On a Nonuniform Parabolic Equation with Mixed Boundary Condition

Abstract

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 ${u_t}$ is nonnegative and the coefficient matrix $({a_{ij}}(t,x))$ of the elliptic part is not necessarily positive definite. When $b(t,x) \equiv 0$, 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.