Nonlinear diffusion equations as asymptotic limits of Cahn--Hilliard systems on unbounded domains via Cauchy's criterion

TL;DR

This paper establishes an abstract theory for nonlinear diffusion equations on unbounded domains, proving existence and uniqueness of solutions without growth restrictions on the nonlinear term, and includes the Stefan problem as a special case.

Contribution

It removes the growth condition on the nonlinear function in nonlinear diffusion equations, extending the applicability to unbounded domains and the Stefan problem.

Findings

Proved existence and uniqueness of solutions without growth restrictions.

Confirmed Cauchy's criterion for approximate solutions.

Included the Stefan problem within the theoretical framework.

Abstract

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g \quad \mbox{in}\ \Omega\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $\beta$ even though the Stefan problem was excluded from examples of (P). This paper completely removes the growth condition for $\beta$ by confirming Cauchy's criterion for solutions of the following approximate problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$: \[ \frac{\partial u_{\varepsilon}}{\partial t} + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon}) + \pi_{\varepsilon}(u_{\varepsilon})) = g \quad \mbox{in}\ \Omega\times(0, T), \] which is called the Cahn--Hilliard system, even if $\Omega \subset \mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be seen that the Stefan problem is covered in the framework of this paper.