The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

Author(s)

Caraballo, Tomas ; Carvalho, Alexandre N. ; Langa, Jose A. ; Oliveira-Sousa, Alexandre N.

• Document type: Journal article
• Source: Journal of Mathematical Analysis and Applications; v. 500, n. 2, p. 27-pg., 2021-03-15.

Abstract

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation. (C) 2021 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 18/10997-6 - Robustness of attractors under autonomous or non-autonomous perturbatinos: Structural Stability
• Grantee: Alexandre Nolasco de Carvalho
• Support Opportunities: Scholarships abroad - Research
• FAPESP's process: 18/10633-4 - A study of structural stability for random attractors
• Grantee: Alexandre do Nascimento Oliveira Sousa
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• FAPESP's process: 17/21729-0 - A study about structural stability of atrators for random dynamical systems
• Grantee: Alexandre do Nascimento Oliveira Sousa
• Support Opportunities: Scholarships in Brazil - Doctorate