Stationary Solutions and Connecting Orbits for p-Laplace Equation

Ćwiszewski, Aleksander; Maciejewski, Mateusz

doi:10.1007/s10884-017-9573-7

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…It is known (see e.g. [16,Th. 3.5]) that, for any u 0 ∈ X , any integral solution of (5.11) has the following properties…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Invariance and Strict Invariance for Nonlinear Evolution Problems with Applications

Ćwiszewski¹,

Gabor²,

Kryszewski³

2020

Preprint

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Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) m-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraint functional are expressed in terms of the Dini derivative of that functional, outside the considered sublevel set in directions determined by the governing m-accretive operator. An approach for non-reflexive Banach spaces is developed and some result improving a recent paper [7] is presented. Applications to nonlinear obstacle problems and agestructured population models are presented in spaces of continuous functions where advantages of that approach are taken. Moreover, some new abstract criteria for the so-called strict invariance are derived and their direct applications to problems with barriers are shown.

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“…It is known (see e.g. [16,Th. 3.5]) that, for any u 0 ∈ X , any integral solution of (5.11) has the following properties…”

Section: 2mentioning

confidence: 99%

“…It is well known that A L 2 is m-accretive (see e.g. [16,Lem. 3.4]) and clearly the part of A L 2 in X is equal to A.…”

Section: 2mentioning

confidence: 99%

Invariance and Strict Invariance for Nonlinear Evolution Problems with Applications

Ćwiszewski¹,

Gabor²,

Kryszewski³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In addition, the global dynamics of the Swift-Hohenberg model has been uncovered using a Morse decomposition based approach in [17], and the existence of periodic orbits for the ill-posed Boussinesq equation was established in [7,15]. Finally, the existence of a heteroclinic orbit between the trivial and a nontrivial stationary state has been established for the one-dimensional p-Laplace equation in [66]. Also, private communication revealed that Zgliczyński could verify the existence of a heteroclinic connection in the Kuramoto-Sivashinsky equation using similar techniques in the recent work in progress [64].…”

Section: Introductionmentioning

confidence: 99%

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Cyranka¹,

Wanner²

2018

SIAM J. Appl. Dyn. Syst.

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We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state and local and global energy minimizers.The proof of the above statement is conceptually simple, and combines several techniques from some of the authors' and Zgliczyński's works. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This propagation has to lead to small interval bounds, while at the same time entering the basin of attraction of the stable fixed point. For interesting parameter values the global attractor exhibits a complicated equilibrium structure, and the dynamical equation is rather stiff. This leads to a time-consuming numerical propagation of error bounds, with many integration steps. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic at various nontrivial parameter values.

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Invariance and strict invariance for nonlinear evolution problems with applications

2022

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Stationary Solutions and Connecting Orbits for p-Laplace Equation

Abstract: We deal with one dimensional p-Laplace equation of the formunder Dirichlet boundary condition, where p > 2 and f :

Cited by 4 publications

References 15 publications

Invariance and Strict Invariance for Nonlinear Evolution Problems with Applications

Invariance and Strict Invariance for Nonlinear Evolution Problems with Applications

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Invariance and strict invariance for nonlinear evolution problems with applications

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