Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors

In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation ∂h∂t=∂2h∂x2+e-h-1hβ,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial h}{\partial t}=\frac{\partial ^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^\beta },\quad \beta >0 \end{aligned}$$\end{document}subject to initial boundary value conditions h(·,0)=h0>0,h(±1,t)=1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} h(\cdot ,0)=h_0>0,\quad h(\pm 1,t)=1. \end{aligned}$$\end{document}We derive the intermediate extinction profile with refined asymptotics, and with extinction time T and extinction point 0, the gradient profile behaves as x→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow 0$$\end{document} like limt→T(∇h)(x,t)∼12βx|x|1|log|x||(β+1)28β|x|2|log|x||1β+1-12,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{t\rightarrow T}\,(\nabla h)(x,t)\quad \sim \quad \frac{1}{\sqrt{2\beta }}\frac{x}{|x|}\frac{1}{\sqrt{|\log |x||}} \left[ \frac{(\beta +1)^2}{8\beta }\frac{|x|^2}{|\log |x||}\right] ^{\frac{1}{\beta +1}-\frac{1}{2}}, \end{aligned}$$\end{document}agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions.