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The Dynamics of Localized Solutions of Nonlocal Reaction-Diffusion Equations

Published online by Cambridge University Press:  20 November 2018

Michael J. Ward*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2
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Abstract

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Many classes of singularly perturbed reaction-diffusion equations possess localized solutions where the gradient of the solution is large only in the vicinity of certain points or interfaces in the domain. The problems of this type that are considered are an interface propagation model from materials science and an activator-inhibitor model of morphogenesis. These two models are formulated as nonlocal partial differential equations. Results concerning the existence of equilibria, their stability, and the dynamical behavior of localized structures in the interior and on the boundary of the domain are surveyed for these two models. By examining the spectrum associated with the linearization of these problems around certain canonical solutions, it is shown that the nonlocal term can lead to the existence of an exponentially small principal eigenvalue for the linearized problem. This eigenvalue is then responsible for an exponentially slow, or metastable, motion of the localized structure.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Alikakos, N., Bronsard, L. and Fusco, G., Slow motion in the Gradient Theory of Phase Transitions via Energy and Spectrum. Calc. Var. Partial Differential Equations (1) 6 (1998), 3966.Google Scholar
[2] Alikakos, N., Chen, X. and Fusco, G., Motion of a Drop by Surface Tension Along the Boundary. J. Geometrie Analysis, 1999, to appear.Google Scholar
[3] Alikakos, N. and Fusco, G., Some Aspects of the Dynamics of the Cahn-Hilliard Equation. Resenhas (4) 1 (1994), 517530.Google Scholar
[4] Alikakos, N. and Fusco, G., Slow Dynamics for the Cahn-Hilliard Equation in Higher Spatial Dimensions, Part 2: the Motion of Bubbles. Arch. Rational Mech. Anal. 141 (1998), 161.Google Scholar
[5] Alikakos, N., Fusco, G. and Kowalczyk, M., Finite Dimensional Dynamics and Interfaces Intersecting the Boundary: Equilibria and Quasi-Invariant Manifold. Indiana Univ. Math. J. (4) 45 (1996), 11191135.Google Scholar
[6] Bronsard, L. and Stoth, B., Volume Preserving Mean Curvature Flow as a Limit ofa Nonlocal Ginzburg-Landau Equation. SIAM J. Math. Anal. (4) 28 (1997), 769807.Google Scholar
[7] Bronsard, L. and Wetton, B., A Numerical Method for Tracking Curved Networks Moving with Curvature Motion. J. Comp. Phys. (1) 120 (1995), 6687.Google Scholar
[8] Chen, X., Hillhorst, D. and Logak, E., Asymptotic Behavior of Solutions of an Allen-Cahn Equation with a Nonlocal Term. Nonlinear Anal. (7) 28 (1997), 12831298.Google Scholar
[9] Chen, X. and Kowalczyk, M. M., Existence of Equilibria for the Cahn-Hilliard Equation via Local Minimizers of the Perimeter. Comm. Partial Differential Equations (7-8) 21 (1996), 12071233.Google Scholar
[10] Gage, M., On an Area-Preserving Evolution For Plane Curves. Contemp. Math. 51 (1986), 5162.Google Scholar
[11] Gage, M. and Hamilton, R. S., The Heat Equation Shrinking Convex Plane Curves. J. Differential Geom. 23 (1986), 6996.Google Scholar
[12] Gierer, A. and Meinhardt, H., A Theory of Biological Pattern Formation. Kybernetika 12 (1972), 3039.Google Scholar
[13] Gui, C., Multi-Peak Solutions for a Semilinear Neumann Problem. Duke Math. J. (3) 84 (1996), 739769.Google Scholar
[14] Gui, C. and Wei, J., Multiple Interior Peak Solutions for some Singularly Perturbed Neumann Problems. J. Differential Equations, 1999, to appear.Google Scholar
[15] Harrison, L. and Holloway, D., Order and Localization in Reaction-Diffusion Pattern. Phys. A 222 (1995), 210233.Google Scholar
[16] Iron, D. and Ward, M. J., A Metastable Spike Solutionfor a Non Local Reaction-Diffusion Model. SIAM J. Appl. Math., 1999, to appear.Google Scholar
[17] Iron, D. and Ward, M. J., Dynamics of Boundary Spikes for a Nonlocal Reaction-Diffusion Model. Submitted to European J. Appl. Math., 1999.Google Scholar
[18] Kowalczyk, M., Multiple Spike Layers in the Shadow Gierer-Meinhardt System: Existence of Equilibria and Approximate Invariant Manifold. Duke Math J. (1) 98 (1999), 59111.Google Scholar
[19] Kowalczyk, M., Exponentially Slow Dynamics and Interfaces Intersecting the Boundary. J. Differential Equations (1) 138 (1997), 5585.Google Scholar
[20] Ni, W. M. and Takagi, I., On the Shape of Least-Energy Solutions to a Semilinear Neumann Problem. Comm. Pure Appl. Math 44 (1991), 819851.Google Scholar
[21] Ni, W. M., Diffusion, Cross-Diffusion, and their Spike-Layer Steady-States. Notices Amer. Math. Soc. (1) 45 (1998), 918.Google Scholar
[22] Rubinstein, J. and Sternberg, P., Nonlocal Reaction-Diffusion Equations and Nucleation. IMA J. Appl. Math. 48 (1992), 249264.Google Scholar
[23] Stafford, D., Ward, M. J. and Wetton, B., The Dynamics of Drops and Attached Interfaces for the Constrained Allen-Cahn Equation. Submitted to European J. Appl. Math., 1999.Google Scholar
[24] Turing, A., The Chemical Basis of Morphogenesis. Philos. Trans. Roy. Soc. B 327 (1952), 3772.Google Scholar
[25] Ward, M. J., Exponential Asymptotics and Convection-Diffusion-Reaction Models. In: Analysis of Multiscale Phenomena using Singular Perturbation Methods (ed. R. O'Malley), AMS short course series in Applied Mathematics 56 (1999), 151184.Google Scholar
[26] Ward, M. J., Metastable Bubble Solutions for the Allen-Cahn Equation with Mass Conservation. SIAM J. Appl. Math. (5) 56 (1996), 12471279.Google Scholar
[27] Ward, M. J., An Asymptotic Analysis of Localized Solutions for some Reaction-Diffusion Models in Multi-Dimensional Domains. Stud. Appl. Math. (2) 97 (1996), 103126.Google Scholar
[28] Ward, M. J. and Stafford, D., Metastable Dynamics and Spatially Inhomogeneous Equilibria in Dutnbell-Shaped Domains. Stud. Appl. Math., 1999, to appear.Google Scholar
[29] Wei, J., On the Inferior Peak Solution to a Singularly Perturbed Neumann Problem. Tohoku Math. J. 50 (1998), 159178.Google Scholar
[30] Wei, J., On Single Inferior Spike Solutions of the Gierer Meinhardt System: Uniqueness and Spectrum Estimates. European J. Appl. Math. 10(1999), Part 4, 353378.Google Scholar