| Abstract: |
| We consider a domains $\Omega_a\subseteq\mathbb{R\!}^{\,2}$ with ramified boundary $\Gamma^{\infty}_a$, for $a$ a parameter with $1/2\leq a\leq a^{\ast}\simeq0.593465$. This domain represents an idealization of bronchial trees in the lungs system. Since the exchanges between the lungs and the circulatory system take place only in the last generation of the bronchial trees, an accurate model for diffusion of oxygen may involve inhomogeneous Robin boundary conditions over $\Gamma^{\infty}_a$. Therefore, we investigate the realization of the diffusion equation
$$\frac{\partial u}{\partial t}-\mathcal{A}u+\alpha u\,=\,f(x,t)\,\,\,\,\,\,\,\,\,\,\textrm{in}\,\,\,\Omega_a\times(0,\infty)$$ with mixed boundary conditions $$\frac{\partial u}{\partial\nu_{_{\mathcal{A}}}}+\beta u\,=\,g(x,t)\,\,\,\textrm{on}\,\,\Gamma^{\infty}_a\times(0,\infty);\,\,\,\,\,\,\,\,\,\,\,\,u=0\,\,\,\textrm{in}\,\,\,(\partial\Omega_a\setminus\Gamma^{\infty}_a)\times(0,\infty),$$ and $u(x,0)=u_0\in C(\overline{\Omega}_a)$, where $\mathcal{A}$ stand as a linear (possibly non-symmetric) divergence-type differential operator, $\frac{\partial u}{\partial\nu_{_{\mathcal{A}}}}$ represents a generalized notion of a normal derivative over irregular surfaces, $\alpha\in L^r(\Omega_a)$, $\beta\in L^s_{\mu}(\Gamma^{\infty}_a)^+$ with $\displaystyle{\textrm{ess}\inf_{x\in\Gamma^{\infty}_a}}|\beta(x)|\geq\beta_0$ for some constant $\beta_0>0$ large enough, where $\min\{r,s\}>1$. We show unique solvability of this diffusion equation, and moreover we establish that weak solution of this model equation are globally continuous in space and in time. |
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