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

Abstract

In the current paper, we consider the following parabolic-elliptic semilinear Keller-Segel model on ${\mathbb{R}}^N$, \begin{equation*} \begin{cases} u_{t}=\nabla\cdot (\nabla u-χu\nabla v)+a u -b u^2, \quad x\in\mathbb{R}^N,\,\, t>0\cr 0=(Δ- I)v+ u, \quad x\in\mathbb{R}^N,\,\, t>0, \end{cases} \end{equation*} where $\chi >0,a\ge 0,b>0$ are constant real numbers and $N$ is a positive integer. We first prove the local existence and uniqueness of classical solutions $(u(x,t;u_0),v(x,t;u_0))$ with $u(x,0;u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under some conditions on the constants $a,b,\chi$ and the dimension $N$, we prove the global existence and boundedness of classical solution $(u(x,t;u_0),v(x,t;u_0))$ for given initial functions $u_0(x)$. Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants $a,b,\chi$ and the dimension $N$, we show that for every strictly positive initial function $u_0(\cdot )$, $$\underset{t\to \infty }{\lim }\underset{x\in {\mathbb{R}}^N}{\sup }[|u(x,t;u_0)-\frac{a}{b}|+|v(x,t;u_0)-\frac{a}{b}|]=0,$$ and that for every nonnegative initial function $u_0(\cdot )$ with non-empty and compact support $\mathrm{supp}(u_0)$, there are $0<c_{\mathrm{low}}^{\ast }(u_0)\le c_{\mathrm{up}}^{\ast }(u_0)<\infty$ such that $$\underset{t\to \infty }{\lim }\underset{|x|\le ct}{\sup }[|u(x,t;u_0)-\frac{a}{b}|+|v(x,t;u_0)-\frac{a}{b}|]=0\forall 0<c<c_{\mathrm{low}}^{\ast }(u_0)$$ and $$\underset{t\to \infty }{\lim }\underset{|x|\ge ct}{\sup }[u(x,t;u_0)+v(x,t;u_0)]=0\forall c>c_{\mathrm{up}}^{\ast }(u_0).$$

References in corpus (1)

• Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds

Cited by in corpus (3)

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