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MATHEMATICA BOHEMICA, Vol. 126, No. 2, pp. 363-377 (2001)
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# Homogenization of diffusion equation

with scalar hysteresis operator

## Jan Francu

* Jan Francu*, Brno University of Technology, Dept. of Mathematics, Technicka 2, 616 69 Brno, Czech Republic, e-mail: ` francu@um.fme.vutbr.cz`

**Abstract:**
The paper deals with a scalar diffusion equation $ c u_t = ({\F }[u_x])_x + f, $ where $\F $ is a Prandtl-Ishlinskii operator and $c, f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic data $c^\e $ and $\eta ^\e $ when the spatial period $\e $ tends to zero. The homogenized characteristics $c^*$ and $\eta ^*$ are identified and the convergence of the corresponding solutions to the solution of the homogenized equation is proved.

**Keywords:** hysteresis, Prandtl-Ishlinskii operator, material with periodic structure, nonlinear diffusion equation, homogenization

**Classification (MSC2000):** 35B27, 47J40, 34C55

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