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

TL;DR

This paper proves the existence of solutions for a nonlinear heat equation with mixed boundary conditions in a perforated domain, using fixed-point theory and Schauder estimates.

Contribution

It establishes the existence of solutions for a nonlinear mixed boundary value problem in a perforated domain, extending previous results to more complex boundary conditions.

Findings

Existence of solutions proven using Leray Schauder Fixed-Point Theorem.

Solutions exist in parabolic Schauder space with specified regularity.

Applicable to heat equations with nonlinear Robin boundary conditions.

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 $\Omega\subseteq\mathbb{R}^n$ 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 $u \in C_{0}^{\frac{1+\alpha}{2}; 1+\alpha}([0,T] \times (\overline{\Omega} \setminus \omega))$.