Asymptotics of solutions of non-linear dissipative equations

We study the large-time asymptotic behaviour of solutions of the Cauchy problem for a system of non-linear evolution equations with dissipation. In the case when the initial data are small, we construct solutions using the contraction-mapping principle. When the initial data are large, we obtain the large-time asymptotics of solutions by taking into account a certain symmetry of the non-linear terms.