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: u t = ∇ ⋅ ( u θ − 1 ∇ u ) − χ ∇ ⋅ u v ∇ v , x ∈ Ω , t > 0 , v t = Δ v − 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 > 3 2 \theta \gt \frac{3}{2} and (1.10)–(1.11). This result improves our recent work.