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Almost global existence of elastic waves of finite amplitude arising from small initial disturbances. (English) Zbl 0635.35066
This paper deals with strict solutions $$u_ i(t,x_ 1,x_ 2,x_ 3)$$ of a system of quasi-linear equations $(\partial ^ 2u_ i/\partial t^ 2)-\sum [c^ 2_ 2D_ kD_ ku_ i+(c^ 2_ 1-c^ 2_ 2)D_ iD_ ku_ k]\quad =\sum ^{k}_{k,r,s}C_{ikrs}(u')D_ kD_ su_ r$ with $$D_ k=(\partial /\partial x_ k)$$, $$u'(D_ ku_ i)$$ and i, k, r, s ranging over 1,2,3. The equations of motion for a homogeneous, isotropic, hyper-elastic material are of this form. For given initial conditions $u_ i=\epsilon f_ i(x_ 1,x_ 2,x_ 3),\quad \partial u_ i/\partial t=\epsilon g_ i(x_ 1,x_ 2,x_ 3)\quad for\quad t=0,$ the life span T is the supremum of all $$t>0$$, to which the $$u_ i$$ can be extended as strict solutions for all $$(x_ 1,x_ 2,x_ 3)$$. For fixed functions $$f_ i$$, $$g_ i$$, $$C_{ikrs}$$ in $$C_ 0$$ with $$C_{ikrs}=C_{rsik}$$ and constants $$c$$ $$2_ 2<c$$ $$2_ 1$$ the life- span only depends on the positive parameter $$\epsilon$$. Earlier [Commun. Pure Appl. Math. 30, 421-446 (1977; Zbl 0404.73023)] the author proved that $$\lim _{\epsilon \to 0} \inf (\epsilon \log (1/\epsilon))\quad 4T(\epsilon)>0.$$
He now derives the much stronger result that $$\lim _{\epsilon \to 0} \inf \epsilon \log T(\epsilon)>0.$$
The analogue for the case of single scalar quasi-linear wave equation had been proved by S. Klainerman [ibid. 37, 269-288 (1984; Zbl 0583.35068)] and F. John and S. Klainerman [ibid. 37, 443-455 (1984; Zbl 0599.35104)].
Reviewer: F.John

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 74B20 Nonlinear elasticity 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
 [1] Klainerman, Comm. Pure Appl. Math. 38 pp 321– (1985) [2] John, Comm. Pure Appl. Math. 39 (1986) [3] Klainerman, Comm. Pure Appl. Math. 37 pp 269– (1984) [4] John, Comm. Pure Appl. Math. 37 pp 443– (1984) [5] John, Comm. Pure Appl. Math. 30 pp 421– (1977) [6] John, Comm. Pure Appl. Math. 36 pp 1– (1983) [7] Formation of Singularities in Elastic Waves, Lecture Notes in Physics 190-214, Springer Verlag 1984, pp. 190–214. [8] John, Comm. Pure Appl. Math. 34 pp 29– (1981) [9] Topics in Finite Elasticity, Regional Conference Series in Applied Mathematics 35, Society for Industrial and Applied Mathematics, 1981,.
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