GLOBAL BOUNDED SOLUTIONS OF THE HIGHER-DIMENSIONAL KELLER-SEGEL SYSTEM UNDER SMALLNESS CONDITIONS IN OPTIMAL SPACES

In this paper, the fully parabolic Keller-Segel system { u(t) = Delta u = del . (u del v), (x, t) is an element of Omega x (0, T), v(t) = Delta v - v + u, (x, t) is an element of Omega x (0, T), (star) is considered under Neumann boundary conditions in a bounded domain Omega subset of R-n with smooth boundary, where n >= 2. We derive a smallness condition on the initial data in optimal Lebesgue spaces which ensure global boundedness and large time convergence. More precisely, we shall show that one can find epsilon(0) > 0 such that for all suitably regular initial data (u(0), v(0)) satisfying parallel to u(0)parallel to(Ln/2 (Omega)) < epsilon(0) and parallel to del v(0)parallel to(Ln (Omega)) < epsilon(0), the above problem possesses a global classical solution which is bounded and converges to the constant steady state (m, m) with m := 1/vertical bar Omega vertical bar integral(Omega) u(0). Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with (star). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.