  We study the initial-boundary value problem for a class of diffusion
equations with nonmonotone diffusion flux functions, including forward-backward
parabolic equations and the gradient flows of nonconvex energy functionals,
under the framework of partial differential inclusions using the method of
convex integration and Baire's category. In connection with rank-one convex
hulls of the corresponding matrix sets, we introduce a structural condition on
the diffusion flux function, called Condition (OC), and establish the
nonuniqueness and instability of Lipschitz solutions to the initial-boundary
value problem under this condition.
