Asymptotic integration of the linear parabolic problem with the high frequency coefficients in the critical case

A concrete (illustrative) example of the linear parabolic problem with two independent variables and high-frequency coefficients in time is considered. The corresponding stationary homogeneous averaged problem is degenerate. Using the method developed recently for an ordinary differential equations [1] and the method of boundary layer a full formal asymptotics of the periodic solution in time is constructed.