论文标题
管中多孔介质方程的相对误差的收敛性
Convergence in relative error for the Porous Medium equation in a tube
论文作者
论文摘要
给定一个有界的域$ d \ subset \ mathbb {r}^n $和$ m> 1 $，我们研究了在管中\ [\ partial_tu =δu=δu=δu=δu^m \ quad \ quad \ quad \ text {in} d \ times \ time \ homog mathbbbbbbbbbbbbbbbbbbbbbbbbb; dirichlet边界条件在边界$ \ partial d \ times \ times \ mathbb {r} $和$ t = 0 $的适当初始基准。在以前的两幅作品中，Vázquez和Gilding＆Goncerzewicz证明，在以对数的时间尺度计算并适当地重新归一化时，一类宽类的解决方案表现出波动波的行为。在本文中，我们表明，在很大程度上，解决方案相对误差与友好的巨人（即，在管的$ d $（带有均匀的dirichlet边界条件）中提出的PME的独特非负解决方案，具有特殊的自相似形式。此外，给出了解决方案自由边界位置的敏捷收敛速率和均匀边界。
Given a bounded domain $D \subset \mathbb{R}^N$ and $m > 1$, we study the long-time behaviour of solutions to the Porous Medium equation (PME) posed in a tube \[ \partial_tu = Δu^m \quad \text{ in } D \times \mathbb{R}, \quad t > 0, \] with homogeneous Dirichlet boundary conditions on the boundary $\partial D \times \mathbb{R}$ and suitable initial datum at $t=0$. In two previous works, Vázquez and Gilding & Goncerzewicz proved that a wide class of solutions exhibit a traveling wave behaviour, when computed at a logarithmic time-scale and suitably renormalized. In this paper, we show that, for large times, solutions converge in relative error to the Friendly Giant, i.e., the unique nonnegative solution to the PME posed in the section $D$ of the tube (with homogeneous Dirichlet boundary conditions) having a special self-similar form. In addition, sharp rates of convergence and uniform bounds for the location of the free boundary of solutions are given.
