Green’s Function Estimates for Time-Fractional Evolution Equations
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Received: 3 June 2019 / Revised: 18 June 2019 / Accepted: 20 June 2019 / Published: 25 June 2019
PDF [446 KB, uploaded 27 June 2019]
We look at estimates for the Green’s function of time-fractional evolution equations of the form
is a Caputo-type time-fractional derivative, depending on a Lévy kernel
with variable coefficients, which is comparable to
, and L
is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of
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
is a pseudo-differential operator with constant coefficients that is homogeneous of order
. Thirdly, we obtain local two-sided estimates for the Green’s function of
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
, 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
is a Caputo-type operator with variable coefficients.
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MDPI and ACS Style
Johnston, I.; Kolokoltsov, V. Green’s Function Estimates for Time-Fractional Evolution Equations. Fractal Fract 2019, 3, 36.
Johnston I, Kolokoltsov V. Green’s Function Estimates for Time-Fractional Evolution Equations. Fractal and Fractional. 2019; 3(2):36.
Johnston, Ifan; Kolokoltsov, Vassili. 2019. "Green’s Function Estimates for Time-Fractional Evolution Equations." Fractal Fract 3, no. 2: 36.
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