Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation

Abstract

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 problemwhere Ω is a bounded and smooth domain of R n , for n ≥ 2, and f (u) and g(u) are reasonably regular functions generalizing, respectively, the prototypes f (u) = u α and g(u) = u l , with proper α, l > 0. After having shown that any sufficiently smooth u(x, 0) = u 0 (x) ≥ 0, v(x, 0) = v 0 (x) ≥ 0 emanate 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, 2 n ) and, Tmax = ∞ and u and v are actually uniformly bounded in time. This paper is in line with the contribution by Horstmann and Winkler ([6]) and, moreover, extends the result by Liu and Tao ([12]). Indeed, in the first work it is proved that for g(u) = u the value α = 2 n 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 < l < 2 n .