Motivated by a problem in foam generation and propagation in porous media, we develop necessary and sufficient conditions for. more general quasilinear hyperbolic problem, involving reactions of finite rate, to admit as solution a steady state following the initial equilibrium state. It is found that for such a solution to exist (and to be physically well-posed), it is necessary that the generalized flux vector F and the generalized concentration vector C be colinear, namely that F = Lambda(C)C, where Lambda(C) is a scalar function. If an additional condition for the overlap of characteristics is valid, the two conditions are sufficient and necessary.