## 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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### References:

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