An improvement toward global boundedness in a fully parabolic chemotaxis with singular sensitivity in any dimension

We are concerned with the following chemotaxis system {equation*} {cases} u_t = Δu -χ∇ · ( u/v ∇ v ), v_t = Δv -u+ v, {cases} {equation*} in an open bounded domain with smooth boundary Ω⊂ Rⁿ with n ≥ 3. It is known that solutions are globally bounded when 0<χ< √ 2/n. In this paper, by considering the following energy functional {align*} F_λ(u,v)= ∫_Ωu^p/v^r + λ∫_Ω{|∇ v|²ᵖ}{vᵖ⁺ʳ} + ∫_Ωvᵖ⁻ʳ {align*} for $p>1$, r= p-1/2, and $λ>0$, we improve this result by showing that there exists χ₀>√2/n such that for any χ∈ (0,χ₀), solutions exist globally and remain bounded.