Solvability boundary value problem for degenerate quasilinear parabolic equation in a domain with nontube boundary

In this paper, we consider a quasilinear parabolic equation in nontube domain, which degenerate on a solution. We suppose the essential boundedness of the derivative of the function that define the curvilinear boundary, and prove an existence and uniqness theorems for the first boundary-value problem. We use compactness methods for functions from Banach space scale. At the Preliminary part establish abstract theorems about completeness certain system of function in nontube domain.