Global solvability for the heat equations in two half spaces and an interface

Abstract

This paper considers the existence of a global-in-time strong solution to the heat equations in the two half spaces $\mathbb{R}^3_+(=\mathbb{R}^2 \times (0,\infty))$, $\mathbb{R}^3_-(= \mathbb{R}^2 \times (-\infty ,0))$, and the interface $\mathbb{R}^2 \times \{ 0 \} (\cong \mathbb{R}^2)$. We introduce and study some function spaces in the two half spaces and the interface. We apply our function spaces and the 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. The key idea of constructing strong solutions to our system is 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, we derive our heat equations in the two half spaces and the interface from an energetic point of view.