# Article

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Keywords:
hysteresis; Prandtl-Ishlinskii operator; material with periodic structure; nonlinear diffusion equation; homogenization; initial-boundary value problem; spatially periodic data
Summary:
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 $c^\varepsilon$ and $\eta ^\varepsilon$ when the spatial period $\varepsilon$ 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.
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