Green's function estimates for time fractional evolution equations

DOI10.3390/FRACTALFRACT3020036arXiv1906.12157MaRDI QIDQ6321215

Vassili Kolokoltsov, Ifan Johnston

Publication date: 28 June 2019

Abstract: We look at estimates for the Green's function of time-fractional evolution equations of the form $D_{0+*}^{u}u=Lu$, where $D_{0+*}^u$ is a Caputo-type time-fractional derivative, depending on a L'evy kernel $u$ with variable coefficients, which is comparable to $$ for $$, and $L$ is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green's function of $$ in the case that $L$ is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green's function of $$ where $Psi$ is a pseudo-differential operator with constant coefficients that is homogeneous of order $alpha$. Thirdly, we obtain local two-sided estimates for the Green's function of $$ 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 $Psi$, 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 $D_0^{(u,t)}u=Lu$, where $D^{(u,t)}$ is a Caputo-type operator with variable coefficients.

Mathematics Subject Classification ID

Mittag-Leffler functions and generalizations (33E12) Laplace transform (44A10) Stable stochastic processes (60G52) Transition functions, generators and resolvents (60J35) Green's functions for elliptic equations (35J08)

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