On the initial value problem for the hyperbolic Keller-Segel equations in Besov spaces

Abstract

In this paper, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from u0 is discontinuous at t=0 in the metric of B2,∞s(R), s>32. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces Bp,11+dp(Rd), 1≤p<∞, which improves the local theory proved by [Zhou, Zhang & Mu, J. Differ. Equ., 302(2021), pp.662-679]. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity ϵΔu as ϵ→0 in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory.