Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 3

Unique continuation results for abstract quasi-linear evolution equations in Banach spaces

Pages: 179 – 195

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n3.a1


Igor Leite Freire (Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom; and Departamento de Matemática, Universidade Federal de São Carlos, SP, Brasil)


Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions in a suitable Banach space. The second one considers well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the potential and $\pi\textrm{–Camassa–Holm}$; generalised Boussinesq equations; and the modified Euler–Poisson system.


conserved quantities, unique continuation of solutions, local wellposedness

2010 Mathematics Subject Classification

Primary 35A01. Secondary 35Q51, 37K40, 74G25.

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The author’s work was supported by CNPq (grant no. 310074/2021-5) and FAPESP (grant no. 2020/02055-0) for financial support.

Received 28 March 2023

Published 19 May 2023