doi: 10.3934/dcds.2022079
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Large time behavior for a nonlocal nonlinear gradient flow

Department of Mathematics, Uppsala University, Box 480,751 06 Uppsala, Sweden

*Corresponding author: Erik Lindgren

In honour of Juan Luis Vázquez's 75th birthday

Received  January 2022 Revised  May 2022 Early access June 2022

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

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.
Citation: Feng Li, Erik Lindgren. Large time behavior for a nonlocal nonlinear gradient flow. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022079
References:
[1]

B. AbdellaouiA. AttarR. Bentifour and I. Peral, On fractional p-Laplacian parabolic problem with general data, Ann. Mat. Pura. Appl., 197 (2018), 329-356.  doi: 10.1007/s10231-017-0682-z.

[2]

A. Banerjee, P. Garain and J. Kinnunen, Some local properties of subsolutons and supersolutions for a doubly nonlinear nonlocal parabolic p-laplace equation, preprint, 2020, arXiv: 2010.05727.

[3]

A. Banerjee, P. Garain and J. Kinnunen, Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p-laplace equations, Communications in Contemporary Mathematics, preprint, 2021, arXiv: 2101.10042. doi: 10.1142/S0219199722500328.

[4]

L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional $p$-Laplace equation in the superquadratic case, Adv. Math., 304 (2017), 300-354.  doi: 10.1016/j.aim.2016.03.039.

[5]

L. BrascoE. Lindgre and A. Schikorra, Higher Hölder regularity for the fractional $p$-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.

[6]

L. BrascoE Lindgren and M. Strömqvist, Continuity of solutions to a nonlinear fractional diffusion equation, J. Evol. Equ., 21 (2021), 4319-4381.  doi: 10.1007/s00028-021-00721-2.

[7]

L. Bungert and M. Burger, Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type, J. Evol. Equ., 20 (2020), 1061-1092.  doi: 10.1007/s00028-019-00545-1.

[8]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[9]

M. Ding, C. Zhang and S. Zhou, Local boundedness and Hölder continuity for the parabolic fractional $p$-Laplace equations, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 38, 45 pp. doi: 10.1007/s00526-020-01870-x.

[10]

J. Giacomoni and S. Tiwari, Existence and global behavior of solutions to fractional $p$-Laplacian parabolic problems, Electron. J. Differential Equations, 2018 (2018), Paper No. 44, 20 pp.

[11]

R. Hynd and E. Lindgren, Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution, Anal. PDE, 9 (2016), 1447-1482.  doi: 10.2140/apde.2016.9.1447.

[12]

R. Hynd and E. Lindgren, Large time behavior of solutions of Trudinger's equation, J. Differential Equations, 274 (2021), 188-230.  doi: 10.1016/j.jde.2020.11.050.

[13]

A. IannizzottoS. LiuK. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.

[14]

H. Ishii and G. Nakamura, A class of integral equations and approximation of $p$-Laplace equations, Calc. Var. Partial Differential Equations, 37 (2010), 485-522.  doi: 10.1007/s00526-009-0274-x.

[15]

P. Juutinen and P. Lindqvist, Pointwise decay for the solutions of degenerate and singular parabolic equations, Adv. Differential Equations, 14 (2009), 663-684. 

[16]

J. KorvenpääT. Kuusi and E. Lindgren, Equivalence of solutions to fractional $p$-Laplace type equations, J. Math. Pures Appl., 132 (2019), 1-26.  doi: 10.1016/j.matpur.2017.10.004.

[17]

E. Lindgren, Hölder estimates for viscosity solutions of equations of fractional $p$-Laplace type, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 55, 18 pp. doi: 10.1007/s00030-016-0406-x.

[18]

J. M. MazónJ. D. Rossi and J. Toledo, Fractional $p$-Laplacian evolution equations, J. Math. Pures Appl., 105 (2016), 810-844.  doi: 10.1016/j.matpur.2016.02.004.

[19]

V. Maz'ya and T. Shaposhnikova, On the bourgain, brezis, and mironescu theorem concerning limiting embeddings of fractional sobolev spaces, J. Func. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[20]

D. Puhst, On the evolutionary fractional $p$-Laplacian, Appl. Math. Res. Express. AMRX, 2015 (2015), 253-273.  doi: 10.1093/amrx/abv003.

[21]

M. Strömqvist, Harnack's inequality for parabolic nonlocal equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1709-1745.  doi: 10.1016/j.anihpc.2019.03.003.

[22]

M. Strömqvist, Local boundedness of solutions to non-local parabolic equations modeled on the fractional $p$-Laplacian, J. Differential Equations, 266 (2019), 7948-7979.  doi: 10.1016/j.jde.2018.12.021.

[23]

J. L. Vázquez, The Dirichlet problem for the fractional $p$-Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038-6056.  doi: 10.1016/j.jde.2015.12.033.

[24]

J. L. Vázquez, The evolution fractional $p$-Laplacian equation in $\Bbb R^N$. Fundamental solution and asymptotic behaviour, Nonlinear Anal., 199 (2020), 112034, 32 pp. doi: 10.1016/j.na.2020.112034.

[25]

J. L. Vázquez, The fractional $p$-Laplacian evolution equation in ${\Bbb R}^N$ in the sublinear case, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 140, 59 pp. doi: 10.1007/s00526-021-02005-6.

[26]

M. Warma, Local Lipschitz continuity of the inverse of the fractional $p$-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129-157.  doi: 10.1016/j.na.2016.01.022.

show all references

References:
[1]

B. AbdellaouiA. AttarR. Bentifour and I. Peral, On fractional p-Laplacian parabolic problem with general data, Ann. Mat. Pura. Appl., 197 (2018), 329-356.  doi: 10.1007/s10231-017-0682-z.

[2]

A. Banerjee, P. Garain and J. Kinnunen, Some local properties of subsolutons and supersolutions for a doubly nonlinear nonlocal parabolic p-laplace equation, preprint, 2020, arXiv: 2010.05727.

[3]

A. Banerjee, P. Garain and J. Kinnunen, Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p-laplace equations, Communications in Contemporary Mathematics, preprint, 2021, arXiv: 2101.10042. doi: 10.1142/S0219199722500328.

[4]

L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional $p$-Laplace equation in the superquadratic case, Adv. Math., 304 (2017), 300-354.  doi: 10.1016/j.aim.2016.03.039.

[5]

L. BrascoE. Lindgre and A. Schikorra, Higher Hölder regularity for the fractional $p$-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.

[6]

L. BrascoE Lindgren and M. Strömqvist, Continuity of solutions to a nonlinear fractional diffusion equation, J. Evol. Equ., 21 (2021), 4319-4381.  doi: 10.1007/s00028-021-00721-2.

[7]

L. Bungert and M. Burger, Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type, J. Evol. Equ., 20 (2020), 1061-1092.  doi: 10.1007/s00028-019-00545-1.

[8]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[9]

M. Ding, C. Zhang and S. Zhou, Local boundedness and Hölder continuity for the parabolic fractional $p$-Laplace equations, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 38, 45 pp. doi: 10.1007/s00526-020-01870-x.

[10]

J. Giacomoni and S. Tiwari, Existence and global behavior of solutions to fractional $p$-Laplacian parabolic problems, Electron. J. Differential Equations, 2018 (2018), Paper No. 44, 20 pp.

[11]

R. Hynd and E. Lindgren, Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution, Anal. PDE, 9 (2016), 1447-1482.  doi: 10.2140/apde.2016.9.1447.

[12]

R. Hynd and E. Lindgren, Large time behavior of solutions of Trudinger's equation, J. Differential Equations, 274 (2021), 188-230.  doi: 10.1016/j.jde.2020.11.050.

[13]

A. IannizzottoS. LiuK. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.

[14]

H. Ishii and G. Nakamura, A class of integral equations and approximation of $p$-Laplace equations, Calc. Var. Partial Differential Equations, 37 (2010), 485-522.  doi: 10.1007/s00526-009-0274-x.

[15]

P. Juutinen and P. Lindqvist, Pointwise decay for the solutions of degenerate and singular parabolic equations, Adv. Differential Equations, 14 (2009), 663-684. 

[16]

J. KorvenpääT. Kuusi and E. Lindgren, Equivalence of solutions to fractional $p$-Laplace type equations, J. Math. Pures Appl., 132 (2019), 1-26.  doi: 10.1016/j.matpur.2017.10.004.

[17]

E. Lindgren, Hölder estimates for viscosity solutions of equations of fractional $p$-Laplace type, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 55, 18 pp. doi: 10.1007/s00030-016-0406-x.

[18]

J. M. MazónJ. D. Rossi and J. Toledo, Fractional $p$-Laplacian evolution equations, J. Math. Pures Appl., 105 (2016), 810-844.  doi: 10.1016/j.matpur.2016.02.004.

[19]

V. Maz'ya and T. Shaposhnikova, On the bourgain, brezis, and mironescu theorem concerning limiting embeddings of fractional sobolev spaces, J. Func. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.

[20]

D. Puhst, On the evolutionary fractional $p$-Laplacian, Appl. Math. Res. Express. AMRX, 2015 (2015), 253-273.  doi: 10.1093/amrx/abv003.

[21]

M. Strömqvist, Harnack's inequality for parabolic nonlocal equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1709-1745.  doi: 10.1016/j.anihpc.2019.03.003.

[22]

M. Strömqvist, Local boundedness of solutions to non-local parabolic equations modeled on the fractional $p$-Laplacian, J. Differential Equations, 266 (2019), 7948-7979.  doi: 10.1016/j.jde.2018.12.021.

[23]

J. L. Vázquez, The Dirichlet problem for the fractional $p$-Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038-6056.  doi: 10.1016/j.jde.2015.12.033.

[24]

J. L. Vázquez, The evolution fractional $p$-Laplacian equation in $\Bbb R^N$. Fundamental solution and asymptotic behaviour, Nonlinear Anal., 199 (2020), 112034, 32 pp. doi: 10.1016/j.na.2020.112034.

[25]

J. L. Vázquez, The fractional $p$-Laplacian evolution equation in ${\Bbb R}^N$ in the sublinear case, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 140, 59 pp. doi: 10.1007/s00526-021-02005-6.

[26]

M. Warma, Local Lipschitz continuity of the inverse of the fractional $p$-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129-157.  doi: 10.1016/j.na.2016.01.022.

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