caa a22 4500 605515727 CHVBK 20210128100710.0 cr unu---uuuuu 210128e20151001xx s 000 0 eng 10.1007/s00205-015-0861-2 doi (NATIONALLICENCE)springer-10.1007/s00205-015-0861-2 A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains [Elektronische Daten] [Matteo Bonforte, Juan Vázquez] We investigate quantitative properties of the nonnegative solutions $${u(t,x)\geq 0}$$ u ( t , x ) ≥ 0 to the nonlinear fractional diffusion equation, $${\partial_t u + \mathcal{L} (u^m)=0}$$ ∂ t u + L ( u m ) = 0 , posed in a bounded domain, $${x\in\Omega\subset \mathbb{R}^N}$$ x ∈ Ω ⊂ R N , with m>1 for t>0. As $${\mathcal{L}}$$ L we use one of the most common definitions of the fractional Laplacian $${(-\Delta)^s}$$ ( - Δ ) 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. Springer-Verlag Berlin Heidelberg, 2015 Bonforte Matteo Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain aut Vázquez Juan Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/1(2015-10-01), 317-362 0003-9527 218:1<317 2015 218 205 https://doi.org/10.1007/s00205-015-0861-2 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0861-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bonforte Matteo Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain aut NATIONALLICENCE 700 1- Vázquez Juan Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/1(2015-10-01), 317-362 0003-9527 218:1<317 2015 218 205