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

We consider the system u(t) = Deltau + v(p), v(t) = Deltav, x epsilon R-+(N), t > 0, -partial derivativeu/partial derivativex(1) = 0, -partial derivativev/partial derivativex(1) = u(q), x(1) = 0, t > 0, u(x, 0) = u(0)(x), v(x,0) = v(0)(x), x epsilon R-+(N), where R-+(N) = {(x(1),x'): x' epsilon RN-1, x(1) > 0}, p, q are positive numbers, and functions u(0), v(0) in the initial conditions are nonnegative and bounded. We show that nonnegative solutions are unique if pq greater than or equal to 1. We also find a nontrivial nonnegative solution with vanishing initial values when pq < 1. (C) 2004 Elsevier Inc. All rights reserved.