Long-time behavior of solutions to the generalized Allen–Cahn model with degenerate diffusivity

The generalized Allen–Cahn equation, ut=ε2(D(u)ux)x-ε22D′(u)ux2-F′(u),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t=\varepsilon ^2(D(u)u_x)_x-\frac{\varepsilon ^2}{2}D'(u)u_x^2-F'(u), \end{aligned}$$\end{document}with nonlinear diffusion, D=D(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D = D(u)$$\end{document}, and potential, F=F(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F = F(u)$$\end{document}, of the form D(u)=|1-u2|m,orD(u)=|1-u|m,m>1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D(u) = |1-u^2|^{m}, \quad \text {or} \quad D(u) = |1-u|^{m}, \quad m >1, \end{aligned}$$\end{document}and F(u)=12n|1-u2|n,n≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(u)=\frac{1}{2n}|1-u^2|^{n}, \qquad n\ge 2, \end{aligned}$$\end{document}respectively, is studied. These choices correspond to a reaction function that can be derived from a double well potential, and to a generalized degenerate diffusivity coefficient depending on the density u that vanishes at one or at the two wells, u=±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u = \pm 1$$\end{document}. It is shown that interface layer solutions that are equal to ±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 1$$\end{document} except at a finite number of thin transitions of width ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} persist for an either exponentially or algebraically long time, depending upon the interplay between the exponents n and m. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg–Landau type are derived.