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{equation}{problemAbstract}\{{array}{ll}u_t=Δ u-∇·(u∇ v),&(x,t)∈ Ω× (0,T),_t=Δ v-v+u, &(x,t)∈Ω× (0,T),\\{array}.{}{equation}is considered under Neumann boundary conditions in a boundeddomain Ωⁿ 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 ε₀>0 such that for all suitably regular initial data(u₀,v₀) satisfying $\|u_0\|_{Ln/2(Ω)} 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.