Existence of Traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

Abstract

We study traveling wave solutions of the following chemotaxis systems,$$\{\begin{array}{l}{u}_t=\Delta u-{\chi }_1\nabla (u\nabla v_1)+{\chi }_2\nabla (u\nabla v_2)+u(a-bu),x\in {\mathbb{R}}^N\\ 0=\Delta v_1-{\lambda }_1{v}_1+{\mu }_{1}u,x\in {\mathbb{R}}^N,\\ 0=\Delta v_2-{\lambda }_2{v}_2+{\mu }_{2}u,x\in {\mathbb{R}}^N,\end{array}$$where $u(x,t),v_1(x,t)$ and $v_2(x,t)$ represent the population densities of a mobile species, a chemoattractant, and a chemo-repulsion, respectively. In an earlier work, we proved that there is a constant $K\ge 0$ such that if $b+{\chi }_2{\mu }_2>{\chi }_1{\mu }_1+K$, then the steady solution $(\frac{a}{b},\frac{a{\mu }_1}{b{\lambda }_1},\frac{a{\mu }_2}{b{\lambda }_2})$ is asymptotically stable with respect to positive perturbations. In this paper, we prove that if $b+{\chi }_2{\mu }_2>{\chi }_1{\mu }_1+K$, then there exist a number $c^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)\ge 2\sqrt{a}$ such that for every $c\in (c^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2),\infty )$ and $\xi \in S^{N-1}$, the system has a traveling wave solution $(u,v_1,v_2)=(U(x\cdot \xi -ct),V_1(x\cdot \xi -ct),V_2(x\cdot \xi -ct))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{a{\mu }_1}{b{\lambda }_1},\frac{a{\mu }_2}{b{\lambda }_2})$ and $(0,0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt{a}$. Moreover we prove that$$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+})}{\lim }{c}^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=\{\begin{array}{l}2\sqrt{a}\text{if}a\le \min \{{\lambda }_1,{\lambda }_2\}\\ \frac{a+{\lambda }_1}{\sqrt{{\lambda }_1}}\text{if}{\lambda }_1\le \min \{a,{\lambda }_2\}\\ \frac{a+{\lambda }_2}{\sqrt{{\lambda }_2}}\text{if}{\lambda }_2\le \min \{a,{\lambda }_1\}\end{array},\forall {\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2>0,$$and$$\underset{x\to \infty }{\lim }\frac{U(x)}{e^{-\sqrt{a}\mu x}}=1,$$ where $\mu$ solves $\sqrt{a}(\mu +\frac{1}{\mu })=c$ in the interval $(0,\min \{1,\sqrt{\frac{{\lambda }_1}{a}},\sqrt{\frac{{\lambda }_2}{a}}\})$.

arXiv commentarXiv admin note: text overlap with arXiv:1610.05215, arXiv:1609.05387

References in corpus (4)

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• Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on R^N
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• Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems