Boundedness in a fully parabolic quasilinear repulsion chemotaxis model of higher dimension

Abstract

We deal with the boundedness of solutions to a class of fully parabolic quasilinear repulsion Chemotaxis systems

$$\begin{cases}u_{t}= \nabla \cdot(\phi(u)\nabla{u})+\nabla\cdot(\psi(u)\nabla{v}), & (x,t)\in\Omega\times(0,T),\\v_{t}=\Delta{v}-v+u, & (x,t)\in\Omega\times(0,T),\end{cases}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ ℝ^N(N ≥ 3), where 0 < ψ(u) ≤ K(u + 1)^α, K₁(s + 1)^m ≤ ϕ(s) ≤ K₂(s + 1)^m with α, K, K₁, K₂ > 0 and m ∈ ℝ. It is shown that if $\alpha - m < {4 \over {N + 2}}$, then for any sufficiently smooth initial data, the classical solutions to the system are uniformly-in-time bounded. This extends the known result for the corresponding model with linear diffusion.