Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

Article type: Research Article

Authors: Caraballo, Tomás^a | Carvalho, Alexandre N.^{b; *} | Langa, José A.^a | Oliveira-Sousa, Alexandre N.^{a; b}

Affiliations: [a] Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain. E-mails: caraball@us.es, langa@us.es | [b] Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Brazil. E-mails: andcarva@icmc.usp.br, alexandrenosousa@gmail.com

Correspondence: [*] Corresponding author. E-mail: andcarva@icmc.usp.br.

Abstract: In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.

Keywords: Nonuniform exponential dichotomy, robustness, permanence of hyperbolic equilibria

DOI: 10.3233/ASY-211719

Journal: Asymptotic Analysis, vol. 129, no. 1, pp. 1-27, 2022