Article
The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations
Author/s | Caraballo Garrido, Tomás
Carvalho, Alexandre Nolasco Langa Rosado, José Antonio Oliveira Sousa, Alexandre N. |
Department | Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
Publication Date | 2021-01-01 |
Deposit Date | 2021-03-19 |
Published in |
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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 ... 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 dfferential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation. |
Citation | Caraballo Garrido, T., Carvalho, A.N., Langa Rosado, J.A. y Oliveira Sousa, A.N. (2021). The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations. Journal of Mathematical Analysis and Applications, 500 (2), 125134-1-125134-28. |
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