$(A,\vec 0)$-parabolic equations with a weak degeneracy

For quasilinear parabolic equations, admitting a weak fixed parabolicity degeneracy, one establishes theorems for the existence and the uniqueness of generalized solutions of the general (in particular, the first, second, and third) boundaryvalue problem. One considers in special the case of linear parabolic equations with a nonnegative characteristic form.