ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXIII, 2 (2004)
p. 279 - 293

On Uniqueness for a System of Heat Equations Coupled in the Boundary Conditions
M. Kordos

Abstract.  We consider the system \begin{align*} u_t\eq&\lap u, & v_t\eq&\lap v, & x&\in\mathbb R^N_+, & t&>0,\\ -\frac{\partial u}{\partial x_1}\eq&v^p, & -\frac{\partial v}{\partial x_1}\eq&u^q, & x_1&\eq0, & t&>0,\\ u(x,0)\eq&u_0(x), & v(x,0)\eq&v_0(x), & x&\in\mathbb R^N_+, && \end{align*} where $\mathbb R^N_+\eq\left\{(x_1,x'): x'\in\mathbb R^{N-1},x_1>0\right\}$, $p$, $q$ are positive numbers, and functions~$u_0$, $v_0$ in the initial conditions are nonnegative and bounded. We show that nonnegative solutions are unique if~$pq\vr1$ or if~$(u_0,v_0)$ is nontrivial. In the case of zero initial data and~$pq<1$, we find all nonnegative nontrivial solutions.
Keywords: Parabolic system, uniqueness.

AMS Subject classification: 35K50; Secondary: 35K60