Global in time results for a parabolic equation solution in non-rectangular domains

This article deals with the parabolic equation
$$ \partial _{t}w-c(t)\partial_{x}^{2} w=f \text{in} D, D=\left\{(t,x)\in\mathbb{R}^{2}:t>0, \varphi_{1} \left( t\right)<x<\varphi_{2}(t)\right\} $$
with $\varphi_{i}: [0,+\infty[\rightarrow \mathbb{R}, i=1, 2$ and $c: [0,+\infty[\rightarrow \mathbb{R}$ satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin 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 $w$ such that $w, \partial _{t}w, \partial ^{j}w\in L^{2}(D), j=1, 2.$ Notice that the case of bounded non-rectangular domains is studied in [9]. The proof is based on energy estimates after transforming the problem in a strip region combined with some interpolation inequality. This work complements the results obtained in [Sad2] in the case of Cauchy-Dirichlet boundary conditions.