Well-posedness and asymptotic behaviour for the convection problem in

Abstract

We analyse the well-posedness of the initial value problem for a convection problem. Mild solutions are obtained in the weak- spaces and the existence of self-similar solutions is shown, while the only small self-similar solution in the Lebesgue space is the null solution. The asymptotic stability of solutions is analysed and, as a consequence, a criterion of self-similarity persistence at large times is obtained.