Properties of Green's function of third mixed problem for parabolic equation in non-cylindrical domain (Q1603106)

This paper deals with the problem for the parabolic second-order equation \[ \begin{aligned} {\partial u\over\partial t}& -\sum^n_{i,j=1} {\partial\over \partial x_i}a_{ij} (t,x){\partial u\over\partial x_j} =f(t,x),\;(t,x)\in \Omega, \\ v_0 u& -\sum^n_{i,j=1} a_{ij}(t,x) {\partial u\over\partial x_j}\nu_j= \nu_0\varphi(x),\;(t,x)\in\Gamma, \end{aligned}\tag{1} \] where \(\Gamma\) is the boundary of \(\Omega=\{t>0\} \times\{x\in \mathbb{R}^n\}\), \(\nu=(\nu_0,\nu_1, \dots, \nu_n)\) is the unit vector of the exterior (with repsect to \(\Omega)\) normal to \(\Gamma\). Under suitable assumptions on \(f\), \(a_{ij}\) the authors construct and study the properties of Green's function corresponding to (1).