Diffusion over ramified domains: solvability and fine regularity

Abstract

We consider a domain $\Omega\subseteq\mathbb{R\!}^{\,2}$ with branched fractal boundary $\Gamma^{\infty}$ and parameter $\tau\in[1/2,\tau^{\ast}]$ introduced by Achdou and Tchou \cite{ACH08}, for $\tau^{\ast}\simeq 0.593465$, which acts as an idealization of the bronchial trees in the lungs systems. For each $\tau\in[1/2,\tau^{\ast}]$, the corresponding region $\Omega$ is a non-Lipschitz domain, which attains its roughest structure at the critical value $\tau=\tau^{\ast}$ in such way that in this endpoint parameter the region $\Omega$ fails to be an extension domain, and its ramified boundary $\Gamma^{\infty}$ is not post-critically finite. Then, we investigate a model equation related to the diffusion of oxygen through the bronchial trees by considering the realization of a generalized diffusion equation with inhomogeneous mixed-type boundary conditions. Under minimal assumptions, we first show that the stationary version of the above diffusion equation in uniquely solvable, and that the corresponding weak solution in globally H\"older continuous on $\overline{\Omega}$. Since we are including the critical case $\tau=\tau^{\ast}$, this is the first time in which global uniform continuity of weak solutions of a Robin-type boundary value problem is attained over a non-extension domain. Furthermore, after two transitioning procedures, we prove the unique solvability of the inhomogeneous time-dependent diffusion equation, and we show that the corresponding weak solution is globally uniformly continuous over $[0,T]\times\overline{\Omega}$ for each fixed parameter $T>0$.