Jana Kopfová

Hysteresis in biological models

The paper gives an overview of results for partial Differential equations with hysteresis whose motivation comes from biology.

Magnetohydrodynamic Flow with Hysteresis

We consider a model system describing the two-dimensional flow of a conducting fluid surrounded by a ferromagnetic solid under the influence of the hysteretic response of the surrounding medium. We assume that this influence can be represented by the Preisach hysteresis operator. Existence and uniqueness of solutions for the resulting system of PDEs with hysteresis nonlinearities is established in the convexity domain of the Preisach operator.

Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions

Abstract We consider the nonlocal KPP-Fisher equation u t ( t , x ) = u x x ( t , x ) + u ( t , x ) ( 1 − ( K ⁎ u ) ( t , x ) ) which describes the evolution of population density u ( t , x ) with respect to time t and location x. The non-locality is expressed in terms of the convolution of u ( t , ⋅ ) with kernel K ( ⋅ ) ≥ 0 , ∫ R K ( s ) d s = 1 . The restrictions K ( s ) , s ≥ 0 , and K ( s ) , s ≤ 0 , are responsible for interactions of an individual with his left and right neighbors, respectively. We show that these two parts of K play quite different roles as for the existence and uniqueness of traveling fronts to the KPP-Fisher equation. In particular, if the left interaction is dominant, the uniqueness of fronts can be proved, while the dominance of the right interaction can induce the co-existence of monotone and oscillating fronts. We also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on K.

Uniqueness theorem for a Cauchy problem with hysteresis

The Cauchy problem for an ordinary differential equation coupled with a hysteresis operator is studied. Under physically reasonable assumptions on the forcing term, uniqueness of solutions is shown without assuming Lipschitz continuity of the hysteresis curves. The result is true for any kind of hysteresis operators with monotone curves of motion.

BV-norm continuity of sweeping processes driven by a set with constant shape

We prove the BV-norm well-posedness of sweeping processes driven by a moving convex set with constant shape, namely the BV-norm continuity of the so-called play operator of elasto-plasticity.

On a parabolic equation with hysteresis and convection: a uniqueness result

A uniqueness result for a parabolic partial differential equation with hysteresis and convection is established. This equation is a part of a model system which describes the magnetohydrodynamic (MHD) flow of a conducting fluid between two ferromagnetic plates. The result of this paper complements the content of [6], where existence of the solution has been proved under fairly general assumptions on the hysteresis operator and the uniqueness was obtained only for a restricted class of hysteresis operators