A free boundary characterization of measure-valued solutions for forward-backward diffusion (Q955890)

Consider the equation \( \partial _t u= \nabla q(\nabla u),\) with \( q=\Phi '\) and \(\Phi\) non-convex; \(u\) is assumed scalar and \(\Phi\) to have quadratic growth at infinity. Two notions of solutions have been used to work with this equation: \textit{M. Slemrod}'s singular perturbation (SP) solutions [J. Dyn. Differ. Equations 3, No.~1, 1--28 (1991; Zbl 0747.35013)], and \textit{S. Demoulini}'s Young measure solutions obtained via a time discretization and energy minimization (EM) at each step [SIAM J. Math. Anal. 27, No.~2, 376--403 (1996; Zbl 0851.35066)]. In this interesting paper the authors compare these two types of solutions, and find a number of results in doing this. Among them are: {\parindent=4mm \begin{itemize}\item[--] The two notions produce different solutions. \item[--] The SP solutions coincide with classical ones if no backward diffusion is encountered, the EM solutions don't necessarily. \item[--] Small perturbations of initial values may lead to sequences of SP solutions that converge to the EM solution (even uniformly) rather than to the SP solution for the unperturbed data (the EM solution is more stable, the SP solution does not depend on data in a stable way). \item[--] The EM solution is characterized by a free boundary problem in which all gradients of non-optimal energy are avoided. \end{itemize}} The paper also contains results about the asymptotic behavior of EM solutions, as well as a number of technical results leading to those in the summary above.