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Large time behavior for a nonlocal nonlinear gradient flow

  • *Corresponding author: Erik Lindgren

    *Corresponding author: Erik Lindgren

In honour of Juan Luis Vázquez's 75th birthday

Erik Lindgren was supported by the Swedish Research Council, 2017-03736

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  • We study the large time behavior of the nonlinear and nonlocal equation

    $ v_t+(- \Delta_p)^sv = f \, , $

    where $ p\in (1, 2)\cup (2, \infty) $, $ s\in (0, 1) $ and

    $ (- \Delta_p)^s v\, (x, t) = 2 \, {\rm{P.V.}} \int_{ \mathbb{R}^n}\frac{|v(x, t)-v(x+y, t)|^{p-2}(v(x, t)-v(x+y, t))}{|y|^{n+sp}}\, dy. $

    This equation arises as a gradient flow in fractional Sobolev spaces. We obtain sharp decay estimates as $ t\to\infty $. The proofs are based on an iteration method in the spirit of J. Moser previously used by P. Juutinen and P. Lindqvist.

    Mathematics Subject Classification: Primary: 35K55, 35K65, 35K67, 35R11, 35B40.


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