Global existence and asymptotic behavior for a fully cross-diffusive predator-prey model

Abstract

In this paper, we consider the fully cross-diffusive predator-prey model{ut=Δu−χ1∇⋅(u∇v)+uv−θu,x∈Ω,t>0,vt=Δv+χ2∇⋅(v∇u)−auv,x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, where Ω⊂RN(1≤N≤3) is a bounded domain with smooth boundary and χ1,χ2,a>0 and θ≥0 are constants. Based on the Lp−Lq estimates for Neumann heat semigroup and energy estimate, we prove the global existence and asymptotic behavior for classical solutions to the model under a certain small condition for the initial data.