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We study some non-parabolic diffusion problems in one-space dimension, wherethe diffusion flux exhibits forward and backward nature of the Perona-Malik,H\"ollig or non-Fourier type. Classical weak solutions to such problems areconstructed 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 equationsinvolved as an inhomogeneous partial differential inclusion and on constructingsolutions from the differential inclusion by a combination of the convexintegration and Baire's category methods. In doing so, we introduce theappropriate notion of subsolutions of the partial differential inclusion andtheir transition gauge, which plays a pivotal role in dealing with somespecific features of the constructed weak solutions.

Non-parabolic diffusion problems in one space dimension