Localization in space and its application in the bounded solution of Cauchy problem to the parabolic-parabolic Keller-Segel system with logistic growth

We study global-in-time well-posedness and the boundedness of the solution to Cauchy problem in the classical Keller-Segel system with logistic term ∂_tn−Δn=−χ∇⋅(n∇c)+λn−μn²τ∂_tc−Δc=−c+n}inR^d×R⁺, where d≥1, τ,χ,μ>0 and λ≥0. It is inspired by a previous result [31], where the global-in-time boundedness of the above Keller-Segel system in smooth bounded convex domains is established for large μ. However, his approach in bounded domain ceases to directly apply in the entire space R^d, and then they raised an interesting question whether a similar global-in-time boundedness statement remains true of Cauchy problem. In this paper, we answer this open problem by developing local-in-space estimates. More precisely, we prove that the above Keller-Segel system possesses a uniquely global-in-time bounded solution for any τ>0 under the assumption that μ is large. The key point of our proof heavily relies on localization in space of solution, which correspond to “local effect” of L^∞(R^d)-norm.