Nonexistence results for the Cauchy problem of some systems of hyperbolic equations
Volume 78 / 2002
Abstract
We consider the systems of hyperbolic equations $$ \leqalignno{\quad &\cases{u_{tt} = {\mit\Delta} (a(t, x)u) + {\mit\Delta} (b(t, x)v) + h(t, x) \vert v \vert^{p},\hskip.5pt\kern-5pt& t>0,\, x \in {\mathbb R}^N,\cr v_{tt} = {\mit\Delta} (c(t, x)v) + k(t, x) \vert u \vert^{q},\kern-5pt& t>0,\, x \in {\mathbb R}^N,\cr} \kern-5pt&{\rm(S1)}\cr \kern-5pt&\cases{u_{tt} = {\mit\Delta} (a(t, x)u) + h(t, x) \vert v \vert^{p},\kern-5pt& t>0,\, x \in {\mathbb R}^N,\cr v_{tt} = {\mit\Delta} (c(t, x)v) +l(t, x)\vert v \vert^{m} + k(t, x) \vert u \vert^{q},\hskip5.9pt\kern-5pt&t>0,\, x \in {\mathbb R}^N,\cr}\kern-5pt&{\rm(S2)} \cr \kern-5pt&\cases{ u_{tt} = {\mit\Delta} (a(t, x)u) + {\mit\Delta} (b(t, x)v) + h(t, x) \vert u \vert^{p},\kern-5pt&t>0,\, x \in {\mathbb R}^N,\cr v_{tt} = {\mit\Delta} (c(t, x)v) + k(t, x) \vert v \vert^{q},\kern-5pt& t>0,\, x \in {\mathbb R}^N,\cr}\kern-5pt&{\rm(S3)}\cr} $$ in $(0, \infty) \times{ \mathbb R}^N $ with $ u(0, x)=u_0(x), v(0, x)=v_0(x), u_t(0, x)=u_1(x)$, $ v_t(0, x)=v_1(x)$. We show that, in each case, there exists a bound ${\bf B}$ on $N$ such that for $1 \leq N \leq {\bf B}$ solutions to the systems blow up in finite time.
