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
Download:
Adobe PDF 
Compressed Postscript 
Version to read:
Adobe PDF
Acta Mathematica Universitatis Comenianae
Institute of Applied Mathematics
Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic
Telephone: + 421-2-60295755 Fax: + 421-2-65425882
e-Mail: amuc@fmph.uniba.sk
Internet: www.iam.fmph.uniba.sk/amuc
©
Copyright 2004, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE