Asymptotic behaviour for a parabolic system with nonlinear boundary conditions

In this paper we obtain the blow-up rate for positive solutions of a system of two heat equations, $u_t=\Delta u, v_t=\Delta v$, in a bounded smooth domain $\Omega$, with boundary conditions $\frac{\partial u}{\partial n}=v^p$, $\frac{\partial v}{\partial n}=u^q$. Under some assumptions on the initial data $u_0, v_0$ and $p,q$ subcritical, we find that the behaviour of $u$ and $v$ is given by $\parallel u(\cdotp,t)\parallel_\infty\sim(T-t)^{-\frac{p+1}{2(pq-1)}}$ and $\parallel v(\cdotp,t)\parallel_\infty\sim(T-t)^{-\frac{q+1}{2(pq-1)}}$ . As a corollary of the blow-up rate we obtain the localization of the blow-up set at the boundary of the domain. The main tool in the proof, is a nonexistence theorem for an elliptic system; we prove that the only nonnegative classical solution of the system $\Delta u=0, \Delta v=0$ in $\mathbb{R}^n_+$, with boundary conditions $\frac{\partial u}{\partial n}=v^p, \frac{\partial v}{\partial n}=u^q$ on $\partial\mathbb{R}^n_+$ is the trivial solution $u\equiv 0, v\equiv 0$, when $p\leq \frac{n}{n-2} , q < \frac{n}{n-2}$ and $pq>1$