Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on ${\mathbb{R}}^N$

Abstract

We consider the following chemotaxis systems $$\{\begin{array}{l}{u}_t=\Delta u-{\chi }_1\nabla (u\nabla v_1)+{\chi }_2\nabla (u\nabla v_2)+u(a-bu),x\in {\mathbb{R}}^N,t>0,\\ 0=(\Delta -{\lambda }_{1}I)v_1+{\mu }_{1}u,x\in {\mathbb{R}}^N,t>0,\\ 0=(\Delta -{\lambda }_{2}I)v_2+{\mu }_{2}u,\text{in}x\in {\mathbb{R}}^N,t>0,\\ u(\cdot ,0)=u_0,x\in {\mathbb{R}}^N,\end{array}$$where ${\chi }_i,{\lambda }_i,{\mu }_i,i=1,2$ and $a,b$ are positive constant real numbers and $N$ is a positive integer. Under some conditions on the parameters, we prove the global existence and boundedness of classical solutions $(u(x,t;u_0),v_1(x,t;u_0),v_2(x,t;u_0))$ for nonnegative, bounded, and uniformly continuous initials $u_0(x)$. Next, we show that, for every strictly positive initial \,$u_0(x)$,$$\underset{t\to \infty }{\lim }\left[\Vert u(\cdot ,t;u_0)-\frac{a}{b}{\Vert }_{\infty }+\Vert {\lambda }_1{v}_1(\cdot ,t;u_0)-\frac{a}{b}{\mu }_1{\Vert }_{\infty }+\Vert {\lambda }_2{v}_2(\cdot ,t;u_0)-\frac{a}{b}{\mu }_2{\Vert }_{\infty }\right]=0.$$ Finally, we explore the spreading properties of the global solutions and prove that there are two positive numbers $0<c_{-}^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)<c_{+}^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)$ such that for every nonegative initial $u_0(x)$ with nonempty and compact support, $$\underset{t\to \infty }{\lim }\left[\underset{|x|\le ct}{\sup }|u(x,t;u_0)-\frac{a}{b}|+\underset{|x|\le ct}{\sup }|{\lambda }_1{v}_1(x,t;u_0)-\frac{a}{b}{\mu }_1|+\underset{|x|\le ct}{\sup }|{\lambda }_2{v}_2(x,t;u_0)-\frac{a}{b}{\mu }_2|\right]=0$$whenever $0\le c<c_{-}^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)$,\ and$$\underset{t\to \infty }{\lim }\left[\underset{|x|\ge ct}{\sup }|u(x,t;u_0)|+\underset{|x|\ge ct}{\sup }|v_1(x,t;u_0)|+\underset{|x|\ge ct}{\sup }|v_2(x,t;u_0)|\right]=0$$whenever $c>c_{+}^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)$. Furthermore we show that$$\underset{({\chi }_1,{\chi }_2)\to (0,0)}{\lim }{c}_{-}^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=\underset{({\chi }_1,{\chi }_2)\to (0,0)}{\lim }{c}_{+}^{\ast }({\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=2\sqrt{a}.$$

References in corpus (2)

• Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on R^N
• Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on ${\mathbb{R}}^N$

Cited by in corpus (5)

• Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain
• Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II. Spreading-vanishing dichotomy in a domain with a free boundary
• Existence of Traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source
• Global existence and boundedness in a fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities without logistic source
• Traveling waves of a full parabolic attraction-repulsion chemotaxis systems with logistic sources