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
Full text of the article: