Rate of convergence to separable solutions of the fast diffusion equation

Abstract

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a subcritical exponent. We show that separable solutions are stable in some suitable sense by finding a class of functions which belong to their domain of attraction. For solutions in this class we establish optimal rates of convergence to separable solutions.

Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

“…In [FW1] and [FW2] they proved the sharp rate of convergence of solutions of (1.1) in R n with n > 4 and 0 < m ≤ n−4 n−2 to the Barenblatt solutions as the extinction time is approached. In [FW3] they also proved the rate of convergence of solutions of (1.1) in R n to separable solutions of (1.1) when n > 10 and 0 < m < (n−2)(n−10) (n−2) 2 −4n+8 √ n−1 . In [FW4] they found an explicit dependence of the slow temporal growth rate of solutions of (1.1) in R n on the initial spatial growth rate.…”

“…For the subcritical case m < (n − 2) + /n, M. Fila and M. Winkler [9][10][11][12] have obtained a lot of subtle phenomena for the solutions of (1.1). In [9,10] they proved the sharp rate of convergence of solutions of (1.1) in R n with n > 4 and 0 < m n − 4/n − 2 to the Barenblatt solutions as the extinction time is approached.…”

Asymptotic behavior of solutions of a k-Hessian evolution equation

“…See e.g. [2,6,7,8,9,10,15,17,19,23]. However, relatively little is known in the case of fully nonlinear parabolic equations on the entire space, which includes different parabolic analogues of the k-Hessian equation.…”

Asymptotic large time behavior of singular solutions of the fast diffusion equation

“…By the uniqueness result of Theorem 1.1 and the scaling property above, the solution f of (7) in R n \{0} which satisfies (8) and (10) for given constants A > 0 and D A > 0 coincides with the rescaled function f λ given by (12)…”

“…Hui [18,19,20], M. Fila, J.L. Vazquez, M. Winkler, E. Yanagida [10,11,12,13,27], A. Blanchet, M. Bonforte, E. Chasseigne, J. Dolbeault, G. Grillo, J.L. Vazquez [2,3,4], etc.…”