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 $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ we remove a cavity $\omega \subseteq \mathrm{\Omega }$. On the exterior boundary of $\mathrm{\Omega }\setminus \bar{\omega }$ 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 $u\in C_0^{\frac{1+\alpha }{2};1+\alpha }([0,T]\times (\bar{\mathrm{\Omega }}\setminus \omega ))$.