Global solutions near homogeneous steady states in a fully cross-diffusive predator-prey system with density-dependent motion

This paper investigates a fully cross-diffusive predator-prey system with density-dependent motion {u(t) = del(phi(1)(v)del u - chi(1)(u)u del u) + mu(1)u(lambda(1) - u + a(1)v), {v(t) = del(phi 2(v)del u - chi(2)(u)u del u) + mu(2)u(lambda(2) - u + a(2)v) in a bounded domain Omega subset of R-n, n is an element of {1, 2, 3} with homogeneous Neumann boundary conditions, where mu(1), mu(2) >= 0, lambda(1), lambda(2), a(1), a(2) > 0, and the functions phi(1)(v), phi(2)(u), chi(1)(v), chi(2)(u) satisfy the following assumptions: phi(1), phi(2), chi(1), chi(2) is an element of C-3([0, infinity)), phi(1), phi(2) > 0 and chi(1), chi(2) >= 0 in [0, infinity). We prove that there exists epsilon(0) > 0 such that if u(0), v(0) is an element of W-2,W-2(Omega O) are nonnegative with partial derivative(nu)u(0) = partial derivative(v)nu(0) = 0 in the sense of traces and parallel to u(0) - u(*)parallel to(W2,2(Omega)) + parallel to v(0) - v(*)parallel to (W2,2(Omega)) < epsilon(0), then there exists a global classical solution (u, v) of (star) converging to constant stable steady state (u(star), v(star)) in W-2,W-2(Omega), where (u(star), v(star)) = (1/vertical bar Omega vertical bar integral u, 1/vertical bar Omega vertical bar integral(Omega) v) for mu(1) = mu(2) = 0, (u(*), v(*)) = (lambda(1)+lambda(2)a(1)/1+a(1)a(2), lambda 2+lambda(2)a(2)/1+a(1)a(2)) for mu(1), mu(2) > 0 and lambda(2) > lambda(1)a(2), (u(*), v(*)) = (lambda(1), 0) for mu(1), mu(2) > 0 and lambda(2) <= lambda(1)a(2). Moreover, the convergence rate is exponential, except for the case lambda(2) = lambda(1)a(2), where it is only algebraical.