Global in space regularity results for the heat equation with Robin–Neumann type boundary conditions in time-varying domains

This article deals with the heat equation
$$\partial _{t}u-\partial _{x}^{2} u=f\; \text{in}\; D,\; D =\left\{ \left( t,x\right) \in \mathbb{R}^{2}:a<t<b,\psi \left( t\right) <x<+\infty\right\}$$
with the function $\psi$ satisfying some conditions and the problem is supplemented with boundary conditions of Robin-Neumann type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for $f\in L^{2}(D)$ there exists a unique solution $u$ such that $u,\; \partial_{t}u,\; \partial_{x}^{j}u\in L^{2}\left( D\right),j=1,\;2.$ The proof is based on the domain decomposition method. This work complements the results obtained in [10].