Uniqueness of transverse solutions for reaction-diffusion equations with spatially distributed hysteresis

Abstract

The paper deals with reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. Such problems describe biological processes and chemical reactions in which diffusive and nondiffusive substances interact according to hysteresis law. Under the assumption that the initial data are spatially transverse, we prove a theorem on the uniqueness of solutions. The theorem covers the case of non-Lipschitz hysteresis branches arising in the theory of slow-fast systems.