An Introduction to Semiflows

Milani, Albert; Koksch, N.

doi:10.1201/9781420035117

Cited by 17 publications

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“…This Theorem is also reproduced in [8]. Observe that η is restricted to be positive, but otherwise can be large.…”

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confidence: 85%

“…The numbers c j appearing in (8) are computed by using the Taylor series of g(ξ) and with this and some elementary inequalities, one proves that M is a contraction. The fact that [c 1+η , c] ⊂ |σ(M )| ⊂ [c 1+η , c + r] comes from the fact that the spectrum of J is the unit disc, N 1−r is an automorphism of the unit disc and the Spectral Mapping Theorem.…”

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confidence: 98%

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Known Results and Open Problems on C1 linearization in Banach Spaces

Rodrigues

Solà-Morales

2012

São Paulo J. Math. Sci.

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Abstract. The purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimensional Banach spaces, especially in the C 1 case, and also to present some open problems that we believe that are still important for the understanding of the theory.The purpose of the present paper is to make a review of results obtained by the authors in the field of linearization of dynamical systems in infinite dimensional Banach spaces, with a focuss on C 1 linearization of contractions, with the intention of stating in the right context a number of open problems we believe it would be worth to be solved.1991 Mathematics Subject Classification. AMS Classification. Primary: 37C15, 35B05 Secondary: 34C20, 37D05.

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“…This Theorem is also reproduced in [8]. Observe that η is restricted to be positive, but otherwise can be large.…”

mentioning

confidence: 85%

mentioning

confidence: 98%

Known Results and Open Problems on C1 linearization in Banach Spaces

Rodrigues

Solà-Morales

2012

São Paulo J. Math. Sci.

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“…Based on Section 3.4 and the work [18,Section 4], the upper-semicontinuity result may follow by assuming only f, g ∈ C(R) satisfy (1.5)-(1.8). However, with additional regularity properties from the global attractors, the more traditional arguments used to show upper-semicontinuity of global attractors in [28,29] (also see [32,Theorem 3.31]) may also prove successful. On the other hand, one may inquire about applying a perturbation parameter ε to the inertial terms u tt , then letting this approach zero.…”

Section: Discussionmentioning

confidence: 99%

Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions

Graber

Shomberg

2016

Nonlinearity

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Abstract. We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying operator is analytic, α > 0, or only of Gevrey class, α = 0. We establish the existence of a global attractor for each α ∈ [0, 1], and we show that the family of global attractors is upper-semicontinuous as α → 0. Furthermore, for each α ∈ [0, 1], we show the existence of a weak exponential attractor. A weak exponential attractor is a finite dimensional compact set in the weak topology of the phase space. This result insures the corresponding global attractor also possess finite fractal dimension in the weak topology; moreover, the dimension is independent of the perturbation parameter α. In both settings, attractors are found under minimal assumptions on the nonlinear terms.

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“…The argument utilizes the sequential characterization of the global attractor (cf., e.g. [40,Proposition 2.15]). The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, solutions of the hyperbolic problem are defined for (u 0 , u 1 ) ∈ H s+1 (Ω) × H s (Ω), s ∈ {0, 1}, while solutions of the parabolic problem make sense only in spaces like L 2 (Ω) × L 2 (Γ) and H s+1 (Ω) × H s+1/2 (Γ) , respectively (see (1.10)).…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic relaxation of reaction-diffusion equations with dynamic boundary conditions

Gal

Shomberg

2015

Quart. Appl. Math.

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Abstract. Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation,on a bounded domain Ω ⊂ R 3 , with ε ∈ (0, 1] and the prescribed dynamic condition,on the boundary Γ := ∂Ω. We also consider the limit parabolic problem (ε = 0) with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. Because of the nature of the boundary condition, fractional powers of the Laplace operator are not well-defined. The precompactness property required by the hyperbolic semiflows for the existence of the global attractors is gained through the approach of [44]. In this case, the optimal regularity for the global attractors is also readily established. In the parabolic setting, the regularity of the global attractor is necessary for the semicontinuity result. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global attractors is given at ε = 0. Finally, we also establish the existence of a family of exponential attractors.

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An Introduction to Semiflows

Cited by 17 publications

References 0 publications

Known Results and Open Problems on C1 linearization in Banach Spaces

Known Results and Open Problems on C1 linearization in Banach Spaces

Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions

Hyperbolic relaxation of reaction-diffusion equations with dynamic boundary conditions

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