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Quantitative approximation properties for the fractional heat equation

  • * Corresponding author: Angkana Rüland

    * Corresponding author: Angkana Rüland 
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  • In this article we analyse quantitative approximation properties of a certain class of nonlocal equations: Viewing the fractional heat equation as a model problem, which involves both local and nonlocal pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain qualitative approximation results from [9]. Using propagation of smallness arguments, we then provide bounds on the cost of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss generalizations of these results to a larger class of operators involving both local and nonlocal contributions.

    Mathematics Subject Classification: Primary: 35R30, 35A35, 35S11; Secondary: 58E20.


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  • Figure 1.  The annulus from the proof of Corollary 1. The shaded area corresponds to the annulus $ A_{R_1(\tilde{k}_1),R_2(\tilde{k}_1)}(\tilde{k}_1) $ with $ \tilde{k}_1 $ chosen to be $ i|k_0| $. The bold red line (see the online version for a colour figures) indicates the discontinuity line of the function $ e^{s_1 \log(k_1^2 + |k_0|^2)} $

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