Nonexistence results for the Cauchy problem of some systems of hyperbolic equations

Mokhtar Kirane¹ ; Salim Messaoudi²

¹ Université de Picardie Jules Verne Faculté de Mathématiques et d'Informatique LAMFA UPRES A 6119, 33 rue Saint Leu 80039 Amiens Cedex 1, France
² Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia

Annales Polonici Mathematici, Tome 78 (2002) no. 1, pp. 39-47

Résumé

We consider the systems of hyperbolic equations $$ \leqalignno{\quad \cases{u_{tt} = {\mit\Delta} (a(t, x)u) + {\mit\Delta} (b(t, x)v) + h(t, x) \vert v \vert^{p},\hskip.5pt\kern-5pt t>0,\, x \in {\mathbb R}^N,\cr v_{tt} = {\mit\Delta} (c(t, x)v) + k(t, x) \vert u \vert^{q},\kern-5pt t>0,\, x \in {\mathbb R}^N,\cr} \kern-5pt{\rm(S1)}\cr \kern-5pt\cases{u_{tt} = {\mit\Delta} (a(t, x)u) + h(t, x) \vert v \vert^{p},\kern-5pt t>0,\, x \in {\mathbb R}^N,\cr v_{tt} = {\mit\Delta} (c(t, x)v) +l(t, x)\vert v \vert^{m} + k(t, x) \vert u \vert^{q},\hskip5.9pt\kern-5pt>0,\, x \in {\mathbb R}^N,\cr}\kern-5pt{\rm(S2)} \cr \kern-5pt\cases{ u_{tt} = {\mit\Delta} (a(t, x)u) + {\mit\Delta} (b(t, x)v) + h(t, x) \vert u \vert^{p},\kern-5pt>0,\, x \in {\mathbb R}^N,\cr v_{tt} = {\mit\Delta} (c(t, x)v) + k(t, x) \vert v \vert^{q},\kern-5pt t>0,\, x \in {\mathbb R}^N,\cr}\kern-5pt{\rm(S3)}\cr} $$ 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.

Détail

DOI : 10.4064/ap78-1-5

Keywords: consider systems hyperbolic equations leqalignno quad cases mit delta mit delta vert vert hskip kern mathbb mit delta vert vert kern mathbb kern kern cases mit delta vert vert kern mathbb mit delta vert vert vert vert hskip kern mathbb kern kern cases mit delta mit delta vert vert kern mathbb mit delta vert vert kern mathbb kern infty times mathbb each there exists bound leq leq solutions systems blow finite time