Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening

This paper deals with nonnegative solutions of the Neumann initial–boundary value problem for the parabolic chemotaxis system{ut=Δu−χ∇⋅(u∇v)+u−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, in bounded convex domains Ω⊂RnΩ⊂Rn, n≥1n≥1, with smooth boundary.It is shown that if the ratio μχ is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by u=v≡1μ is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data (u0,v0)(u0,v0) such that u0≢0u0≢0, the above problem possesses a uniquely determined global classical solution (u,v)(u,v) with (u,v)|t=0=(u0,v0)(u,v)|t=0=(u0,v0) which satisfies‖u(⋅,t)−1μ‖L∞(Ω)→0and‖v(⋅,t)−1μ‖L∞(Ω)→0 as t→∞t→∞.