Boundedness in a chemo-repulsion model with nonlinear sensitivity and logistic source

This paper investigates the Neumann initial-boundary value problem of the chemo-repulsion model: uₜ = Δ u+ ∇·(uχ(v)∇ v)+au-bu²,\:vₜ = Δ v-v+u, where $ a,b>0 $ and χ(·)∈ C²([0,∞)) satisfies 0≤ χ(s)≤ χ₀(1+s)ᵏ for all s≥0, with some χ₀>0. By constructing a family of functionals: Eₚ(t): = ∫Ω uv²ᵖ +∫Ω |∇ vᵖ⁺¹|² for p≥1, the global boundedness of classical solutions is proved for all k≥0 in two-dimensional.