This paper discusses the existence of weak solutions for an initial boundary-value problem of a nonuniform second order parabolic equation in which the coefficient b(t, x) of 22 is nonnegative and the coefficient matrix" (a . .(t, x)) of the elliptic part is not necessarily positive definite. When b(t, x) = 0, this problem is reduced to a degenerate elliptic system. A discussion of the existence of weak solutions for the degenerate elliptic boundary-value problem from the parabolic system is included.