Reaction-Diffusion Equations with Spatially Distributed Hysteresis

Abstract

We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis in the right-hand side. The input of hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the discontinuity of hysteresis. These conditions are formulated in terms of geometry of manifolds defining hysteresis thresholds and the graph of initial data.

Periodic solutions to a class of biological diffusion models with hysteresis effect

“…The reaction-diffusion equations with hysteresis have been considerably studied during the last decades (see [18,19,11,[20][21][22]14,23] and references therein). In particular, some research have been done on the hysteretic reaction-diffusion equations coming from biochemistry, see Section 5.5 of [5] and [18,19,24,6,25].…”

“…In particular, some research have been done on the hysteretic reaction-diffusion equations coming from biochemistry, see Section 5.5 of [5] and [18,19,24,6,25]. It is remarked that P. Gurevich et al [20,21] studied the reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. They mainly discussed the existence and uniqueness of solutions as well as their continuous dependence on initial data for a wide class of initial data.…”

Pattern formation in parabolic equations containing hysteresis with diffusive thresholds

“…Differential equations with non-ideal relays have been previously used to model reaction and diffusion processes in spatially distributed colonies of bacteria [21,22,38,19]. In contrast to our setting, all relays in these models have been assumed identical (that is, having the same bi-stability range), while spatial patterns appeared, in particular, due to the spatial diffusion of environmental substances.…”

Existence and Uniqueness of Solutions to Heat Equations with Hysteresis Coupled with Navier–Stokes Equations in 2D and 3D

“…How would the solutions behave if the hysteresis operator converged to a non-ideal relay (i.e., the functions ψ 1 and ψ 2 approached step functions)? In order to resolve the above questions we might refer Gurevich and Tikhomirov [19], where parabolic problems with non-ideal relay are treated. …”

“…The author would like to express deep gratitude to the referees for giving valuable comments and drawing the author attention to a recent paper [19] and the book [5]. The author also thanks the editor for introducing the references [13] and [32].…”