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 Title On the Cauchy problem for weakly coupled system of damped wave equations
 Name Kenji Nishihara Country Japan Email kenji@waseda.jp Co-Author(s) Kenji Nishihara Submit Time 2014-02-22 21:05:34 Session Special Session 60: Recent advances in evolutionary equations
 Contents \centerline{\large{\bf On the Cauchy problem for weakly coupled system }} \vspace{3mm} \centerline{\large{\bf of damped wave equations}} \vspace{3mm} \centerline{\bf Kenji Nishihara} \vspace{1mm} \centerline{Faculty of Political Science and Economics, Waseda University} \vspace{1mm} \centerline{kenji@waseda.jp} \vspace{5mm} 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.