Boundedness in a chemotaxis system with nonlinear diffusion and singular sensitivity

This paper studies the following chemotaxis system with nonlinear diffusion and singular sensitivity \[ {cases} u_t = ∇ · ( (u+a)ᵐ⁻¹ ∇ u ) - χ∇ · ( u ∇ v/v^k ), \\[4pt] v_t = Δv - v + u, {cases} \] in a smoothly bounded domain \(Ω⊂ R^n\), \(n ≥ 2\), subject to no-flux boundary conditions, where \(χ,k,a,m>0\). For suitably regular initial data, we prove global existence and uniform-in-time boundedness of classical solutions if one of the following conditions holds: \[ {aligned} & n=2, k∈(0,∞), m>1;\\ & n≥ 3, k∈(0,12), m>1+(1-k)(n-2)/n;\\ & n≥ 3, k∈[12,1), m>3n-2/2n;\\ & n≥ 3, k∈[1,∞), m>max\{1,3n-2/2n+2\}. {aligned} \] The main contribution is the physically relevant three-dimensional case. In particular, for every \(χ>0\), \(a>0\), \(m>1\) and \(k≥ 1\), the system admits a global bounded classical solution. Thus, in dimension three, nonlinear diffusion of porous-medium type removes the smallness restriction on the chemotactic sensitivity $χ$ that earlier results require in the case \(m=1\) and \(k≥ 1\).