Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type

Boundary value problems

Yilong Wang¹ 

[1] School of Sciences, Southwest Petroleum University, Chengdu, China

关键词: chemotaxis;    attraction-repulsion;    boundedness;    global existence;    nonlinear diffusion;    35K55;    35Q92;    35B35;    92C17;   

DOI  :  10.1186/s13661-016-0518-6

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

This paper deals with the global existence and boundedness of solutions to the following quasilinear attraction-repulsion chemotaxis system:{ut=∇⋅(D(u)∇u)−∇⋅(χu∇v)+∇⋅(ξu∇w),x∈Ω,t>0,0=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,$$ \left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot (\xi u\nabla w), & x\in\Omega, t>0,\\ 0=\Delta v+\alpha u-\beta v,& x\in\Omega, t>0,\\ 0=\Delta w+\gamma u-\delta w,& x\in\Omega, t>0, \end{array}\displaystyle \right . $$under homogeneous Neumann boundary conditions in a bounded domainΩ⊂Rn$\Omega\subset\mathbb{R}^{n}$(n≥2$n\geq2$) with smooth boundary, whereD(u)≥cDum−1$D(u)\geq c_{D} u^{m-1}$withm≥1$m\geq1$and some constantcD>0$c_{D}>0$. It is proved that ifξγ−χα>0$\xi\gamma-\chi\alpha>0$orm>2−2n$m>2-\frac{2}{n}$, then for any sufficiently regular initial data, this system possesses a unique global bounded classical solution for the case of nondegenerate diffusion (i.e.,D(u)>0$D(u)>0$for allu≥0$u\geq 0$), whereas for the case of degenerate diffusion (i.e.,D(u)≥0$D(u)\geq0$for allu≥0$u\geq0$), it is shown that there exists a global bounded weak solution under the same assumptions.

【 授权许可】

CC BY