Large-time asymptotic behaviour of solutionsof non-linear Sobolev-type equations

摘要

The large-time asymptotic behaviour of solutions of the Cauchyproblem is investigated for a non-linear Sobolev-type equation with dissi-pation. For small initial data the approach taken is based on a detailedanalysis of the Green's function of the linear problem and the use of thecontraction mapping method. The case of large initial data is also closelyconsidered. In the supercritical case the asymptotic formulae are quasi-linear. The asymptotic behaviour of solutions of a non-linear Sobolev-typeequation with a critical non-linearity of the non-convective kind differs bya logarithmic correction term from the behaviour of solutions of the corre-sponding linear equation. For a critical convective non-linearity, as well asfor a subcritical non-convective non-linearity it is proved that the leadingterm of the asymptotic expression for large times is a self-similar solu-tion. For Sobolev equations with convective non-linearity the asymptoticbehaviour of solutions in the subcritical case is the product of a rarefactionwave and a shock wave.