Structure of solution manifolds in a strongly coupled elliptic system

López-Gómez, Julián; Eilbeck, J. C.; Molina, Mauricio Godoy; Duncan, Kirsteen

doi:10.1093/imanum/12.3.405

Cited by 31 publications

(31 citation statements)

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“…Furthermore, as the number of modes increases the approximated compact component converges to the corresponding continuous one (cf. [LEDM92] and the references therein).…”

Section: Pseudo-spectral Methods Coupled With Path Followingmentioning

confidence: 99%

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The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction–Diffusion Equations

Gómez-Reñasco

López-Gómez

2000

Journal of Differential Equations

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In this paper we analyze how the dynamics of a class of superlinear indefinite reaction diffusion equations varies as the nodal behavior of a coefficient changes.To perform this analysis we use both theoretical and numerical tools. The analysis aids the numerical study, and the numerical study confirms and completes the analysis. The numerics in addition provides us with some further results for whichat-first glance analytical tools are not available yet. Our main analytical result shows that the problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution. This result is a relevant feature for superlinear indefinite problems, since our numerical computations show how these models can have an arbitrarily large number of positive solutions if the trivial state is unstable. 2000Academic Press

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“…Furthermore, as the number of modes increases the approximated compact component converges to the corresponding continuous one (cf. [LEDM92] and the references therein).…”

Section: Pseudo-spectral Methods Coupled With Path Followingmentioning

confidence: 99%

“…To compute the global solution curves of (4.1) as well as the dimensions of the unstable manifolds of their solutions we have used standard path-following techniques as given in [DK81,Ei86,LEDM92] and the references therein, to which we send the reader for further details.…”

Section: Pseudo-spectral Methods Coupled With Path Followingmentioning

confidence: 99%

The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction–Diffusion Equations

Gómez-Reñasco

López-Gómez

2000

Journal of Differential Equations

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“…In all our numerical calculations we have used trigonometric modes and the collocation points have been taken to be equidistant, with the number of modes equal to the number of collocation points. The details of our numerical scheme may be found in [14] and the references therein. We refer the interested reader to [14].…”

Section: On the Existence Of Positive Solutionsmentioning

confidence: 99%

“…The details of our numerical scheme may be found in [14] and the references therein. We refer the interested reader to [14]. calculations were carried out for the choices In Figure 2.1 we have divided the (λ, · ∞ )-diagram of non-negative solutions into three pieces, those in the first column, corresponding to a different range of values of λ.…”

Section: On the Existence Of Positive Solutionsmentioning

confidence: 99%

“…This explains as well why a high number of modes might be necessary to get a reasonable approximation when spatially varying coefficients arise in the formulation of the model, in strong contrast with the case of constant coefficients where in general a low number of modes are sufficient to get good numerical approximations even when dealing with systems (cf. [14] and the references therein).…”

Section: On the Existence Of Positive Solutionsmentioning

confidence: 99%

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Pointwise Growth and Uniqueness of Positive Solutions for a Class of Sublinear Elliptic Problems where Bifurcation from Infinity Occurs

García-Melián

Gómez-Reñasco

López-Gómez

et al. 1998

Archive for Rational Mechanics and Analysis

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In this paper we analyze the uniqueness and the pointwise growth of the positive solutions of a nonlinear elliptic boundary-value problem of general sublinear type with a weight function multiplying the nonlinearity. When this function vanishes on some subdomain, the problem exhibits a bifurcation from infinity. In this case almost nothing is known about the pointwise growth of the positive solutions as the parameter approaches the critical value where the bifurcation from infinity occurs. In this work we show that the positive solutions grow to infinity in the region where the weight function vanishes and that on its support they stabilize to the minimal positive solution of the original equation subject to infinite Dirichlet boundary conditions. This behavior provides us with the uniqueness of the positive solution near the value of the parameter where the bifurcation from infinity occurs. Also, we solve the problem using spectral collocation methods coupled with path-following techniques to show how the main uniqueness result is optimal. Throughout the paper the mathematical analysis aids the numerical study, and the numerical study confirms and illuminates the analysis.

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In the blink of an Eye

López-Gómez

Molina-Meyer

Progress in Nonlinear Differential Equations and Their Applications

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Structure of solution manifolds in a strongly coupled elliptic system

Cited by 31 publications

References 0 publications

The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction–Diffusion Equations

The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction–Diffusion Equations

Pointwise Growth and Uniqueness of Positive Solutions for a Class of Sublinear Elliptic Problems where Bifurcation from Infinity Occurs

In the blink of an Eye

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