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| \centerline{\large{\bf On the Cauchy problem for weakly coupled system
}}
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\centerline{\large{\bf of damped wave equations}}
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\centerline{\bf Kenji Nishihara}
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\centerline{Faculty of Political Science and Economics, Waseda University}
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\centerline{kenji@waseda.jp}
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In this talk we consider the Cauchy problem for the weakly coupled system of
damped wave equations
$$
\left \{ \begin{array}{l}
u_{tt} - \Delta u + b_1\cdot u_t = |v|^p, \\
v_{tt} - \Delta v + b_2 \cdot v_t = |u|^q,
\end{array}\right.
\ \ (t,x) \in {\bf R}_+ \times {\bf R}^N. \leqno{(P)}
$$
Our main interest is concerning to the critical exponent of $p,q$. Note that, when $b_1=b_2 \equiv 1$, the critical exponent is given by
$$
A:= \max \left( \frac{p+1}{pq-1}, \frac{q+1}{pq-1} \right)
\left\{ \begin{array}{cl}
< N/2 & :\, \mbox{supercritical}, \\
= N/2 & :\, \mbox{critical}, \\
> N/2 & :\, \mbox{subcritical},
\end{array}\right.
$$
in the sense that, in the supercritical case $(P)$ has a globa in time solution $(u,v)$ for a small data, while the local solution $(u,v)$ with some data blows up within a finite time in the critical and subcritical cases. |
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