usual maximal monotone Heavyside graph. We introduce the notion of w-weak and of weak solution. Adapting suitably the variational techniques introduced in a paper of Tilli and Scianna (see $[5]$) we get a discrete in time approximation scheme involving, step by step, one-phase Hele-Shaw problems, the approximate solutions having a-priori estimates which pass to the limit; existence, uniqueness, monotonicity and regularity results for w-weak and weak solutions are then obtained, at last.

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