The Dirichlet Problem for the fractional p-Laplacian evolution equation

By Juan Luis V'azquez. 2015

94 citations recorded by the source corpus.

We consider a model of fractional diffusion involving the natural nonlocal version of the $p$-Laplacian operator. We study the Dirichlet problem posed in a bounded domain $\Omega$ of ${\mathbb{R}}^N$ with zero data outside of $\Omega$, for which the existence and uniqueness of strong nonnegative solutions is proved, and a number of quantitative properties are established. A main objective is proving the existence of a special separate variable solution $U(x,t)=t^{-1/(p-2)}F(x)$, called the friendly giant, which produces a universal upper bound and explains the large-time behaviour of all nontrivial nonnegative solutions in a sharp way. Moreover, the spatial profile $F$ of this solution solves an interesting nonlocal elliptic problem. We also prove everywhere positivity of nonnegative solutions with any nontrivial data, a property that separates this equation from the standard $p$-Laplacian equation.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/cd1fe9915a44264170d12d3682ef6d820198f34b

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In Applied Mathematics

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Citations in the corpus

On fractional p-Laplacian parabolic problem with general data (2016, background): “In [20] and [16] for p 6= 2 and f ≡ 0, the authors obtained the existence of energy solution for all u0 ∈ L (2)(Ω), explaining the asymptotical behavior with respect to properties of the corresponding Barenblatt type solution (for p > 2).”
Nonlinear commutators for the fractional p-Laplacian and applications (2015, unknown): “The fractional p-Laplacian has recently received quite some interest, for example we refer to [2, 9, 10, 21, 18, 16, 13, 17, 23].”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “ity and asymptotic behaviour. There is a developing literature on nonlocal nonlinear diusion equations on domains. 8.3 Fractional diusion equations of p-Laplacian type We report here about our work [250]. It deals with a model of fractional diusion involving a nonlocal version of the p-Laplacian ope”
A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian (2016, background): “See for instance [10, 5, 6, 12, 21, 23, 24, 31, 19] and the references therein.”