Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production

Abstract

In this paper we study the asymptotic behaviors of global solutions to the fully parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇v)+ru−μu1+σ, vt=Δv−v+uγ, subject to the homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥2), where parameters μ,σ,γ>0, r∈R, and the nonlinearity D,S∈C2([0,∞)) are supposed to generalize the prototypesD(u)≥a0(u+1)−α,0≤S(u)≤b0u(u+1)β−1 with a0,b0>0 and α,β∈R. We first consider the case of r>0 and provide a boundedness result under α+β+γ<2n, or β+γ<1+σ, or β+γ=1+σ with large μ>0. The main result is concerned with the asymptotic stability when damping effects of logistic source are strong enough. Specifically, there is μ0>0 independent of initial data, such that the bounded classical solution (u,v) satisfies (u,v)→((rμ)1σ,(rμ)γσ) in L∞(Ω) exponentially under conditions of μ>μ0 and r>0. For the case of r<0, the trivial constant equilibria in the model is obtained in a priori way, that is, any bounded solution (u,v) satisfies (u,v)→(0,0) in L∞(Ω) exponentially, regardless of the size of μ>0.