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Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations

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Abstract

In this paper, we study the boundedness and Hölder continuity of local weak solutions to the following nonhomogeneous equation

$$\begin{aligned} \partial _tu(x,t)+\mathrm{P.V.}\int _{{\mathbb {R}}^N}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}\big (u(x,t)-u(y,t)\big )dy= f(x,t,u) \end{aligned}$$

in \(Q_T=\Omega \times (0,T)\), where the symmetric kernel K(x, y, t) has a generalized form of the fractional p-Laplace operator of s-order. We impose some structural conditions on the function f and use the De Giorgi-Nash-Moser iteration to establish the boundedness of local weak solutions in the a priori way. Based on the boundedness result, we also obtain Hölder continuity of bounded solutions in the superquadratic case. These results can be regarded as a counterpart to the elliptic case due to Di Castro et al. (Ann Inst H Poincaré Anal Non Linéaire, 2016).

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous reviewer for providing us several important reference papers and many helpful suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Mengyao Ding & Shulin Zhou

  2. School of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Chao Zhang

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  1. Mengyao Ding
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  2. Chao Zhang
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  3. Shulin Zhou
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Correspondence to Chao Zhang.

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Communicated by L. Caffarelli.

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Supported by the National Natural Science Foundation of China (12071098, 11671111, 12071009).

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Ding, M., Zhang, C. & Zhou, S. Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations. Calc. Var. 60, 38 (2021). https://doi.org/10.1007/s00526-020-01870-x

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