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Dipartimento di Matematica ''F. Casorati''

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Global Harnack Principle for a class of fast diffusion equations

Nikita Simonov, Universidad Autonoma de Madrid

Aula Beltrami, Dipartimento di Matematica - Thursday, January 16, 2020 h.16:00


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 Rd,
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.


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