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 operatorF: ∂∂t[u+F(u)]+∂u∂x=fin 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.