Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2023-12-18 , DOI: 10.1007/s00028-023-00932-9 Yi C. Huang , Hatem Zaag
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
$$\begin{aligned} \frac{\partial h}{\partial t}=\frac{\partial ^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^\beta },\quad \beta >0 \end{aligned}$$subject to initial boundary value conditions
$$\begin{aligned} h(\cdot ,0)=h_0>0,\quad h(\pm 1,t)=1. \end{aligned}$$We derive the intermediate extinction profile with refined asymptotics, and with extinction time T and extinction point 0, the gradient profile behaves as \(x\rightarrow 0\) like
$$\begin{aligned} \lim _{t\rightarrow T}\,(\nabla h)(x,t)\quad \sim \quad \frac{1}{\sqrt{2\beta }}\frac{x}{|x|}\frac{1}{\sqrt{|\log |x||}} \left[ \frac{(\beta +1)^2}{8\beta }\frac{|x|^2}{|\log |x||}\right] ^{\frac{1}{\beta +1}-\frac{1}{2}}, \end{aligned}$$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.
中文翻译：
II型超导体中涡线与边界重新连接的梯度分布
在最近的一项工作中，Duong、Ghoul 和 Zaag 确定了（应该是）一般情况下具有功率非线性的标准半线性热方程的爆炸解的梯度分布。他们的方法改进了 Bricmont 和 Kupiainen 引入并由 Merle 和 Zaag 进一步发展的构造技术。在本文中，我们将其改进扩展到平面近似下涡线与 II 型超导体中边界重新连接的问题，这是由 Chapman、Hunton 和 Ockendon 推导的物理模型，具有非线性热方程的有限时间淬灭特征
$$\begin{对齐} \frac{\partial h}{\partial t}=\frac{\partial ^2 h}{\partial x^2}+e^{-h}-\frac{1}{ h^\beta },\quad \beta >0 \end{对齐}$$受初始边界值条件影响
$$\begin{对齐} h(\cdot ,0)=h_0>0,\quad h(\pm 1,t)=1。 \end{对齐}$$我们通过精细渐近推导出中间消光剖面，并且消光时间为 T 和消光点 0，梯度剖面表现为 就像 \(x\rightarrow 0\)
$$\begin{对齐} \lim _{t\rightarrow T}\,(\nabla h)(x,t)\quad \sim \quad \frac{1}{\sqrt{2\beta }}\frac {x}{|x|}\frac{1}{\sqrt{|\log |x||}} \left[ \frac{(\beta +1)^2}{8\beta }\frac{| x|^2}{|\log |x||}\right] ^{\frac{1}{\beta +1}-\frac{1}{2}}, \end{对齐}$$与 Merle 和 Zaag 先前得出的消光剖面的梯度一致。我们的结果适用于一般边界条件和更高维度。
京公网安备 11010802027423号