Global solution of a diffusive predator–prey model with prey-taxis

We consider the prey-taxis system: ut=d1Δu−χ∇⋅(u∇v)+u(a−μu)+buf(v),x∈Ω,t>0,vt=d2Δv+v(c−βv)−uf(v),x∈Ω,t>0in a smoothly bounded domain Ω⊂Rn, with zero-flux boundary condition, where a,d1,d2,χ,μ,b,c are positive constants and β is a non-negative constant. We first investigate the global existence and local boundedness of solution for the case β=0. Moreover, when β>0, we show that the solution exists globally and is uniformly bounded provided μ is large enough.