Quantitative approximation properties for the fractional heat equation

• Published in: Mathematical control and related fields Vol. 10; no. 1; pp. 1 - 26
• Main Authors: Rüland, Angkana, Salo, Mikko
• Format: Journal Article
• Language: English
• Published: 01.03.2020
• Subjects: Fractional parabolic Calderón problem; nonlocal operators; Runge approximation; cost of approximation; weak unique continuation
• ISSN: 2156-8499; 2156-8472
• DOI: 10.3934/mcrf.2019027

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.