Large Time Behavior via the Method of ℓ-Trajectories

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Long-time dynamics of Kirchhoff wave models with strong nonlinear damping (2010, methodology): “] denotes the integer part of T. Now we can apply the stabilizability estimate in (68) with t = k for each k and obtain (72). Proof of Theorem 3.11 We use the idea due to M´alek–Neˇcas [29] (see also [30] and [13]). For some T ≥ 1 which we specify latter and for some l > max{d,6}/4 we consider th”
Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains (2008, methodology): “Another powerful and useful method to prove the finite dimensionality of the global attractor is based on the so-called l-trajectories: one needs minimal regularity on the solutions in order to apply this method, see [32,53,54,152–154,173,190], and [192].”
On Finite Fractal Dimension of the Global Attractor for the Wave Equation with Nonlinear Damping (2002, methodology): “In a subsequent paper [ 7 ] this method is also shown to yield the existence of an exponential attractor, and the ‘‘trick’’ with considering the end-points of trajectories is used to transfer the results back to the phase space of initial conditions.”