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Bonforte MCorresponding Author

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Fine properties of solutions to the Cauchy problem for a fast diffusion equation with Caffarelli–Kohn–Nirenberg weights

Publicated to:ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE. 40 (1): 1-59 - 2023-01-01 40(1), DOI: 10.4171/AIHPC/42

Authors: Bonforte, M.; Simonov, N.

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Abstract

We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the fast diffusion equation with weights (WFDE) ut = |x| div(|x|-β ∇um) posed on (0, +1) × ℝd, with d ≥ 3, in the so-called good fast diffusion range mc < m < 1, within the range of parameters γ, β which is optimal for the validity of the so-called Caffarelli–Kohn–Nirenberg inequalities. It is natural to ask in which sense such solutions behave like the Barenblatt B (fundamental solution): for instance, asymptotic convergence, i.e. ∥u(t) - B(t)∥Lp(ℝd)t→∞ 0, is well known for all 1 ≤ p ≤ 1, while only a few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data X ⊂ L1+(ℝd) that produces solutions which are pointwise trapped between two Barenblatt (global Harnack principle), and uniformly converge in relative error (UREC), i.e. d∞(u(t)) = ∥u(t)=B(t) - 1∥L∞(ℝd)t→∞ 0. Such a characterization is in terms of an integral condition on u(t = 0). To the best of our knowledge, analogous issues for the linear heat equation, m = 1, do not possess such clear answers, but only partial results. Our characterization is also new for the classical, nonweighted FDE. We are able to provide minimal rates of convergence to B in different norms. Such rates are almost optimal in the nonweighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in L1+(ℝd) n X, preserve the same “fat” spatial tail for all times, hence UREC fails and d∞(u(t)) = 1, even if ∥u(t) - B(t)∥L1(ℝd)t→∞ 0.

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