Forward–backward parabolic equations and hysteresis

  • Augusto Visintin
Original article

DOI: 10.1007/s005260100120

Cite this article as:
Visintin, A. Calc Var (2002) 15: 115. doi:10.1007/s005260100120


A forward-backward parabolic problem is obtained by coupling the equation \({\partial\over\partial t}(u+w) -\Delta u= f\) 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 P.D.E., 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Augusto Visintin
    • 1
  1. 1.Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo di Trento, Italia (e-mail: IT

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