00-193 V. Rottsch\"afer, C. E. Wayne

Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation (290K, Postscript) Apr 20, 00

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Abstract. We study traveling wave solutions to a general fourth-order differential equation that is a singular perturbation of the Fisher-Kolmogorov equation. We apply the geometric method for singularly perturbed systems to show that for every positive wavespeed there exists a traveling wave. We also find that there exists a critical wave speed $c^*$ which divides these solutoins into monotonic ($c\ge c^*$) and oscillatory ($c < c^*$) solutions. We show that the monotonic fronts are locally stable under perturbations in appropriate weighted Sobolev spaces by using various energy functionals.

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