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

Author

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

Abstract

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.

Keywords

conserved quantities, unique continuation of solutions, local wellposedness

2010 Mathematics Subject Classification

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

Full Text (PDF format)

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