A Widder’s Type Theorem for the Heat Equation with Nonlocal Diffusion

By Begoña Barrios, I. Peral, F. Soria, Enrico Valdinoci. 2013

98 citations recorded by the source corpus.

AbstractThe main goal of this work is to prove that every non-negative strong solutionu(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ut+(-Δ)α/2u=0for(x,t)∈Rn×(0,T),0<α<2,can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$u(x,t)=∫RnPt(x-y)u(y,0)dy,where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$Pt(x)=1tn/αPxt1/α,and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$P(x):=∫Rneix·ξ-|ξ|αdξ.This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/ff633db3614779e343098989029f9938ab568f51

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Citations in the corpus

Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, methodology): “As far as I1 and I2 are concerned, up to the smooth cut-off function θn, these integrals have been considered in [6], Section 2 (I1 formula (2.”
Nonlocal Diffusion and Applications (2015, background): “Many fundamental topics slipped completely out of these notes: just to name a few, the topological methods and the fine regularity theory in the fractional cases are not presented here, the fully nonlinear or singular/degenerate equations are not taken into account, only very few applications are di”
Quantitative local and global a priori estimates for fractional nonlinear diffusion equations (2012, result): “This is an interesting result that is not present in the literature to our knowledge and complements the uniqueness results of [4].”
Optimal results for the fractional heat equation involving the Hardy potential (2014, background): “A result on the uniqueness of positive solution to the linear problem can be found in [10].”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “Likewise, we may study the fractional heat equation in classes of (maybe) large functions and pose the question: given a solution of the initial value problem posed in the whole space RN , is it representable by the convolution formula? The paper [25] by B.”
Asymptotic method of moving planes for fractional parabolic equations (2020, background): “INTRODUCTION Reaction-diffusion equations and systems have been studied very extensively in the past few years as models for many problems arising in applications such as, quasi-geostrophic flow [4], general shadow and activator-inhibitor system [36], nonlocal diffusions [1, 2], and the fractional D”