GLOBAL BOUNDEDNESS VERSUS FINITE-TIME BLOW-UP OF SOLUTIONS TO A QUASILINEAR FULLY PARABOLIC KELLER-SEGEL SYSTEM OF TWO SPECIES

摘要

This paper deals with two-species quasilinear parabolic-parabolic Keller-Segel system u(it) = del . (phi(i)(u(i))del(ui)) - V . (psi(i))(u(i))Delta(v)), i = 1, 2, v(t) = Delta(v - v) vertical bar u(1)vertical bar u(2) in Omega x (0,T), subject to the homogeneous Neumann boundary conditions, with bounded domain Omega subset of R-n, n >= 2. We prove that if psi(i)(u(i))/phi(i)(u(i)) <= C(i)u(i)(alpha i) for u(i) > 1 with 0 < alpha(i) < 2 and C-i > 0, i = 1,2, then the solutions are globally bounded, while if psi(1)(u(1))(n)/ phi(1)(u(1)) >= C(1)u(1)(alpha 1) for u(1) > 1 with Omega - BR, alpha(1) > 2, then for any radial u(20) epsilon C-0(Omega) and m(1) > 0, there exists positive radial initial data u(10) with integral(Omega) (u10) = m(1) such that the solution blows up in a finite time T-max in the sense limt(t)-> T-max parallel to U-1(., t) + U-2 (., t)parallel to L infinity(Omega) = infinity. In particular, if alpha(1) > 2 with 0 < alpha(2) < 2 the finite time blow-up for the species u(1) is obtained under suitable initial data, a new phenomenon unknown yet even for the semilinear Keller-Segel system of two species.