Global solvability of the heat equations in two unbounded domains and the interface

Abstract

This paper considers the existence of a unique global-in-time strong solution to the heat equations in two unbounded domains $Ω_A$, $Ω_B$ and the interface $Γ(= \partial Ω_A \cap \partial Ω_B)$. We introduce and study some function spaces in the two unbounded domains and the interface. We apply our function spaces and maximal $L^p$-regularity for Hilbert space-valued functions to show the existence of a local-in-time strong solution to our heat equations. By using an energy equality of our heat system, we prove the existence of a unique global-in-time strong solution to the system with large initial data when the slope of the interface is gentle and our parameters are limited. The key ideas for showing the existence of our strong solutions are to transform our system into a system of equations in two half spaces $\mathbb{R}^3_+, \mathbb{R}^3_-$ and the whole space $\mathbb{R}^2$, and to make use of nice properties of the heat semigroups and kernels for $\mathbb{R}^3_+$, $\mathbb{R}^3_-$, and $\mathbb{R}^2$. In Appendix (I), we derive our heat equations in the two unbounded domains and the interface from an energetic point of view. In Appendix (II), we study representation formulas for differential operators on unbounded domains and surfaces.