Existence result for a nonlinear mixed boundary value problem for the heat equation

In this paper we study the existence of solutions in parabolic Schauder space of a nonlinear mixed boundary value problem for the heat equation in a perforated domain. From a given regular open set $\Omega\subseteq\mathbb{R}^n$ we remove a cavity $\omega\subseteq \Omega$. On the exterior boundary of $\Omega\setminus\overline{\omega}$ we prescribe a Neumann boundary condition, while on the interior boundary we set a nonlinear Robin-type condition. Under suitable assumptions on the data and by means of Leray Schauder Fixed-Point Theorem, we prove the existence of (at least) one solution $u \in C_{0}^{\frac{1+\alpha}{2}; 1+\alpha}([0,T] \times (\overline{\Omega} \setminus \omega))$.