Boundedness in a fully parabolic chemotaxis system with nonlinear diffusion and sensitivity, and logistic source

Abstract

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{ t}=\nabla \cdot ((u+1)^{m-1} \nabla u-(u+1)^αχ(v)\nabla v) + ku-μu^2 & x\in Ω, t>0, \\ v_{t} = Δv-vu & x\in Ω, t>0,\\ \end{cases} \end{equation*} $\Omega$ being a bounded and smooth domain of ${\mathbb{R}}^n$, $n\ge 1$, and where $m,k\in \mathbb{R}$, $\mu >0$ and $\alpha <\frac{m+1}{2}$. For any $v\ge 0$ the chemotactic sensitivity function is assumed to behave as the prototype $\chi (v)=\frac{{\chi }_0}{(1+av{)}^2}$, with $a\ge 0$ and ${\chi }_0>0$. We prove that for nonnegative and sufficiently regular initial data $u(x,0)$ and $v(x,0),$ the corresponding initial-boundary value problem admits a global bounded classical solution provided $\mu$ is large enough.

References in corpus (1)

• Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption