On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions

We consider initial boundary value problem for uniformly $2$-parabolic differential operator of second order in cylinder domain in ${\mathbb R}^n$ with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo–Galerkin method we prove that problem has unique solution in special Bochner space.