Global boundedness and large time behavior of a chemotaxis system with indirect signal absorption

Abstract

The paper considers the following system{ut=Δu−∇⋅(u∇v)+f(u),x∈Ω,t>0,vt=Δv−vw,x∈Ω,t>0,wt=−δw+u,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn(n≥2) with smooth boundary, where δ>0 and f(s)=μ(s−sr) with μ≥0,r>1.When μ>0 we proved for any r>n2 the system admits a globally bounded classical solution. However, when f(u)=μ(u−u2) with μ>0 and n≥4, we establish the global existence and the boundedness of the solution if μ is suitably large. Moreover, by constructing the functional, we proved the solution converges to (1,0,1δ) in L∞(Ω) as t→∞ under the basic assumptions concerning to the boundedness on μ. Furthermore, by constructing the another functional, we proved the convergence is exponential if μ is suitably large. Finally, we showed the solution of the system with μ=0 is globally bounded if n≥3 and 0<‖v0‖L∞(Ω)≤πn, which improves the result of [5].