Boundedness of a Chemotaxis-Convection Model Describing Tumor-Induced Angiogenesis

This paper is concerned with the parabolic-parabolic-elliptic system { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ 1 ∇ ⋅ ( u m ∇ w ) , x ∈ Ω , t > 0 , u t = Δ v + ξ 2 ∇ ⋅ ( v ∇ w ) + u − v , x ∈ Ω , t > 0 , 0 = Δ w + u − 1 | Ω | ∫ Ω u , ∫ Ω w = 0 , x ∈ Ω , t > 0 , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω in a bounded domain Ω ⊂ ℝ n with a smooth boundary, where the parameters χ, ξ 1 , ξ 2 are positive constants and m ≥ 1. Based on the coupled energy estimates, the boundedness of the global classical solution is established in any dimensions ( n ≥ 1) provided that m > 1.