Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis

The following degenerate parabolic system modelling chemotaxis is considered. (KS) \ arrayllll & uₜ = ∇ · ( ∇ uᵐ - u ∇ v ), & x ∈ RN, \ 0 1, τ=0 or 1, and N ≥ 1. Our aim in this paper is to prove the existence of a global weak solution of (KS) under some appropriate conditions on m without any restriction on the size of the initial data. Specifically, we show that a solution ($u,v$) of (KS) exists globally in time if either (i) m ≥ 2 for large initial data or (ii) 1 < m ≤ 2-2/N for small initial data. In the case of (ii), the decay properties with the optimal rate of the solution ($u,v$) are also discussed.