A new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source

We deal with the boundedness of solutions of a class of quasilinear chemotaxis systems generalizing the prototype (⁎) { u t = ∇ ⋅ ( ϕ ( u ) ∇ u ) − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , ∂ u ∂ ν = ∂ v ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω , where R N ( N ≥ 2 ) is a bounded domain with zero-flux boundary condition, μ > 0 is a positive parameter, and ϕ ( u ) = ( u + 1 ) − α . It is shown that for all reasonably regular initial data, system (⁎) admits a global existence and boundedness of solutions when α α 0 for some α 0 > 2 − N N + 2 , thereby proving that the value α = 2 − N N + 2 is not critical in this regard. This extends previous results that rely on a different energy-type inequality.