Abstract :
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.
