Abstract
This work deals with an initial- and boundary-value problem for a quasilinear parabolic equation that includes a possibly discontinuous hysteresis operator, \({\cal F}\):
In particular the case of \({\cal F}\) equal to a so-called relay operator is studied. Well-posedness is proved, as well as regularity of the solution and its robustness w.r.t. perturbations of \({\cal F}\). The large-time behaviour is studied; asymptotic stability and compactness are shown. For a time-periodic f, existence of a periodic solution is also established.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bénilan, P.: Equations d'évolution dans un espace de Banach quelconque et applications. Thèse, Orsay (1972)
Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, Berlin Heidelberg New York (1996)
Crandall, M.G.: The semigroup approach to first order quasilinear equations in several space variables. Israel J. Math. 12, 108–132 (1972)
Hilpert, M.: On uniqueness for evolution problems with hysteresis. In: Rodrigues, J.-F. (ed.) Mathematical Models for Phase Change Problems, pp. 377–388. Birkhäuser, Basel (1989)
Kenmochi, N., Visintin, A.: Asymptotic stability for nonlinear P.D.E.s with hysteresis. Euro. J. Appl. Math. 5, 39–56 (1994)
Krasnosel'skiĭ, M.A., Darinskiĭ, B.M., Emelin, I.V., Zabreĭko, P.P., Lifsic, E.A., Pokrovskiĭ, A.V.: Hysterant operator. Soviet Math. Dokl. 11, 29–33 (1970)
Krasnosel'skiĭ, M.A., Pokrovskiĭ, A.V.: Systems with Hysteresis. Springer, Berlin Heidelberg New York (1989). Russian edition: Nauka, Moscow (1983)
Krejčí, P.: Periodic solutions to a parabolic equation with hysteresis. Math. Z. 194, 61–70 (1987)
Krejčí, P.: Hysteresis and periodic solutions of semi-linear and quasi-linear wave equations. Math. Z. 193, 247–264 (1986)
Krejčí, P.: Convexity, Hysteresis and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo (1997)
Kružkov, S.N.: Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order. Soviet Math. Dokl. 10, 785–788 (1969)
Kružkov, S.N.: First order quasilinear equations in several independent variables. Math. U.S.S.R. Sbornik 10, 217–243 (1970)
Mayergoyz, I.D.: Mathematical Models of Hysteresis and Their Applications. Elsevier, Amsterdam (2003)
Preisach, F.: Über die magnetische Nachwirkung. Z. Physik 94, 277–302 (1935)
Visintin, A.: A model for hysteresis of distributed systems. Ann. Math. Pure Appl. 131, 203–231 (1982)
Visintin, A.: A phase transition problem with delay. Control Cybernetics 11, 5–18 (1982)
Visintin, A.: Differential Models of Hysteresis. Springer, Berlin Heidelberg New York (1994)
Visintin, A.: Forward—backward parabolic equations and hysteresis. Calc. Var. 15, 115–132 (2002)
Visintin, A.: Quasi-linear hyperbolic equations with hysteresis. Ann. Inst. H. Poincaré. Analyse non linéaire 19, 451–476 (2002)
Visintin, A.: Mathematical models of hysteresis. In: Bertotti, G., Mayergoyz, I. (eds.) The Science of Hysteresis. Academic Press, San Diego (2005)
Visintin, A.: Maxwell's equations with vector hysteresis. Arch. Rat. Mech. Anal. 175, 1–38 (2005)
Visintin, A.: Quasilinear first-order P.D.E.s with hysteresis. J. Math. Anal. Appl. (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000) 35K60, 35R35, 47J40
Rights and permissions
About this article
Cite this article
Visintin, A. Quasilinear parabolic P.D.E.s with discontinuous hysteresis. Annali di Matematica 185, 487–519 (2006). https://doi.org/10.1007/s10231-005-0164-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-005-0164-6