  We study some non-parabolic diffusion problems in one-space dimension, where
the diffusion flux exhibits forward and backward nature of the Perona-Malik,
H\"ollig or non-Fourier type. Classical weak solutions to such problems are
constructed in a way to capture some expected and unexpected properties,
including anomalous asymptotic behaviors and energy dissipation or allocation.
Specific properties of solutions will depend on the type of the diffusion flux,
but the primary method of our study relies on reformulating diffusion equations
involved as an inhomogeneous partial differential inclusion and on constructing
solutions from the differential inclusion by a combination of the convex
integration and Baire's category methods. In doing so, we introduce the
appropriate notion of subsolutions of the partial differential inclusion and
their transition gauge, which plays a pivotal role in dealing with some
specific features of the constructed weak solutions.
