Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces

In this paper, the fully parabolic Keller-Segel systemequationproblemAbstract\array{lluₜ= u- (u v), & (x, t) (0, T), \ₜ= v-v+u, & (x, t) (0, T), \. equationis considered under Neumann boundary conditions in a boundeddomain ⁿ with smooth boundary, where n 2. We derive a smallness condition on the initial data in optimal Lebesgue spaceswhich ensure global boundedness and large time convergence. More precisely, we shall show that one can find ₀>0 such that for all suitably regular initial data (u₀, v₀) satisfying \|u₀\|₋^₍{₂ () } 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 (). For such systems, we prove a global existence and boundedness result under corresponding smallness conditionson the initially present total mass of cells and the chemical gradient.