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November/December 2012 Traveling waves of a non-local conservation law
Xinfu Chen, Huiqiang Jiang
Differential Integral Equations 25(11/12): 1143-1174 (November/December 2012). DOI: 10.57262/die/1356012255


We consider traveling waves related to shocks for a non-local scalar conservation law \[ u_{t}+(f(u))_{x}=K\ast u-u, \] where $f$ is an arbitrary convex function, $K\ast u$ stands for the convolution in the spatial variable $x$, and $K$ is an arbitrary non-negative kernel with unit mass centered at the origin. Given a pair of values at $\pm\infty$ with $u_{-} > u_{+}$, we establish here the existence and uniqueness of a traveling-wave solution $u=\phi\left( x-ct\right) $ such that the wave speed \[ c=\frac{f\left( u_{+}\right) -f\left( u_{-}\right) }{u_{+}-u_{-}} \] and $\phi$, satisfying $\phi\left( \pm\infty\right) =u_{\pm}$, is a smooth decreasing function that can have at most one jump discontinuity at the origin. Our approach here is to consider a family of ``truncated" problems on $\left[ -n,n\right] $ which can be solved using Schauder's fixed-point theorem and then sending $n\rightarrow\infty$.


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Xinfu Chen. Huiqiang Jiang. "Traveling waves of a non-local conservation law." Differential Integral Equations 25 (11/12) 1143 - 1174, November/December 2012.


Published: November/December 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1274.35230
MathSciNet: MR3013408
Digital Object Identifier: 10.57262/die/1356012255

Primary: 35L65 , 35L67 , 35R09

Rights: Copyright © 2012 Khayyam Publishing, Inc.


Vol.25 • No. 11/12 • November/December 2012
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