Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source

Abstract

This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source, $$\displaylines{ u_t=\Delta u -\chi\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N,\cr \tau v_t=\Delta v-\lambda v +\mu u,\quad x\in\mathbb{R}^N, }$$ where \(\chi, \mu,\lambda,a,b\) are positive numbers, and \(\tau\ge 0\). Among others, it is proved that if \(b>2\chi\mu\) and \(\tau \geq \frac{1}{2}(1-\frac{\lambda}{a})_{+}\), then for every \(c\ge 2\sqrt{a}\), this system has a traveling wave solution \((u,v)(t,x)=(U^{\tau,c}(x\cdot\xi-ct),V^{\tau,c}(x\cdot\xi-ct))\) (for all \(\xi\in\mathbb{R}^N \)) connecting the two constant steady states \((0,0)\) and \((\frac{a}{b},\frac{\mu}{\lambda}\frac{a}{b})\), and there is no such solutions with speed \(c\) less than \(2\sqrt{a}\), which improves the results established in [30] and shows that this system has a minimal wave speed \(c_0^*=2\sqrt a\), which is independent of the chemotaxis.

For more information see https://ejde.math.txstate.edu/Volumes/2020/53/abstr.html