Boundary value problems for linear parabolic equations degenerate on the boundary of a region

In the strip $\mathrm{Q\{\,0<t\leqslant T,\ 0<x<\infty\,\}}$ we consider a linear second-order parabolic equation which is degenerate on the boundary $\mathrm{t=0}$, $\mathrm{x=0}$. Assuming that the coefficient of the time derivative has a zero of a sufficiently high order at $\mathrm{t=0}$, we find the sufficient conditions to ensure the correctness of certain boundary value problems. One of these problems occurs in the theory of the temperature boundary layer.