Abstract
In this chapter we study uniform trajectory attractors for non-autonomous nonlinear systems. In Sect. 8.1 we establish the existence of uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathéodory’s nonlinearity. Section 8.2 devoted to structural properties of the uniform global attractor for non-autonomous reaction-diffusion system in which uniqueness of Cauchy problem is not guarantied. In the case of translation compact time-depended coefficients it is established that the uniform global attractor consists of bounded complete trajectories of corresponding multi-valued processes. Under additional sign conditions on non-linear term we also prove (and essentially use previous result) that the uniform global attractor is, in fact, bounded set in \(L^{\infty }(\varOmega )\cap H_0^1(\varOmega )\). Section 8.3 devoted to uniform trajectory attractors for nonautonomous dissipative dynamical systems. As applications we may consider FitzHugh–Nagumo system (signal transmission across axons), complex Ginzburg–Landau equation (theory of superconductivity), Lotka–Volterra system with diffusion (ecology models), Belousov–Zhabotinsky system (chemical dynamics) and many other reaction-diffusion type systems from Sect. 2.4.
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Notes
- 1.
i.e. \(V_i\) is a real reflexive separable Banach space continuously and densely embedded into a real Hilbert space H, H is identified with its topologically conjugated space \(H^*\), \(V_i^*\) is a dual space to \(V_i\). So, there is a chain of continuous and dense embeddings: \(V_i\subset H\equiv H^*\subset V_i^*\).
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Zgurovsky, M.Z., Kasyanov, P.O. (2018). Uniform Trajectory Attractors for Non-autonomous Nonlinear Systems. In: Qualitative and Quantitative Analysis of Nonlinear Systems. Studies in Systems, Decision and Control, vol 111. Springer, Cham. https://doi.org/10.1007/978-3-319-59840-6_8
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