A new result for global existence and boundedness of solutions to a parabolic-parabolic Keller-Segel system with logistic source

We consider the following fully parabolic Keller-Segel system with logistic source {u(t) = Delta u - X-del . (u del v) + au - mu u(2), x is an element of Omega, t > 0, vt = Delta v - v + u, x is an element of Omega, t > 0, (KS) over a bounded domain Omega subset of R-N with smooth boundary partial derivative Omega the parameters a is an element of E,mu > 0, x > 0. It is proved that if mu > 0, then (KS) admits a global weak solution, while if a is an element of E,mu > 0, x > 0 then (KS) possesses a global classical solution, which is bounded, where C 1/n/2+1 n/2+1 is a positive constant which is corresponding to the maximal Sobolev regularity. Apart from this, we also show that if a = 0 and mu > (N-2)-/2 chi C 1/n/2+1 n/2+1, then both u(. , t) and v(. , t) decay to zero with respect to the norm in L-infinity (Omega) gas t ->infinity no. (C) 2018 Elsevier Inc. All rights reserved.