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Invariant manifolds for metastable patterns in \(u_ t = \varepsilon{}^ 2 u_{xx} - f(u)\). (English) Zbl 0738.35023
A previous paper of the authors [Commun. Pure Appl. Math. 42, No. 5, 523- 576 (1989; Zbl 0685.35054)] and the present paper can be regarded as part I and part II of a presentation of results on ‘metastable patterns’ in solutions of (1) \(u_ t=\varepsilon^ 2u_{xx}-f(u)\). In the first paper the authors discussed the existence and persistence of such patterns; now the geometry of the flow in function space for equation (1) is in the centre of interest. As originally conjectured by G. Fusco and I. K. Hale [J. Dyn. Differ. Equations 1, No. 1, 75-94 (1989; Zbl 0684.34055)] metastable states with \(N\) layers are associated with the unstable manifold of unstable stationary states of (1) having \(N\) layers.

35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
35B99 Qualitative properties of solutions to partial differential equations
58J32 Boundary value problems on manifolds
Full Text: DOI
[1] DOI: 10.1016/0022-0396(78)90033-5 · Zbl 0338.35055
[2] DOI: 10.1002/cpa.3160420502 · Zbl 0685.35054
[3] Brunovsky, Dynamics Reported 1 pp 57– (1988)
[4] Bates, Dynamics Reported 2 pp 1– (1989)
[5] Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics 68 (1988) · Zbl 0662.35001
[6] DOI: 10.2977/prims/1195188180 · Zbl 0445.35063
[7] DOI: 10.1016/0022-0396(75)90084-4 · Zbl 0304.35008
[8] Henry, Geometric theory of semilinear parabolic equations (1982)
[9] DOI: 10.1016/0022-0396(85)90153-6 · Zbl 0572.58012
[10] Hartman, Ordinary Differential Equations (1964) · Zbl 0125.32102
[11] Hale, Asymptotic Behavior of Dissipative Systems (1988) · Zbl 0642.58013
[12] DOI: 10.1007/BF01048791 · Zbl 0684.34055
[13] DOI: 10.1090/S0894-0347-1988-0943276-7
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