On small-data solution of the chemotaxis–SIS epidemic system with bilinear incidence rate

Abstract

This paper deals with the initial–boundary value problem for chemotaxis susceptible-infected-susceptible epidemic system with bilinear incidence rate ut=d1Δu+χ∇⋅(u∇v)−β(x)uv+γ(x)v,x∈Ω,t>0,vt=d2Δv+β(x)uv−γ(x)v,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn (n≥1), where d1>0, d2>0, χ∈R, 0<β∈C1(Ω¯) and 0<γ∈C1(Ω¯). It is proved that if (u(x,0),v(x,0)) in L∞(Ω)×L1(Ω) is suitably small and ‖v(x,0)‖L∞(Ω)+‖∇v(x,0)‖Lq(Ω)≤K for each K>0 and q>n, then the problem possesses a unique global classical solution which is bounded in Ω×(0,∞). Moreover, we prove that the solution exponentially stabilizes to the disease-free equilibrium N|Ω|,0 with N=∫Ω(u(x,0)+v(x,0)) in L∞(Ω) as t→∞.