On Critical Exponents for a System of Heat Equations Coupled in the Boundary Conditions

In this paper, we consider the system ut = ∆u, vt = ∆v x ∈ R+ , t > 0, − ∂u ∂x1 = v, − ∂v ∂x1 = u x1 = 0, t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ R+ , where R+ = {(x1, x′) | x′ ∈ RN−1, x1 > 0}, p, q > 0, and u0, v0 nonnegative. We prove that if pq ≤ 1 every nonnegative solution is global. When pq > 1 we let α = 1 2 p+1 pq−1 , β = 1 2 q+1 pq−1 . We show that if max(α, β) ≥ N 2 , all nontrivial nonnegative solutions are nonglobal; whereas if max(α, β) < N 2 there exist both global and nonglobal nonnegative solutions. When N = 1, we establish some results for the blow up rate for the nonglobal solutions and some results for the decay rate for the global solutions (in the supercritical case). We also construct a nontrivial solution with vanishing initial values when pq < 1.