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

Abstract

In this paper, we consider the system (Formula Presented) where ℝN+ = {(x1, x′)|x′ ∈ ℝN-1, x1 > 0}, p, q > 0, and u0, v0 are nonnegative and bounded. We prove that if pq ≤ 1 every nonnegative solution is global. When pq > 1 we let α = (p + 2)/2(pq - 1), β = (2q + 1)/2(pq - 1). We show that if max(α, β) > N/2 or max(α, β) = N/2 and p, q ≥ 1, then all nontrivial nonnegative solutions are nonglobal; whereas if max(α, β)