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The fractional p-Laplacian evolution equation in \({\mathbb {R}}^N\) in the sublinear case

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Abstract

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameter \(1<p<2\) and fractional exponent \(s\in (0,1)\). Rather standard theory shows that the Cauchy Problem for data in the Lebesgue \(L^q\) spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case \(p>2\) has been dealt with in a recent paper. We study here the “fast” regime \(1<p<2\) which is more complex. As main results, we construct the self-similar fundamental solution for every mass value M and any p in the subrange \(p_c=2N/(N+s)<p<2\), and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all \(L^q\) spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for \(p_c<p<p_1\), where \(p_1\) is a new critical number that we introduce, \(p_1\in (p_c,2)\). We extend this type of singular solutions to the “very fast range” \(1<p<p_c\). They represent examples of weak solutions having finite-time extinction in that lower p range. We briefly examine the situation in the limit case \(p=p_c\). Finally, we show that very singular solutions are related to fractional elliptic problems of the nonlinear eigenvalue type, of interest in their own right.

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Acknowledgements

Author partially funded by Projects MTM2014-52240-P and PGC2018-098440-B-I00 (Spain). Partially performed as an Honorary Professor at Univ. Complutense de Madrid. The author thanks F. del Teso for the numerical treatment that led to the self-similar profile displayed at the end of Sect. 8. He is also grateful to A. Iannizzotto for information and discussions about his paper [38] that led to the continuity results we present here. The anonymous referee provided a careful reading and a number of useful suggestions.

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  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain

    Juan Luis Vázquez

  2. Real Academia de Ciencias, Madrid, Spain

    Juan Luis Vázquez

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Correspondence to Juan Luis Vázquez.

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Vázquez, J.L. The fractional p-Laplacian evolution equation in \({\mathbb {R}}^N\) in the sublinear case. Calc. Var. 60, 140 (2021). https://doi.org/10.1007/s00526-021-02005-6

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