A General Fractional Porous Medium Equation

By A. de Pablo, F. Quir'os, A. Rodr'iguez, Juan Luis V'azquez. 2011

240 citations recorded by the source corpus.

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion: \input amssym $$\left\{ {\matrix{ {{{\partial u} \over {\partial t}} + \left( { ‐ \Delta } \right)^{\sigma /2} \left( {\left| u \right|^{m ‐ 1} u} \right) = 0,} \hfill & {x \in {\Bbb R} ^N ,\,\,t > 0,} \hfill \cr {u\left( {x,0} \right) = f\left( x \right),} \hfill & {x \in {\Bbb R} ^N .} \hfill \cr } } \right.$$ We consider data \input amssym $f\in L^1(\Bbb{R}^N)$ and all exponents $0<\sigma<2\;and\;m>0$. Existence and uniqueness of a strong solution is established for $ m > {m_\ast}={(N-\sigma)_+}/N$, giving rise to an L1‐contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range ${0 < m} \le {m_\ast}$ existence and uniqueness happen under some restrictions, and the properties of the solutions are different from the ones for the case above m*. We also study the dependence of solutions on f, m, and σ. Moreover, we consider the above questions for the problem posed in a bounded domain. © 2012 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/90ad8bbf99181aa32a116095afe646b5a9ec1094

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Citations in the corpus

Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains (2014, unknown): “…de Cantoblanco, 28049 Madrid, Spain (b) e-mail address: matteo.bonforte@uam.es (c) Université Aix-Marseille, I2M, Centre de Mathématique et Informatique, Technopôle de ChateauGiombert, Marseille, France (d) e-mail address: sire@cmi.univ-mrs.fr (e) e-mail address: juanluis.vazquez@uam.es Contents”
Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations (2015, background): “ 0 in Ω×(0,+∞), (9) u = 0 in RN \Ω. 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]). Actually, taking a family {(uσ,wσ)} of solutions to our problem, and lettin”
The Fisher-KPP problem with doubly nonlinear diffusion (2016, background): “We give in the graph below (Figure 1) a description of the result on finite propagation for different related models of nonlinear diffusion with or without fractional effects, see [21, 10, 3].”
Classical solutions and higher regularity for nonlinear fractional diffusion equations (2013, methodology): “On the other hand, in our paper [17] we prove, for all m > 0, that when the initial value is nonnegative the solution is strictly positive everywhere for positive times.”
A fractional Yamabe flow and some applications (2011, background): “positive constant. 5 Extinction proﬁle of a fractional porous medium equation Let u(x,t) be the solution of (11) and T > 0 be its extinction time. Since u0 is not identically zero, it is proved in [29] that u(x,t) > 0 in Rn × (0,T) and u(x,t) ∈ Cα(Rn × (0,T)) for some α ∈ (0,1). We deﬁne v(F(x”
The Dirichlet Problem for the fractional p-Laplacian evolution equation (2015, background): “Evolution problems of nonlocal diffusion type originated from material science and statistical mechanics, and led to much work on fractional porous medium models, see for instance [7, 22, 23, 20, 43, 44, 6, 10, 52, 60, 25], the work was summarized in [56, 58].”
Quantitative local and global a priori estimates for fractional nonlinear diffusion equations (2012, background): “In [31] there are the global L1 − L∞ smoothing effects which provide global upper bounds for solutions to the Cauchy problem 1.”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, methodology): “Construction of the solutions of the FPME on bounded domains with the SFL was already used in [102, 103] as an approximation to the problem in the whole space, but the regularity or asymptotic properties were not studied.”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, methodology): “The right space for um is the fractional Sobolev space Ḣσ/2(RN ), defined as the completion of C∞0 (RN ) with the norm ‖ψ‖Ḣσ/2 = (∫ RN |ξ|σ|ψ̂|2 dξ )1/2 = ‖(−∆)σ/4ψ‖L2 , where ψ̂ denotes the Fourier transform of ψ. Note that (−∆)σ/4um ∈ L2(RN ) if um ∈ Ḣσ/2(RN ).”
Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, background): “≤ 0 Indeed, to ensure the positivity of the last term, we need the so-called Stroock-Varopoulos inequality, namely Z RN up−1(t,x)(−∆)su(t,x)dx≥ 4(p−1) p2 Z RN (−∆)s2u p 2 (t,x) 2 dx see [40] and also [32]. Of course the above proof needs to be justiﬁed, by approximation, hence the need for a slightl”