Thursday, 16 January 2020, 4 p.m. (sharp),

Dr. Nikita Simonov, Universidad Autónoma de Madrid

at the conference room of Dipartimento di Matematica "F. Casorati" - aula Beltrami,in Pavia, will give a lecture titled:

GLOBAL HARNACK PRINCIPLE FOR A CLASS OF FAST DIFFUSION EQUATIONS

At the end a refreshment will be organized.

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Abstract. We study global properties of non-negative, integrable

solutions to the Cauchy problem of the weighted fast diffusion

equation u_t = |x|^s div(|x|^{-r} ∇u^m ) with (d − 2 − r)/(d − s) < m

< 1. The weights |x|^s and |x|^ {−r} , with s < d and s − 2 < r ≤ s(d

− 2)/d can be both degenerate and singular and need not belong to the

class A_2 , this range of parameters is optimal for the validity of a

class of Caffarelli-Kohn-Nirenberg inequalities.

We characterize the largest class of data for which the so called

Global Harnack Principle (GHP) holds (a global lower and upper bound

in terms of suitable Barenblatt solutions). As a consequence of the

GHP, we prove convergence of the uniform relative error, namely |(u −

B)/B| → 0 as t → ∞ uniformly in R^d, where B is a suitable Barenblatt

solution. In the case with no weights (s = r = 0) and for a special

class of data, we give (almost) sharp rates of convergence to the

Barenblatt profile in the L^1 and the L^∞ topologies, in the radial

case we give sharp rates. We extend some of the results to

non-negative, integrable solutions to the Cauchy problem of the

p-Laplace evolution equation u_t = ∆_p(u), where ∆_p(w) :=

div(|∇w|^(p−2) ∇w), with 2d/(d + 1) < p < 2.

The above results were obtained in collaboration with Prof. M.

Bonforte and D. Stan.