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

TL;DR

This paper proves the uniqueness of solutions for reaction-diffusion equations with spatially distributed hysteresis, relevant to biological and chemical processes, especially when initial data are transverse and hysteresis branches are non-Lipschitz.

Contribution

It establishes a uniqueness theorem for reaction-diffusion equations with complex hysteresis behavior, including non-Lipschitz cases, under spatial transversality conditions.

Findings

Proved uniqueness of solutions with spatially transverse initial data.

Extended the theory to include non-Lipschitz hysteresis branches.

Applicable to biological and chemical reaction models.

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.