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 {u_t=Δu^m−∇⋅(u∇v)+μu−u^r,x∈Ω,t>0,v_t=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⊂R^N(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{>2−2Nif1<r<N+2N,>1+(N+2−2r)+N+2ifN+22≥r≥N+2N,≥1ifr>N+22, and the nonnegative initial data (u₀,v₀)∈C^ι(Ω̄)×W^1,∞(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→∞.