Continuity of solutions of a quasilinear hyperbolic equation with hysteresis. (English) Zbl 1249.35214

Examples show that quasilinear first order hyperbolic PDEs with a hysteresis operator in the constitutive law possess similar properties as in the case without hysteresis, that is, shocks may occur in a finite time. As the main result, however, the author proves that if the hysteresis operator has the counterclockwise convexity property, that is, ascending branches of the hysteresis loops are convex and descending branches are concave, shocks do not occur and the solution constructed by time semidiscretization remains regular.


35L60 First-order nonlinear hyperbolic equations
34C55 Hysteresis for ordinary differential equations
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