GLOBAL EXISTENCE OF SOLUTIONS TO A KELLER-SEGEL MODEL WITH LOGISTIC SOURCE IN R2

In this paper, we consider the following parabolic-elliptic chemotaxis system with logistic source in the whole space R2: { u(t) = triangle u - V center dot (u del v) + au - bu(2), x is an element of R-2, t > 0, - triangle v = u - v, x is an element of R-2, t > 0. We prove that the solution of the model with a > 0, b > 0 exists globally for arbitrary initial data u0 E L-1(R-2) boolean AND L-infinity(R-2). It is different from the classical Keller-Segel model in that the existence of the logistic source makes the model lose the mass conservation property. Instead, we give an upper bound on IIuIIL1(R2) for subsequent estimates. This paper mainly explores logistic source and diffusion term to suppress the concentration term, and uses analysis skills to prove the boundedness of the term of II(1+u) log(1+u)IIL1(R-2), thus obtaining global existence of bounded solutions by the Moser iteration. Compared with the classical Keller-Segel model, the existence of a logistic source prevents blow-up that holds in R-2 for any b > 0. This indicates that the logistic source (b > 0) has broken the threshold between diffusion and aggregation in the model such that the solution exists globally for any initial data.