Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system

AbstractThis paper deals with the boundedness of global solutions to the quasilinear Keller–Segel system $$\left\{\begin{array}{ll}u_t=\nabla\cdot\big(D(u)\nabla u-u\nabla v\big), &\quad x\in\Omega,\,\, t>0,\\ v_t=\Delta v-uf(v),&\quad x\in\Omega, \,\,t>0,\\ \nabla u\cdot \nu=0,\,\, \nabla v\cdot\nu=0,&\quad x\in \partial\Omega,\,\, t>0\end{array}\right.$$ut=∇·(D(u)∇u-u∇v),x∈Ω,t>0,vt=Δv-uf(v),x∈Ω,t>0,∇u·ν=0,∇v·ν=0,x∈∂Ω,t>0in a bounded domain $${\Omega\subset \mathbb{R}^{n}(n\geq 3)}$$Ω⊂Rn(n≥3) with smooth boundary, where D(u) is supposed to satisfy D(u) ≥ D0um-1 with some positive constant D0. It is proved that when $${m>2-\frac{n+2}{2n}}$$m>2-n+22n, the system possesses global bounded weak solutions for any sufficiently smooth nonnegative initial data. In particular, we improved the recent result by Wang et al. (Z Angew Math Phys, 2015. doi:10.1007/s00033-014-0491-9) in the sense that we established the global boundedness of weak solutions. We also removed the convexity assumption on the domain used by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014, 2015).