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 ) b(t,x) of u t {u_t} is nonnegative and the coefficient matrix ( a i j ( t , x ) ) ({a_{ij}}(t,x)) of the elliptic part is not necessarily positive definite. When b ( t , x ) ≡ 0 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.