AIMS Mathematics, 2020, 5(6): 5470-5494. doi: 10.3934/math.2020351

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Long time dynamics for functional three-dimensional Navier-Stokes-Voigt equations

1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, SPAIN
2 Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, Ctra. de Utrera, Km. 1, 41013-Sevilla, SPAIN

In this paper we consider a non-autonomous Navier-Stokes-Voigt model including a variety of delay terms in a unified formulation. Firstly, we prove the existence and uniqueness of solutions by using a Galerkin scheme. Next, we prove the existence and eventual uniqueness of stationary solutions, as well as their exponential stability by using three methods: first, a Lyapunov function which requires differentiability for the delays; next we exploit the Razumikhin technique to weaken the differentiability assumption to just continuity; finally, we use a Gronwall-like type of argument to provide sufficient conditions for the exponential stability in a general case which, in particular, for a situation of variable delay, it only requires measurability of the variable delay function.
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