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Existence for large times of finite amplitude elastic waves arising from small disturbances. (English) Zbl 0646.35059
Summary: This paper deals with strict solutions \(u_ i(t,x_ 1,x_ 2,x_ 3)\) of a system of quasi-linear equations \[ \partial^ 2_ tu_ i- \sum_{k}[c^ 2_ 2\partial_{x_ k}\partial_{x_ k}u_ i+(c^ 2_ 1-c^ 2_ 2)\partial_{x_ k}\partial_{x_ i}u_ k]=\sum_{k,r,s}C_{ikrs}(u')\partial_{x_ k}\partial_{x_ s}u_ r \] with \(u'=(\partial_{x_ i}u_ k)\), and i,k,r,s ranging over 1,2,3. For given initial conditions \[ u_ i=\epsilon f_ i(x_ 1,x_ 2,x_ 3),\quad \partial_ tu_ i=\epsilon g(x_ 1,x_ 2,x_ 3)\quad for\quad t=0 \] the life-span \(T(\epsilon)\) is the supremum of all \(t>0\) to which the \(u_ i\) can be extended as strict solutions for all x. It is shown that \(\liminf_{\epsilon \to 0} \epsilon \log T(\epsilon)>0\) for \(C_{ikrs},f_ i,g_ i\in C_ 0^{\infty}\), and \(C_{ikrs}=C_{rsik},C_{ikrs}(0)=0\).
MSC:
35L70 Second-order nonlinear hyperbolic equations
35B20 Perturbations in context of PDEs
74B20 Nonlinear elasticity
35B40 Asymptotic behavior of solutions to PDEs
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