Source: Osaka J. Math. Volume 49, Number 2
(2012), 331-348.
Consider the Cauchy problem for a system of weakly coupled
heat equations, whose typical one is
\begin{equation*}
\left\{
\begin{array}{@{}ll@{}}
u_{t} - \Delta u = \lvert v \rvert^{p-1}v,\\
v_{t} - \Delta v = \lvert u \rvert^{q-1}u,
& (t, x) \in \mathbf{R}_{+} \times \mathbf{R}^{N},
\end{array}
\right.
\end{equation*}
with $p,q \ge 1$,
$pq > 1$. When $p,q$ satisfy $\max((p+1)/(pq-1),(q+1)/(pq-1))
< N/2$, the exponents $p,q$ are supercritical. In this
paper we assort the supercritical exponent case to two cases.
In one case both $p$ and $q$ are bigger than the Fujita exponent
$\rho_{F}(N)=1+2/N$, while in the other case $\rho_{F}(N)$
is between $p$ and $q$. In both cases we obtain the time-global
and unique existence of solutions for small data and their
asymptotic behaviors. These observation will be applied to
the corresponding system of the damped wave equations in low
dimensional space.
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