Semi-wavefront solutions in models of collective movements with density-dependent diffusivity (Q509291)

This paper deals with the scalar parabolic equation \[ \rho_{t} + f(\rho)_{x} = (D(\rho)\rho_{x})_{x} + g(\rho), \;(x,t) \in {\mathbb R}\times [0,\infty). \eqno{(1)} \] Here \(f\in C^{1}[0,\bar{\rho}]\), \(f(0)=0\), \(g\in C[0,\bar{\rho}]\), \(D\in C^{1}[0,\bar{\rho}]\), \(D>0\) for \(\rho>0\), and \(D\) vanishes at \(\rho=0\), \(\bar{\rho}\) is a positive constant. A traveling wave solution \(u=\varphi(x-ct)\) to the equation (1) with a monotonic \(\varphi(\xi)\) is said to be a semi-wavefront solution whenever either \(D(\varphi)=(\omega_{0},+\infty)\) and \(\lim_{\xi\to +\infty}\varphi(\xi)=l^{+}\) or \(D(\varphi)=(-\infty, \omega_{0})\) and \(\lim_{\xi\to -\infty}\varphi(\xi)=l^{-}\) for some constants \(\omega_{0},l^{\pm}\). The existence of semi-wavefront solutions for every wave speed \(c\) is shown, and their properties are investigated.