Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity (Q395564)

The authors consider a chemotaxis model in the form of the following PDE-ODE hybrid system NEWLINE\[NEWLINE\begin{cases} u_t = [D u_x-\xi u(\log c)_x]_x, \\ c_t = \mu uc \end{cases}\tag{1}NEWLINE\]NEWLINE for \((x,t)\in \mathbb{R}\times [0,\infty)\). Here \(u(x,t)\) denotes the particle density and \(c(x,t)\) the concentration of a non-diffusible chemical signal. By using a Hopf-Cole transformation \(v=\displaystyle{(\log c)_x\over \mu}\) and by letting \(\chi=-\mu \xi\), system (1) is converted to the following conservational parabolic-hyperbolic system NEWLINE\[NEWLINE\begin{cases} u_t-\chi(uv)_x = Du_{xx}, \\ v_t-u_x = 0 \end{cases}\tag{2}NEWLINE\]NEWLINE subject to the initial data \((u,v)(x,0)=(u_0,v_0)(x)\rightarrow (u_{\pm},v_{\pm})\) as \(x\rightarrow \pm\infty\), where \(u_0(x)\geq 0\) and \(u_{\pm}\geq 0\).NEWLINENEWLINEThe main purpose of this paper is to establish the nonlinear stability of traveling wave solutions of the transformed system with \(\chi>0\) and \(u_+=0\), a problem left open in an earlier work by \textit{T. Li} and \textit{Z.-A. Wang} [SIAM J. Appl. Math. 70, No. 5, 1522--1541 (2009; Zbl 1206.35041)]. Spectral analysis is also performed to show that the traveling wave solutions of (2) with \(\chi<0\) are linearly unstable. By transferring the results back to the original chemotaxis system, results on the traveling wave solutions of system (1) are obtained.