Existence of fast traveling waves for some parabolic equations: A dynamical systems approach |
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Authors: | Arnd Scheel |
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Institution: | (1) Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, D-14195 Berlin, Germany |
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Abstract: | We study semilinear elliptic equationsu + cu
x
=f(u,u) and
2
u + cu
x
=f(u,u,
2
u) in infinite cylinders (x,y) ×
n+1
using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limitc, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips. |
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Keywords: | Traveling waves inertial manifolds singular perturbation |
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