The Cauchy problem for an equation of Sobolev type with power non-linearity

This paper deals with the study of the large-time asymptotic behaviour of solutions of the Cauchy problem for a non-linear non-local equation of Sobolev type with dissipation. In the case when the initial data are small our approach is based on a detailed study of the Green's function of the linear problem and the use of the contraction-mapping method. We also consider the case when the initial data are large. In the supercritical case the asymptotics is quasilinear. The asymptotic behaviour of solutions in the critical case differs from the behaviour of solutions of the corresponding linear equation by a logarithmic correction. In the subcritical case we prove that that the principal term of the large-time asymptotics of the solution can be represented by a self-similar solution if the initial data have non-zero total mass.