We consider a domain \(\Omega \subseteq \mathbb {R\!}^{\,2}\) with branched fractal boundary \(\Gamma ^{\infty }\) and parameter \(\tau \in [1/2,\tau ^{*}]\) introduced by Achdou and Tchou [6], for \(\tau ^{*}\simeq 0.593465\) , which acts as an idealization of the bronchial trees in the lungs systems. For each \(\tau \in [1/2,\tau ^{*}]\) , the corresponding region \(\Omega \) is a non-Lipschitz domain, which attains its roughest structure at the critical value \(\tau =\tau ^{*}\) 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 \(\begin{aligned} \frac{\partial u}{\partial t}-{\mathscr {A}} u+{\mathscr {B}}u\,=\,f(t,x)\,\,\,\,\,\,\,\,\text {in}\,\,\,(0,\infty )\times \Omega \end{aligned}\) with inhomogeneous mixed-type boundary conditions \(\begin{aligned} \displaystyle \frac{\partial u}{\partial \nu _{_{{\mathscr {A}}}}}+\beta u\,=\,g(x,t)\,\,\,\,\,\,\,\text {on}\,\,(0,\infty )\times \Gamma ^{\infty },\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,u=0\,\,\,\,\,\,\,\text {in}\,\,\,(0,\infty )\times (\partial \Omega \setminus \Gamma ^{\infty }), \end{aligned}\) and \(u(x,0)=u_0\in C({\overline{\Omega }})\) , where \({\mathscr {A}}\) is an uniformly elliptic second-order (non-symmetric) differential operator with bounded measurable coefficients, \({\mathscr {B}}\) is as a lower-order (non-symmetric) differential operator with unbounded measurable coefficients, \(\displaystyle \frac{\partial u}{\partial \nu _{_{{\mathscr {A}}}}}\) stands as a generalized notion of a normal derivative over rough surfaces (in the sense of Definition 4), and \(\beta \in L^s_{\mu }(\Gamma ^{\infty })^+\) with \(\displaystyle {\text {ess}\inf _{x\in \Gamma ^{\infty }}}|\beta (x)|\ge \beta _0\) for a sufficiently large constant \(\beta _0>0\) , and \(s>1\) . Under minimal assumptions, we first show that the stationary version of the above diffusion equation is uniquely solvable, and that the corresponding weak solution in globally Hölder continuous on \({\overline{\Omega }}\) . Since we are including the critical case \(\tau =\tau ^{*}\) , 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 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 \(T>0\) .
