Abstract
We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If the nonlinearity in the equation does not oscillate too much at the origin, the solution is proved to be moreover Hölder continuous. The results are new even when the Dirichlet form is the one corresponding to the fractional Laplacian.
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Athanasopoulos, I., Caffarelli, L.A.: Continuity of the temperature in boundary heat control problems. Adv. Math. 224(1), 293–315 (2010)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Brändle, C., de Pablo, A.: Nonlocal heat equations: regularizing effect, decay estimates and nash inequalities. Commun. Pure Appl. Anal. 17(3), 1161–1178 (2018)
Caffarelli, L., Chan, C.H., Vasseur, A.: Regularity theory for parabolic nonlinear integral operators. J. Am. Math. Soc. 24(3), 849–869 (2011)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
Caffarelli, L.A., Vasseur, A.: The De Giorgi method for nonlocal fluid dynamics. In: Cabré, X., Soler, J. (eds.) Nonlinear Partial Differential Equations, pp. 1–38. Basel, Advanced Courses in Mathematics-CRM Barcelona, Springer (2012)
De Giorgi, E.: Sulla differenziabilitá e l’analiticitá delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3, 25–43 (1957)
DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)
DiBenedetto, E., Kwong, Y.C.: Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations. Trans. Am. Math. Soc. 330(2), 783–811 (1992)
Felsinger, M., Kassmann, M.: Local regularity for parabolic nonlocal operators. Commun. Partial Differ. Equ. 38(9), 1539–1573 (2013)
Karamata, J.: Sur un mode de croissance régulière des fonctions. Mathematica (Cluj) 4, 38–53 (1930)
Kim, S., Lee, K.-A.: Hölder estimates for singular non-local parabolic equations. J. Funct. Anal. 261(12), 3482–3518 (2011)
de Pablo, A., Quirós, F., Rodríguez, A.: Nonlocal filtration equations with rough kernels. Nonlinear Anal. Ser. A Theory Methods 137, 402–425 (2016)
de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A general fractional porous medium equation. Commun. Pure Appl. Math. 65(9), 1242–1284 (2012)
Varopoulos, N.T.: Hardy–Littlewood theory for semigroups. J. Funct. Anal. 63(2), 240–260 (1985)
Vázquez, J.L., de Pablo, A., Quirós, F., Rodríguez, A.: Classical solutions and higher regularity for nonlinear fractional diffusion equations. J. Eur. Math. Soc. 19(7), 1949–1975 (2017)
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All authors supported by the Spanish Project MTM2014-53037-P.
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Communicated by O. Savin.
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de Pablo, A., Quirós, F. & Rodríguez, A. Regularity theory for singular nonlocal diffusion equations. Calc. Var. 57, 136 (2018). https://doi.org/10.1007/s00526-018-1410-2
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DOI: https://doi.org/10.1007/s00526-018-1410-2