GLOBAL STABILITY OF KELLER-SEGEL SYSTEMS IN CRITICAL LEBESGUE SPACES

This paper is concerned with the initial-boundary value problem for the classical Keller-Segel system {rho(t) - Delta rho = - del. (rho del c), x is an element of Omega, t > 0 (1) gamma c(t )- Delta c + c = rho, x is an element of Omega, t > 0( ) in a bounded domain Omega subset of R-d with d >= 2 under homogeneous Neumann boundary conditions, where gamma >= 0. We study the existence of non-trivial global classical solutions near the spatially homogeneous equilibria rho = c (math) M > 0 with M being any given large constant which is an open problem proposed in [2, p. 1687]. More precisely, we prove that if 0 < M < 1 + lambda(1) with lambda(1) being the first positive eigenvalue of the Neumann Laplacian operator, one can find epsilon(0) > 0 such that for all suitable regular initial data (rho(0), gamma c(0)) satisfying 1/vertical bar Omega vertical bar integral(Omega )rho(0)dx - M = gamma (1/vertical bar Omega vertical bar integral(Omega )c(0)dx - M) = 0 (2) and parallel to rho(0 )- M parallel to(Ld/2 (Omega) )+ gamma parallel to del c(0)parallel to(Ld )(Omega) < epsilon(0), (3) problem (1) possesses a unique global classical solution which is bounded and converges to the trivial state (M, M) exponentially as time goes to infinity. The key step of our proof lies in deriving certain delicate L-p - L-q decay estimates for the semigroup associated with the corresponding linearized system of (1) around the constant steady states. It is well-known that classical solution to system (1) may blow up in finite or infinite time when the conserved total mass m (sic) rho(0)dx exceeds some threshold number if d = 2 or for arbitrarily small mass if d >= 3. In contrast, our results indicates that non-trivial classical solutions starting from initial data satisfying (2)-(3) with arbitrarily large total mass m exists globally provided that vertical bar Omega vertical bar is large enough such that m< (1 + lambda(1))vertical bar Omega vertical bar.