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A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains

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Abstract

We investigate quantitative properties of the nonnegative solutions \({u(t,x)\geq 0}\) to the nonlinear fractional diffusion equation, \({\partial_t u + \mathcal{L} (u^m)=0}\), posed in a bounded domain, \({x\in\Omega\subset \mathbb{R}^N}\), with m > 1 for t > 0. As \({\mathcal{L}}\) we use one of the most common definitions of the fractional Laplacian \({(-\Delta)^s}\), 0 < s < 1, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. In addition, we obtain similar estimates for fractional semilinear elliptic equations. Either the standard Laplacian case s = 1 or the linear case m = 1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems.

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Authors and Affiliations

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

    Matteo Bonforte & Juan Luis Vázquez

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  1. Matteo Bonforte
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  2. Juan Luis Vázquez
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Correspondence to Juan Luis Vázquez.

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Communicated by S. Serfaty

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Bonforte, M., Vázquez, J.L. A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains. Arch Rational Mech Anal 218, 317–362 (2015). https://doi.org/10.1007/s00205-015-0861-2

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  • DOI: https://doi.org/10.1007/s00205-015-0861-2

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