Visintin, Augusto Forward-backward parabolic equations and hysteresis. (English) Zbl 1010.35056 Calc. Var. Partial Differ. Equ. 15, No. 1, 115-132 (2002). Author’s summary: A forward-backward parabolic problem is obtained by coupling the equation \[ \frac{\partial}{\partial t} (u + w) - \Delta u = f \] with a nonmonotone relation \(u = \alpha (w)\). In the framework of a two-scale model, we replace the latter condition by a relaxation dynamics which converges to a hysteresis relation. We provide a suitable formulation of the hysteresis law, approximate it by the relaxation dynamics, couple it with the PDE, derive uniform estimates via an \(L^1\)-technique, and then pass to the limit as the relaxation parameter vanishes. This yields existence of a solution for the modified problem. This procedure is also applied to other equations. Reviewer: Hanna Marcinkowska (Wrocław) Cited in 12 Documents MSC: 35K55 Nonlinear parabolic equations 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) Keywords:hysteresis relation; parabolic equation with nonmonotone relation; dissipation inequality; Hele-Shaw problem with hysteresis; relaxation parameter PDFBibTeX XMLCite \textit{A. Visintin}, Calc. Var. Partial Differ. Equ. 15, No. 1, 115--132 (2002; Zbl 1010.35056) Full Text: DOI