Advances in Differential Equations

Traveling waves for a bistable equation with nonlocal diffusion

Franz Achleitner and Christian Kuehn

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Abstract

We consider a single component reaction-diffusion equation in one dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. Our main result shows the existence, uniqueness (up to translations) and local asymptotic stability of a traveling wave solution connecting two stable homogeneous steady states. In particular, we provide an extension to classical results on traveling wave solutions involving local diffusion. This extension to evolution equations with Riesz-Feller operators requires several technical steps. These steps are based upon an integral representation for Riesz-Feller operators, a comparison principle, regularity theory for space-fractional diffusion equations, and control of the far-field behavior.

Article information

Source
Adv. Differential Equations Volume 20, Number 9/10 (2015), 887-936.

Dates
First available in Project Euclid: 23 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1435064517

Mathematical Reviews number (MathSciNet)
MR3360395

Zentralblatt MATH identifier
1327.35053

Subjects
Primary: 35A01: Existence problems: global existence, local existence, non-existence 35A09: Classical solutions 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 35R09: Integro-partial differential equations [See also 45Kxx] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx]

Citation

Achleitner, Franz; Kuehn, Christian. Traveling waves for a bistable equation with nonlocal diffusion. Adv. Differential Equations 20 (2015), no. 9/10, 887--936. https://projecteuclid.org/euclid.ade/1435064517.


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