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On the parabolic Harnack inequality for non-local diffusion equations. (English) Zbl 1446.35246

In this very interesting paper the authors exhibit a counter-example for dimensions \(d \geq \beta\), where \(\beta \in(0, 2]\) is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times \(t > 0\). This fact shows the big difference between time-fractional diffusion and the purely space-fractional case, where a local Harnack inequality holds. The key observation is that the memory strongly affects the estimates. The authors shows also that if the initial data \(u_0 \in L^q_{\mathrm{loc}}\) for \(q\) larger than the critical value \(\frac{d}{\beta}\) of the elliptic operator \((-\Delta)^{\frac{\beta}{2}}\), a non-local version of the Harnack inequality still holds. Lastly the authors prove that the diffusion behavior is substantially different in higher dimensions than \(d = 1\) provided \(\beta > 1\), since the local Harnack inequality holds if \(d<\beta\).

MSC:

35R11 Fractional partial differential equations
45K05 Integro-partial differential equations
45M05 Asymptotics of solutions to integral equations
35C15 Integral representations of solutions to PDEs
26A33 Fractional derivatives and integrals
35B40 Asymptotic behavior of solutions to PDEs
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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