The Blow-up Rate for a System of Heat Equations with Non-trivial Coupling at the Boundary

We study the blow-up rate of positive radial solutions of a system of two heat equations, (u₁)_t=Δu₁(u₂)_t=Δu₂, in the ball B(0, 1), with boundary conditions Under some natural hypothesis on the matrix P=(p_ij) that guarrantee the blow-up of the solution at time T, and some assumptions of the initial data u_0i, we find that if ∥x₀∥=1 then u_i(x₀, t) goestoinfinitylike(T−t), where the α_i<0 are the solutions of (P−Id)(α₁,α₂)^t=(−1,−1)^t. As a corollary of the blow-up rate we obtain the loclaization of the blow-up set at the boundary of the domain. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.