Sergey Tikhomirov

Reaction-diffusion equations with spatially distributed hysteresis

This paper deals with reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. In particular, such problems describe chemical reactions and biological processes in which diffusive and nondiffusive substances interact according to hysteresis law. We find sufficient conditions that guarantee the existence and uniqueness of solutions as well as their continuous dependence on initial data.

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.

Hölder Shadowing on Finite Intervals

For any

Symmetric Periodic Solutions of Parabolic Problems With Discontinuous Hysteresis

\theta, \omega > 1/2

Spatially discrete reaction-diffusion equations with discontinuous hysteresis

we prove that, if any

Rattling in Spatially Discrete Diffusion Equations with Hysteresis

d

Depinning Asymptotics in Ergodic Media

-pseudotrajectory of length

Recent Advances in Reaction-Diffusion Equations with Non-Ideal Relays

\sim 1/d^{\omega}

Lipschitz shadowing and structural stability of flows

of a diffeomorphism

Oriented Shadowing Property and \Omega -Stability for Vector Fields

f\in C^2