Boundedness for a Fully Parabolic Keller–Segel Model with Sublinear Segregation and Superlinear Aggregation

This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem {ut=Δu−∇⋅(f(u)∇v) in Ω×(0,Tmax),vt=Δv−v+g(u) in Ω×(0,Tmax), where Ω is a bounded and smooth domain of Rn, for n≥ 2 , and f(u) and g(u) are reasonably regular functions generalizing, respectively, the prototypes f(u) = uα and g(u) = ul, with proper α, l> 0. After having shown that any sufficiently smooth u(x, 0) = u(x) ≥ 0 and v(x, 0) = v(x) ≥ 0 produce a unique classical and nonnegative solution (u, v) to problem (◊), which is defined on Ω × (0 , Tmax) with Tmax denoting the maximum time of existence, we establish that for any l∈(0,2n) and 2n≤α<1+1n−l2, Tmax= ∞ and u and v are actually uniformly bounded in time. The paper is in line with the contribution by Horstmann and Winkler (J. Differ. Equ. 215(1):52–107, 2005) and, moreover, extends the result by Liu and Tao (Appl. Math. J. Chin. Univ. Ser. B 31(4):379–388, 2016). Indeed, in the first work it is proved that for g(u) = u the value α=2n represents the critical blow-up exponent to the model, whereas in the second, for f(u) = u, corresponding to α= 1 , boundedness of solutions is shown under the assumption 0<2n.