Abstract
Certain singularly perturbed partial differential equations exhibit a phenomenon known as dynamic metastability, whereby the solution evolves on an asymptotically exponentially long time interval as the singular perturbation parameter e tends to zero. This article illustrates a technique to analyze metastable behavior for a range of problems in multi-dimensional domains. The problems considered include the exit problem for diffusion in a potential well, models of interface propagation in materials science, an activator-inhibitor model in mathematical biology, and a flame-front problem. Many of these problems can be formulated in terms of non-local partial differential equations. This non-local feature is shown to be essential to the existence of metastable behavior.
This work was supported by NSERC grant 5-81541.
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References
N. Alikakos, P.W. Bates, and G. Fusco, Slow Motion for the Cahn-Hilliard Equation in One Space Dimension, J. Diff. Equat. 90 (1991), 81–135.
N. Alikakos, L. Bronsard, and G. Fusco, Slow motion in the Gradient Theory of Phase Transitions via Energy and Spectrum, Calc. Var. Partial Differential Equations 6(1) (1998), 39–66.
N. Alikakos, X. Chen, and G. Fusco, Motion of a Drop by Surface Tension Along the Boundary, Preprint.
N. Alikakos and G. Fusco, Some Aspects of the Dynamics of the Cahn-Hilliard Equation, Resenhas 1(4) (1994), 517–530.
N. Alikakos and G. Fusco, Slow Dynamics for the Cahn-Hilliard Equation in Higher Spatial Dimensions, Part 2: The Motion of Bubbles, to appear, Arch. Rational Mech. Anal. (1998).
N. Alikakos, G. Fusco, and M. Kowalczyk, Finite Dimensional Dynamics and Interfaces Intersecting the Boundary: Equilibria and Quasi-Invariant Manifold, Indiana Univ. Math. J. 45(4) (1996), 1119–1155.
P.W. Bates and G. Fusco, Equilibria with Many Nuclei for the Cahn-Hilliard Equation, Preprint (1998).
P.W. Bates and J. Xun, Metastable Patterns for the Cahn-Hilliard Equation: Parts I and II, J. Diff. Equat. 111 (1994), 421–457; J. Diff. Equat. 117 (1995), 165–216.
H. Berestycki, S. Kamin, and G. Sivashinsky, Nonlinear Dynamics and Metastability in a Burgers Type Equation, Compte Rendus Acad. Sci., Paris t. 321, Serie 1 (1995), 185–190.
J. Carr and R. Pego, Metastable Patterns in Solutions of u t = ε2 u xx − f(u), Comm. Pure Appl. Math. 42 (1989), 523–576.
X. Chen and M.M. Kowalczyk, Existence of Equilibria for the Cahn-Hilliard Equation via Local Minimizers of the Perimeter, Comm. Partial Differential Equat. 21(7–8) (1996), 1207–1233.
G. Fusco and J.K. Hale, Slow Motion Manifolds, Dormant Instability and Singular Perturbations, J. Dyn. Diff. Equat. 1 (1989), 75–94.
M. Gage, On an Area-Preserving Evolution For Plane Curves, Contemp. Math. 51 (1986), 51–62.
M. Gage and R.S. Hamilton, The Heat Equation Shrinking Convex Plane Curves, J. Diff. Geom. 23 (1986), 69–96.
A. Gierer and H. Meinhardt, A Theory of Biological Pattern Formation, Kybernetik 12 (1972), 30–39.
C. Gui, and J. Wei, Multiple Interior Peak Solutions for some Singularly Perturbed Neumann Problems, to appear, Journ. of Diff. Eq. (1999).
D. Holloway, Reaction-diffusion Theory of Localized Structures with Application to Vertebrate Organogenesis, Ph.D Thesis (Chemistry), University of British Columbia, Vancouver (1995).
D. Iron and M.J. Ward, A Metastable Spike Solution for a Non-Local Reaction-Diffusion Model, to appear, SIAM J. Appl. Math. (1998).
M. Kowalczyk, Exponentially Slow Dynamics and Interfaces Intersecting the Boundary, J. Diff. Eq. 138(1) (1997), 55–85.
M. Kowalczyk, Multiple Spike Layers in the Shadow Gierer-Meinhardt System: Existence of Equilibria and Approximate Invariant Manifold, to appear, Duke Math. J. (1999).
G. Kriegsmann, Hot Spot Formation in Microwave Heated Ceramic Fibers, IMA J. Appl. Math. 59 (1997), 123–148.
J. Laforgue and R.E. O’MALLEY, On the Motion of Viscous Shocks and the Super-Sensitivity of Their Steady-State Limits, Methods and Appl. of Anal. 1 (1994), 465–487.
E. von Lavante, and R.A. Strehlow, The Mechanism of Lean Limit Flame Extinction, Combustion and Flame 49 (1983), 123–140.
D. Ludwig, Persistence of Dynamical Systems under Random Perturbations, SIAM Review 17 (1975), 605–640.
R.S. Maier and D.L. Stein, Limiting Exit Location Distributions in the Stochastic Exit Problem, SIAM J. Appl. Math. 57(3) (1997), 752–790.
H. Matano, Asymptotic Behavior and Stability of Semilinear Diffusion Equations, Publ. Res. Inst. Math., Kyoto 15, (1979), 401–451.
B.J. Matkowsky and Z. Schuss, The Exit Problem for Randomly Perturbed Dynamical Systems, SIAM J. Appl. Math. 33 (1977), 365–382.
W.M. Ni, On the Shape of Least-Energy Solutions to a Semilinear Neumann Problem, Comm. Pure Appl. Math. XLIV (1991), 819–851.
W.M. Ni, Diffusion, Cross-Diffusion, and their Spike-Layer Steady-States, Notices of the AMS 45(1) (1998), 9–18.
Z. Rakib and G.I. Sivashinsky, Instabilities in Upward Propagating Flames, Combust. Sci. and Tech. 54 (1987), 69–84.
L.G. Reyna and M.J. Ward, On the Exponentially Slow Motion of a Viscous Shock, Comm. Pure Appl. Math. 48 (1995), 79–120.
L.G. Reyna and M.J. Ward, Metastable Internal Layer Dynamics for the Viscous Cahn-Hilliard Equation, Methods and Appl. of Anal. 2(3) (1995), 285–306.
J. Rubinstein and P. Sternberg, Nonlocal Reaction-Diffusion Equations and Nucleation, IMA J. Appl. Math. 48 (1992), 249–264.
J. Rubinstein, P. Sternberg, and J. Keller, Fast Reaction, Slow Diffusion, and Curve Shortening, SIAM J. Appl. Math. 49(1) (1989), 116–133.
D. Stafford and M.J. Ward, Metastable Dynamics and Spatially Inhomogeneous Equilibria in Dumbell-Shaped Domains, to appear, Studies in Applied Math (1999).
D. Stafford, M.J. Ward, and B. Wetton, The Dynamics of Drops and Attached Interfaces for the Constrained Allen-Cahn Equation, submitted, Europ. J. Appl. Math. (1999).
X. Sun and M.J. Ward, Metastability for a Generalized Burgers Equation with Applications to Propagating Flame-Fronts, European J. Appl. Math. 10 (1999), 27–53.
X. Sun and M.J. Ward, Exponentially Ill-Conditioned Convection-Diffusion Equations in Multi-Dimensional Domains, submitted, Methods and Appl. of Analy. (1999).
A. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. B 327 (1952), 37–72.
M.J. Ward, Metastable Patterns, Layer Collapses, and Coarsening for a One-Dimensional Ginzburg-Landau Equation, Stud. Appl. Math. 91 (1994), 51–93.
M.J. Ward, Metastable Bubble Solutions for the Allen-Cahn Equation with Mass Conservation, SIAM J. Appl. Math. 56(5) (1996), 1247–1279.
M.J. Ward, An Asymptotic Analysis of Localized Solutions for some Reaction-Diffusion Models in Multi-Dimensional Domains, Stud. Appl. Math 97(2) (1996), 103–126.
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Ward, M.J. (2001). Metastable dynamics and exponential asymptotics in multi-dimensional domains. In: Jones, C.K.R.T., Khibnik, A.I. (eds) Multiple-Time-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 122. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0117-2_9
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DOI: https://doi.org/10.1007/978-1-4613-0117-2_9
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