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We introduce and analyze a new, nonlinear fourth-order regularization of forward-backward parabolic equations. In one space dimension, under general assumptions on the potentials, which include those of Perona--Malik type, we prove existence of Radon measure-valued solutions under both natural and essential boundary conditions. If the decay at infinity of the nonlinearities is sufficiently fast, we also exhibit examples of local solutions whose atomic part arises and/or persists (in contrast to the linear fourth-order regularization) and even disappears within finite time (in contrast to pseudoparabolic regularizations).

Measure-Valued Solutions to a Nonlinear Fourth-Order Regularization of Forward-Backward Parabolic Equations