Green’s function estimates for time-fractional evolution equationsстатья
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Дата последнего поиска статьи во внешних источниках: 27 октября 2021 г.
Авторы:
Johnston I. ,
Kolokoltsov V.
Журнал:
Fractal and Fractional
Том:
3
Номер:
2
Год издания:
2019
Издательство:
MDPI Publishing
Местоположение издательства:
Basel, Switzerland, Switzerland
Первая страница:
1
Последняя страница:
38
DOI:
10.3390/fractalfract3020036
Аннотация:
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. © 2019 by the authors. Licensee MDPI, Basel, Switzerland.
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