Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller-Segel system {u(t) = del.(del u - u del v) in R-2 x (0,infinity), v(t) = Delta v - lambda v + u in R-2 x (0,infinity), u(x, 0) = u(0)(x) >= 0, v(x, 0) = v(0)(x) >= 0 inR(2) with a constant lambda >= 0, where (u(0), v(0)) is an element of (L-1(R-2) boolean AND L-infinity(R-2)) x (L-1(R-2) boolean AND H-1(R-2)). Let m(u(0); R-2) = integral(R2) u(0)(x)dx. The same method as in [9] yields the existence of a blowup solution with m(u(0); R-2) > 8 pi. On the other hand, it was recently shown in [7] that under additional hypotheses u(0) log(1+vertical bar x vertical bar(2)) is an element of L-1(R-2) and u(0) log u(0) is an element of L-1(R-2), any solution with m(u(0); R-2) < 8 pi exists globally in time. In [18], the extra assumptions were taken off, but the condition on mass was restricted to m(u(0); R-2) < 4 pi. In this paper, we prove that any solution with m(u(0); R-2) < 8 pi exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u(0) also in the critical case m(u(0); R-2) = 8 pi.