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.