In this talk, we present some recent results on global stability for classical Keller-Segel system of chemotaxis. We will first talk about some related results for the Keller-Segel model, including global boundedness and blow-up results. Then we discuss the stability problem of Keller-Segel equation near spatially homogeneous steady solutions which is an open problem proposed in a recent survey by N. Bellomo et al. By establishing certain delicate L^p-L^q decay estimates for the associated linearized semigroup, we give a partially affirmative answer to the problem. More importantly, our results indicate that nontrivial globally bounded classical solution exists with any given Large total mass provided the domain is sufficiently large. This is the first evidence with rigorous proof for the existence of nontrivial global classical solution with large total mass.

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