Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity

Abstract

This paper deals with the parabolic-elliptic Keller-Segel system with signal-dependent chemotactic sensitivity function, ut=Δu-·(uχ(v)),x∈Ω,t>0,0=Δv-v+u,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn,n≥2, with initial data u0∈C0(Ω) satisfying u0-≥-0 and ∫Ωu0>0. The chemotactic sensitivity function χ(v) is assumed to satisfy 0 0.The global existence of weak solutions in the special case χ(v)=χ0v is shown by Biler (Adv.-Math.-Sci.-Appl.-1999; 9:347-359). Uniform boundedness and blow-up of radial solutions are studied by Nagai and Senba (Adv.-Math.-Sci.-Appl.-1998; 8:145-156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if χ0<2n(k=1);χ0<2n·kk(k-1)k-1γk-1(k>1), where γ->-0 is a constant depending on Ω and u0.