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

Abstract

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 $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^n$ we remove a cavity $\ddot{I}‰\subseteq \hat{I}\text{c◯}$. On the exterior boundary of $\hat{I}\text{c◯}\setminus \overline{\ddot{I}}‰$ 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 $u\in C_0^{\frac{1+\hat{I}\pm }{2};1+\hat{I}\pm }([0,T]\times (\overline{\hat{I}}\text{c◯}\setminus \ddot{I}‰))$.