AI 中文总结
该研究针对有界域上-1<p<0的极快扩散方程，证明其容许解的最优正则性，进而得到解向友好巨人解的精细长时间渐近行为及改进收敛速率。
AI 中文摘要
我们证明了在-1<p<0范围内、经变换后的极快扩散方程容许解的最优整体正则性，该方程定义在具有零狄利克雷边界条件的光滑有界域上，初始数据与距离函数可比。更确切地说，我们建立了解的存在性与唯一性，证明对每个正时间，解在空间上属于C^{1,p+1}(\overline{\Omega})，且在时间上直至边界均为C^∞类；此外，解的所有时间导数均属于C^{1,p+1}(\overline{\Omega})，且指数p+1是最优的。这些正则性估计进一步导出了解向“友好巨人解”的精细长时间渐近行为，包括在C^{1,p+1}(\overline{\Omega})拓扑下的一阶展开，以及在C^{p+1}(\overline{\Omega})中相对误差的改进收敛速率。
英文摘要
We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range $-1<p<0$, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that solutions belong to $C^{1,p+1}(\overlineΩ)$ in space for every positive time and are $C^\infty$ in time uniformly up to the boundary. Moreover, all their time derivatives belong to $C^{1,p+1}(\overlineΩ)$, and the exponent $p+1$ is optimal. These regularity estimates further yield fine long-time asymptotics toward the friendly giant solution, including a first-order expansion in the $C^{1,p+1}(\overlineΩ)$ topology and an improved convergence rate for the relative error in $C^{p+1}(\overlineΩ)$.
Comments41 pages
