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 where Omega 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)=u alpha and g(u)=ul, with proper alpha ,l>0. After having shown that any sufficiently smooth u(x,0)=u0(x)>= 0 and v(x,0)=v0(x)>= 0 produce a unique classical and nonnegative solution (u,v) to problem (lozenge), which is defined on Omega x(0,Tmax) with Tmax denoting the maximum time of existence, we establish that for any l is an element of 0,<= alpha , Tmax=infinity 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 alpha= represents the critical blow-up exponent to the model, whereas in the second, for f(u)=u, corresponding to alpha =1, boundedness of solutions is shown under the assumption 0<l.