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 ΩA, ΩB and the interface Γ(= ∂ ΩA ∩ ∂ ΩB). We introduce and study some function spaces in the two unbounded domains and the interface. We apply our function spaces and maximal Lp-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 ℝ3+, ℝ3- and the whole space ℝ2, and to make use of nice properties of the heat semigroups and kernels for ℝ3+, ℝ3-, and ℝ2. 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.
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BibTeX
@misc{koba2026global,
title = {{Global solvability of the heat equations in two unbounded domains and the interface}},
author = {Hajime Koba},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28764},
eprint = {2609.28764},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28764},
}
RIS
TY - UNPB
TI - Global solvability of the heat equations in two unbounded domains and the interface
AU - Hajime Koba
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28764
UR - https://arxiv.org/abs/2609.28764
AB - 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(= ∂ Ω_A ∩ ∂ 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 ℝ^3_+, ℝ^3_- and the whole space ℝ^2, and to make use of nice properties of the heat semigroups and kernels for ℝ^3_+, ℝ^3_-, and 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.
ER -