Fine Asymptotics for Fast Diffusion Equations
José Antonio Carrillo, Juan L. Vázquez · Communications in Partial Differential Equations · 2003
We investigate the large-time asymptotics of fast diffusion equations, , where . We calculate convergence to Barenblatt profiles with algebraic rates in the exponent interval in dimensions d≥ 2. We cover in this way the gap still existing in the literature concerning the rates of approach to a Barenblatt profile, which have been recently obtained for all exponents . Our main result improves the standard convergence into a rate of decay of the form valid for any and any solution with nonnegative and integrable data satisfying a condition of finite relative entropy. In the formula uis the actual solution, Uthe asymptotic model, and the L 1norm is taken in the space variable. Let us recall that the Barenblatt profiles do not exist for and then the solutions have a quite different large-time evolution. We are also concerned with the presence of certain critical exponents in the asymptotic behavior; we explain the role of the value (d− 1)/d, by looking at the linearized equation. Finally, we obtain a better rate of convergence of the form for radially symmetric solutions with strongly decaying data as . This rate is optimal.