Well-posedness for a class of biological diffusion models with hysteresis effect

This paper is concerned with a class of biological models which consist of nonlinear diffusion equations and a hysteresis operator describing the relationship between some variables of the equations. The existence of solutions to the analogous problem was ever considered by Aiki and Minchev (SIAM J Math Anal 36:2020–2032, 2005) under some assumptions including the global Lipschitz continuity of reaction terms. We show the existence of nonnegative solutions to the problem under consideration using the approximation method when the reaction terms are locally Lipschitz continuous. Moreover, we discuss the continuous dependence of solutions on initial data.