Harnack's inequality for parabolic nonlocal equations
Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1709-1745.
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The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.

DOI : 10.1016/j.anihpc.2019.03.003
Classification : 35K10, 35B65, 35R11
Mots-clés : Nonlocal parabolic equations, Harnack inequalities, Local boundedness 
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     title = {Harnack's inequality for parabolic nonlocal equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1709--1745},
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Strömqvist, Martin. Harnack's inequality for parabolic nonlocal equations. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1709-1745. doi : 10.1016/j.anihpc.2019.03.003. http://www.numdam.org/articles/10.1016/j.anihpc.2019.03.003/

[1] Barlow, Martin T.; Bass, Richard F.; Kumagai, Takashi Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps, Math. Z., Volume 261 (2009) no. 2, pp. 297–320 | MR | Zbl

[2] Bogdan, Krzysztof; Sztonyk, Paweł Harnack's inequality for stable Lévy processes, Potential Anal., Volume 22 (2005) no. 2, pp. 133–150 | MR | Zbl

[3] Bonforte, Matteo; Sire, Yannick; Vázquez, Juan Luis Optimal existence and uniqueness theory for the fractional heat equation, Nonlinear Anal., Volume 153 (2017), pp. 142–168 | MR | Zbl

[4] Bonforte, Matteo; Vázquez, Juan Luis Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. Math., Volume 250 (2014), pp. 242–284 | MR | Zbl

[5] Caffarelli, Luis; Hin Chan, Chi; Vasseur, Alexis Regularity theory for parabolic nonlinear integral operators, J. Am. Math. Soc., Volume 24 (2011) no. 3, pp. 849–869 | MR | Zbl

[6] Di Castro, Agnese; Kuusi, Tuomo; Palatucci, Giampiero Nonlocal Harnack inequalities, J. Funct. Anal., Volume 267 (2014) no. 6, pp. 1807–1836 | MR | Zbl

[7] Di Castro, Agnese; Kuusi, Tuomo; Palatucci, Giampiero Local behavior of fractional p -minimizers, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 5, pp. 1279–1299 | Numdam | MR | Zbl

[8] Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521–573 | MR | Zbl

[9] DiBenedetto, Emmanuele Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993 | DOI | MR | Zbl

[10] Dyda, Bartłomiej; Kassmann, Moritz On weighted Poincaré inequalities, Ann. Acad. Sci. Fenn., Math., Volume 38 (2013) no. 2, pp. 721–726 | MR | Zbl

[11] Felsinger, Matthieu; Kassmann, Moritz Local regularity for parabolic nonlocal operators, Commun. Partial Differ. Equ., Volume 38 (2013) no. 9, pp. 1539–1573 | MR | Zbl

[12] Han, Qing; Lin, Fanghua Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence, RI, 2011 | MR | Zbl

[13] Kassmann, Moritz Harnack inequalities and hölder regularity estimates for nonlocal operators revisited, 2011 https://sfb701.math.uni-bielefeld.de/files/preprints/sfb11015.pdf (SFB 701-preprint no. 11015, Available at)

[14] Kassmann, Moritz A new formulation of Harnack's inequality for nonlocal operators, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 11–12, pp. 637–640 | MR | Zbl

[15] Kassmann, Moritz; Schwab, Russell W. Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N. S.), Volume 5 (2014) no. 1, pp. 183–212 | MR | Zbl

[16] Moser, J. On a pointwise estimate for parabolic differential equations, Commun. Pure Appl. Math., Volume 24 (1971), pp. 727–740 | DOI | MR | Zbl

[17] Saloff-Coste, Laurent Aspects of Sobolev-Type Inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[18] Strömqvist, M. Local boundedness of solutions to nonlocal equations modeled on the fractional p-laplacian (Preprint 2017) | arXiv | MR

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