Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source

This paper is concerned with a class of quasilinear chemotaxis systems generalizing the prototype (0.1){ut=Δum−∇⋅(u∇v)+μu−ur,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x)x∈Ω, in a smooth bounded domain Ω⊂RN(N≥2) with parameters m,r≥1 and μ≥0. The PDE system in (0.1) is used in mathematical biology to model the mechanism of chemotaxis, that is, the movement of cells in response to the presence of a chemical signal substance which is in homogeneously distributed in space. It is shown that if m{>2−2Nif11+(N+2−2r)+N+2ifN+22≥r≥N+2N,≥1ifr>N+22, and the nonnegative initial data (u0,v0)∈Cι(Ω̄)×W1,∞(Ω)(ι>0), then (0.1) possesses at least one global bounded weak solution. Apart from this, it is proved that if μ=0 then both u(⋅,t) and v(⋅,t) decay to zero with respect to the norm in L∞(Ω) as t→∞.