Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit

Mark I. Vishik; Sergey V. Zelik

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Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit

Mark I. Vishik, Sergey V. Zelik

Analysis of PDEs · Submitted 9 Oct 2011

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The authors' main claim We prove that, under natural assumptions on the nonlinear interaction function and the external forces, this problem possesses the uniform attractors and that these attractors tend to the attractor of the limit parabolic equation.

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arXiv ID1110.1834 (version 1)

Pith ID7ZXWBOE5, Pith's permanent record of this paper and its review

CitationMark I. Vishik, Sergey V. Zelik. "Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit." arXiv:1110.1834 (2011). https://pith.science/paper/7ZXWBOE5

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.26 · compares with prior results 0.12 · mentions code or data availability 0.01.

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We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter at the second derivative with respect to the "time" variable corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function and the external forces, this problem possesses the uniform attractors and that these attractors tend to the attractor of the limit parabolic equation. Moreover, in case where the limit attractor is regular, we give the detailed description of the structure of these uniform attractors when the perturbation parameter is small enough, and estimate the symmetric distance between the perturbed and non-perturbed attractors.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter at the second derivative with respect to the "time" variable corresponding to the axis of the cylinder.	contribution 0.55	0.89	0.04	asserted 0.92
s1	We prove that, under natural assumptions on the nonlinear interaction function and the external forces, this problem possesses the uniform attractors and that these attractors tend to the attractor of the limit parabolic equation.	result 0.99	0.95	0.80	asserted 1.00
s2	Moreover, in case where the limit attractor is regular, we give the detailed description of the structure of these uniform attractors when the perturbation parameter is small enough, and estimate the symmetric distance between the perturbed and non-perturbed attractors.	result 0.88	0.95	0.20	asserted 1.00

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Original abstract

We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter at the second derivative with respect to the "time" variable corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function and the external forces, this problem possesses the uniform attractors and that these attractors tend to the attractor of the limit parabolic equation. Moreover, in case where the limit attractor is regular, we give the detailed description of the structure of these uniform attractors when the perturbation parameter is small enough, and estimate the symmetric distance between the perturbed and non-perturbed attractors.

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In this section we prove the existence of a solution to ([ref]) for small $Ṡ_L^2$ in the case of hyperbolic stationary state $S_+=(s_+,0).$ We adapt to our case the methods [ref] , [ref] of construction of stable and unstable invariant manifolds in the hyperbolic case.

On Asymptotic Completeness of Scattering in the Nonlinear Lamb System, II · Incoming trajectories · 0.96

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