TY - JOUR

T1 - Green’s function estimates for time-fractional evolution equations

AU - Johnston, Ifan

AU - Kolokoltsov, Vassili

N1 - Publisher Copyright: © 2019 by the authors. Licensee MDPI, Basel, Switzerland. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019/6

Y1 - 2019/6

N2 - We look at estimates for the Green’s function of time-fractional evolution equations of the form D0+∗ν u = Lu, where Dν0+∗ is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y−1−β for β ∈ (0, 1), and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D0β u = Lu in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D0β u = Ψ(−i∇)u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α. Thirdly, we obtain local two-sided estimates for the Green’s function of D0β u = Lu where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ, as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D0(ν,t) u = Lu, where D(ν,t) is a Caputo-type operator with variable coefficients.

AB - We look at estimates for the Green’s function of time-fractional evolution equations of the form D0+∗ν u = Lu, where Dν0+∗ is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y−1−β for β ∈ (0, 1), and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D0β u = Lu in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D0β u = Ψ(−i∇)u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α. Thirdly, we obtain local two-sided estimates for the Green’s function of D0β u = Lu where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ, as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D0(ν,t) u = Lu, where D(ν,t) is a Caputo-type operator with variable coefficients.

KW - Aronson estimates

KW - Caputo derivative

KW - Fractional evolution

KW - Green’s function

KW - Two-sided estimates

UR - http://www.scopus.com/inward/record.url?scp=85089836131&partnerID=8YFLogxK

U2 - 10.3390/fractalfract3020036

DO - 10.3390/fractalfract3020036

M3 - Article

AN - SCOPUS:85089836131

VL - 3

SP - 1

EP - 38

JO - Fractal and Fractional

JF - Fractal and Fractional

SN - 2504-3110

IS - 2

M1 - 36

ER -