(untitled) (Journal of Partial Differential Equations, 2009)
This paper is concerned with the asymptotic behavior of solution to the following model involving two species all with chemotaxis: ∂p∂t=D_p∇(p∇lnpω), ∂q∂t=D_q∇(q∇lnqω), ∂ω∂t=βp-δω, p∇ln(pω·n=q∇lnqω)·n=0. We prove that the solution exists globally as β ≥ 0. As β < 0, whether the solution exists globally or not depends on the initial data. By function transformation and comparison, the asymptotical behavior of the solution is studied.