On the space-like analyticity in the extension problem for nonlocal parabolic equations

Agnid Banerjee; Nicola Garofalo

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arXiv · 2204.08948

On the space-like analyticity in the extension problem for nonlocal parabolic equations

Agnid Banerjee·Nicola Garofalo

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Abstract

In this note we give an elementary proof of the space-like real analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type $(\partial_t - div_x(B(x)\nabla_x))^s$, $0<s<1$. Our primary interest is in the so-called \emph{extension variable}. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where $B(x)$ is a uniformly elliptic matrix-valued function with real-analytic entries.

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Agnid Banerjee, Nicola Garofalo. 2022-04-19. On the space-like analyticity in the extension problem for nonlocal parabolic equations. https://arxiv.org/abs/2204.08948

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