Abstract
We consider the long-time behaviour of solutions of the Cauchy problem for a quasilinear equation ut + f(u)x = 0 with a strictly convex flux function f(u) and initial function u0(x) having the the one-sided limiting mean values u± that are uniform with respect to translations. The estimates of the rates of convergence to solutions of the Riemann problem depending on the behaviour of the integrals $ \int\limits_a^{a + y} {\left( {{u_0}\left( x \right) - {u^ \pm }} \right)} dx $ as y→±∞ are established. The similar results are obtained for solutions of the mixed problem in the domain x > 0, t > 0 with a constant boundary data u– and initial data having limiting mean value u±.