Asymptotic profiles for the third-grade fluids equations in one space dimension

In this article we consider the third-grade fluids equation in one space dimension. We show that the large time behavior of solutions is given by the very singular self-similar solutions of the associated Burgers equation. The proof of our result relies on the use of fine energy estimates in weighted spaces rewritten in the variables $\xi/\sqrt{\tau}$ and $\ln \tau$.