论文标题
最佳的规律性和有界域中多孔介质方程的精细渐近学
Optimal regularity and fine asymptotics for the porous medium equation in bounded domains
论文作者
论文摘要
我们证明，在某些等待时间$ t^*$之后，在平滑界面域中，非负解决方案对多孔介质方程的最佳全局规律性。更准确地说，我们表明解决方案是空间上的$ c^{2，α}（\overlineΩ）$，带有$α= \ frac1m $，以及$ c^\ infty $ in Time（以$ t> t^*$为$ x \ in \ in \ inline occ. in \ in \ inlineoccyplineΩ$）。此外，这使我们能够在很大程度上完善解决方案的渐近学，以两种方式改善最著名的结果：我们建立了更快的收敛速度$ O（t^{ - 1-1-γ}）$，并且我们证明这种收敛性在$ c^{1，α}（α}（α}}（\ operlinepolline）中）$ topoly $ topolopoly。
We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time $T^*$. More precisely, we show that solutions are $C^{2,α}(\overlineΩ)$ in space, with $α=\frac1m$, and $C^\infty$ in time (uniformly in $x\in \overlineΩ$), for $t>T^*$. Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence $O(t^{-1-γ})$, and we prove that the convergence holds in the $C^{1,α}(\overlineΩ)$ topology.
