On uniqueness for a semilinear parabolic system coupled in an equation and a boundary condition

We consider the system ut = Δu + vp, vt = Δv, x ∈ RN +, t >0, − ∂u ∂x1 = 0, − ∂v ∂x1 = uq, x1 = 0, t >0, u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ RN +, where RN + = {(x1,x ): x ∈ RN−1, x1 > 0}, p, q are positive numbers, and functions u0, v0 in the initial conditions are nonnegative and bounded. We show that nonnegative solutions are unique if pq 1. We also find a nontrivial nonnegative solution with vanishing initial values when pq < 1.  2004 Elsevier Inc. All rights reserved.