Global and bounded solution to a quasilinear parabolic-elliptic pursuit-evasion system in N-dimensional domains

Abstract

In this paper, we deal with the global existence and boundedness of solutions to a pursuit-evasion model{ut=Δu−χ∇⋅(u(u+1)r−1∇w)+u(λ1−μ1ur1−1+av),x∈Ω,t>0,vt=Δv+ξ∇⋅(v∇z)+v(λ2−μ2vr2−1−bu),x∈Ω,t>0,0=Δw−w+v,x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0 in a bounded domain Ω⊂RN(N≥1) with smooth boundary, where r, λ1, λ2, μ1, μ2, χ, ξ, a, b are positive constants as well as ri>1(i=1,2). Under the Neumann boundary conditions, if min⁡{(r1−1)(r2−1),(r1−r)(r2−1)}>(N−2)+N, it is proved that the above system possesses a unique global-in-time and bounded classical solution for all appropriately regular nonnegative initial data. This extends the previous global boundedness result for r=1, ri=2(i=1,2) and N≤3 (see Li-Tao-Winkler [9]), as well as the result for r=1, ri>1(i=1,2) and N≥1 (see Zheng and Zhang [38]).