Uniqueness and non-uniqueness for a system of heat equations with nonlinear coupling at the boundary

Abstract

We study the uniqueness problem for non negative solutions of a system of two heat equations, ut = ∆u, vt = ∆v, in a bounded smooth domain Ω, with nonlinear boundary conditions, ∂u ∂η = v, ∂v ∂η = u. We prove that for identically zero initial data, (u(x, 0), v(x, 0)) ≡ (0, 0), the zero solution is unique if and only if if pq ≥ 1. Moreover, in the case of non-negative non-trivial initial data the solution is always unique.