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Approximately invariant manifolds and global dynamics of spike states. (English) Zbl 1157.37013

It is proved that if an infinite-dimensional dynamical system has the given approximately invariant and approximately normally hyperbolic manifold, then this system has a true invariant manifold nearby. This result is applied to reveal the global dynamics of boundary spike states for the following generalization of the Allen-Cahn equation with small diffusion parameter \(0<\varepsilon\leq 1\), \(u_t=\varepsilon^2\Delta u-u+f(u), x\in \Omega, \frac{\partial u}{\partial n}=0,x\in \partial \Omega.\) Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain with smooth boundary, \(n\) is the outward unit normal vector to \(\partial \Omega\), and \(f\in C^1\) is such that there is a non-degenerate positive radially symmetric ground state of the corresponding rescaled elliptic problem in \(\mathbb{R}^n\).

MSC:

37D10 Invariant manifold theory for dynamical systems
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
35K55 Nonlinear parabolic equations
58J10 Differential complexes
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