Carleman inequality for a class of super strong degenerate parabolic operators and applications

Araújo, Bruno Sérgio, Universidade Federal de Campina Grande, Campina Grande, PB, Brazil
Demarque, Reginaldo, Universidade Federal Fluminense, Rio das Ostras, RJ, Brazil
Viana, Luiz, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, RJ, Brazil

Electron. J. Qual. Theory Differ. Equ. 2023, No. 9, 1-25. DOI: https://doi.org/10.14232/ejqtde.2023.1.9

Communicated by  Mugnai, Dimitri Received on 2022-08-17
Appeared on 2023-04-07
Abstract: In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in general. Additionally, we also apply the aforementioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem.

Keywords: degenerate parabolic equations, Carleman estimates, linear systems in control theory, nonlinear systems in control theory

Mathematics Subject Classification: 35K65, 93B05, 93C05, 93C10
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