Quantitative local and global a priori estimates for fractional nonlinear diffusion equations

By Matteo Bonforte, Juan Luis V'azquez. 2012

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Abstract We establish quantitative estimates for solutions u ( t , x ) to the fractional nonlinear diffusion equation, ∂ t u + ( − Δ ) s ( u m ) = 0 in the whole range of exponents m > 0 , 0 s 1 . The equation is posed in the whole space x ∈ R d . We first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data. In the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions, depending only on the norm of the initial data in a certain ball. The estimates take a different form in three exponent ranges: slow diffusion, good range of fast diffusion, and very fast diffusion. Finally, we show existence and uniqueness of initial traces.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/a44fafb0892d290c36c1873e6a4b3300743124ca

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains (2015, background): “We refer the interested reader to [5,12,13, 33, 63, 66, 67] for further details about possible applications.”
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]).”
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.”
On the Cauchy problem for a general fractional porous medium equation with variable density (2013, background): “Unfortunately, as shown in [3], the decay (4.”
Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, methodology): “1 of [11] since we use it several times in the paper.”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, methodology): “Besides, we use the simplified notation um for data of any sign, instead of the actual “odd power” |u|m−1u, and we will also use such a notation when m is replaced by 1/m.”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, unknown): “Bonforte we have obtained a priori upper and lower estimates of intrinsic, local type for this problem posed in RN , [50].”
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].”