Nonexistence of global solutions of a nonlinear hyperbolic system

Deng, Keng

doi:10.1090/S0002-9947-97-01841-2

Nonexistence of global solutions of a nonlinear hyperbolic system
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Trans. Amer. Math. Soc. 349 (1997), 1685-1696 Request permission

Abstract:

Consider the initial value problem \begin{equation*} \begin {array}{llll} u_{tt} = \Delta u+\vert v\vert ^{p}, & v_{tt} = \Delta v +\vert u\vert ^{q}, &x\in \mathbb {R}^{n},&t>0,\\ u(x,0)=f(x),&v(x,0)=h(x),&{}&{}\\ u_{t}(x,0) = g(x), &v_{t}(x,0) = k(x), \end{array} \end{equation*} with $1\le n\le 3$ and $p,q>0$. We show that there exists a bound $B(n) (\le \infty )$ such that if $1<pq<B(n)$ all nontrivial solutions with compact support blow up in finite time.

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