×

Forward-backward parabolic equations and hysteresis. (English) Zbl 1010.35056

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.

MSC:

35K55 Nonlinear parabolic equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
PDFBibTeX XMLCite
Full Text: DOI