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

By Matteo Bonforte, Juan Luis V'azquez. 2013

111 citations recorded by the source corpus.

We investigate quantitative properties of the nonnegative solutions $${u(t,x)\geq 0}$$u(t,x)≥0 to the nonlinear fractional diffusion equation, $${\partial_t u + \mathcal{L} (u^m)=0}$$∂tu+L(um)=0, posed in a bounded domain, $${x\in\Omega\subset \mathbb{R}^N}$$x∈Ω⊂RN, 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/e558c32e1e345410d8d0f3c63af6f4b8ed01955c

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains (2014, methodology): “8) can be found in [27, 28], for the RFL, while in the case of the SFL they can be easily deduced by the celebrated heat kernel estimates of Davies and Simon [20, 23] (for the case s = 1), see also [10] for more details about the application of the results of [20, 23] to the SFL.”
Representation of solutions and large-time behavior for fully nonlocal diffusion equations (2015, background): “For the regularity theory of nonlocal equations in case α = 1 or β = 2, we refer to [11, 7, 22, 4, 5, 30, 2, 49, 48] and the references therein.”
Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations (2015, background): “(9) This corresponds, for p > 2, to the so-called fractional porous medium equation, recently addressed and studied in a number of contributions (see, among others, [17], [5], [6] and [7]).”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, methodology): “The papers are [46], [51], [52], [41], and [42].”