Homogenization of heat equation with hysteresis

Abstract

The contribution deals with heat equation in the form (c u+W[u]) t= div(a·∇u)+f , where the nonlinear functional operator W[ u] is a Prandtl–Ishlinskii hysteresis operator of play type characterized by a distribution function η. The spatially dependent initial boundary value problem is studied. Proof of existence and uniqueness of the solution is omitted since the proof is a slightly modified proof by Brokate–Sprekels. The homogenization problem for this equation is studied. For ε→0, a sequence of problems of the above type with spatially ε-periodic coefficients c ε , η ε , a ε is considered. The coefficients c *, η * and a * in the homogenized problem are identified and convergence of the corresponding solutions u ε to u * is proved.