## Differential and Integral Equations

- Differential Integral Equations
- Volume 25, Number 11/12 (2012), 1143-1174.

### Traveling waves of a non-local conservation law

#### Abstract

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 $n\rightarrow\infty$.

#### Article information

**Source**

Differential Integral Equations, Volume 25, Number 11/12 (2012), 1143-1174.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012255

**Mathematical Reviews number (MathSciNet)**

MR3013408

**Zentralblatt MATH identifier**

1274.35230

**Subjects**

Primary: 35L65: Conservation laws 35L67: Shocks and singularities [See also 58Kxx, 76L05] 35R09: Integro-partial differential equations [See also 45Kxx]

#### Citation

Chen, Xinfu; Jiang, Huiqiang. Traveling waves of a non-local conservation law. Differential Integral Equations 25 (2012), no. 11/12, 1143--1174. https://projecteuclid.org/euclid.die/1356012255