How far does logistic dampening influence the global solvability of a high-dimensional chemotaxis system?

Abstract

AbstractThis paper deals with the homogeneous Neumann boundary value problem for chemotaxis system $$\begin{aligned} \textstyle\begin{cases} u_{t} = \Delta u - \nabla \cdot (u\nabla v)+\kappa u-\mu u^{\alpha }, & x\in \Omega, t>0, \\ v_{t} = \Delta v - uv, & x\in \Omega, t>0, \end{cases}\displaystyle \end{aligned}$$ { u t = Δ u − ∇ ⋅ ( u ∇ v ) + κ u − μ u α , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}(N\geq 2)$ Ω ⊂ R N ( N ≥ 2 ) , where $\alpha >1$ α > 1 and $\kappa \in \mathbb{R},\mu >0$ κ ∈ R , μ > 0 for suitably regular positive initial data.When $\alpha \ge 2$ α ≥ 2 , it has been proved in the existing literature that, for any $\mu >0$ μ > 0 , there exists a weak solution to this system. We shall concentrate on the weaker degradation case: $\alpha <2$ α < 2 . It will be shown that when $N<6$ N < 6 , any sublinear degradation is enough to guarantee the global existence of weak solutions. In the case of $N\geq 6$ N ≥ 6 , global solvability can be proved whenever $\alpha >\frac{4}{3}$ α > 4 3 . It is interesting to see that once the space dimension $N\ge 6$ N ≥ 6 , the qualified value of α no longer changes with the increase of N.