Asymptotic behavior of a solution to the drift-diffusion equation for a fast-diffusion case

We consider asymptotic behavior of a solution to the drift-diffusion equation for a fast-diffusion case. In the degenerate drift-diffusion equation, it is known that large time behavior of solutions converges to the Zel'dovich-Kompaneetz-Barenblatt (ZKB) function. For a fast-diffusion case, we show that the asymptotic profile for a solution is the generalized ZKB function such as the Talenti function. We use the entropy dissipation method combining the logarithmic Sobolev and the Shannon inequalities for the Renyi entropy that is known as an extension of the Boltzmann-Shannon entropy. (C) 2021 The Authors. Published by Elsevier Inc.