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

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\in \mathbb{N}$), written as $$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+(-\mathrm{\Delta }+1)\beta (u)=gin\mathrm{\Omega }\times (0,T),$$ 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)${}_{\epsilon }$ with approximate parameter $\epsilon >0$: [ \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.