Optimal regularity and fine asymptotics for very fast diffusion equations in bounded domains

Abstract

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 $C^{p+1}(\overlineΩ)$.