Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity

Abstract

In this study, we investigate the two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity: ut=∇⋅(uθ−1∇u)−χ∇⋅uv∇v,x∈Ω,t>0,vt=Δv−v+u+g(x,t),x∈Ω,t>0,(∗)\left\{\begin{array}{ll}{u}_{t}=\nabla \cdot \left({u}^{\theta -1}\nabla u)-\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta v-v+u+g\left(x,t),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ \end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\ast ) in a bounded domain with smooth boundary. We present the global boundedness of weak solutions to the model (∗\ast ) if θ>32\theta \gt \frac{3}{2} and (1.10)–(1.11). This result improves our recent work.

Keywords

• chemotaxis
• singular sensitivity
• global boundedness
• nonlinear diffusion
• primary: 92c17
• secondary: 35q92
• 35k57