Quasilinear first-order PDEs with hysteresis

This paper deals with the Cauchy problem for a quasilinear first-order equation that includes a possibly discontinuous hysteresis operator F: ∂ ∂t u +F(u) + ∂u ∂x = f in R, for t > 0. Existence of a weak solution is proved for F equal to a completed relay operator. In the case of f ≡ 0, an entropy-type condition yields Lipschitz-continuous and monotone dependence on the initial data, hence uniqueness.  2005 Elsevier Inc. All rights reserved