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Almost global existence to nonlinear wave equations in three space dimensions. (English) Zbl 0599.35104

This paper achieves a long-sought goal, and as such represents a major contribution to the theory of hyperbolic equations. The Cauchy problem for the equation \[ u_{tt}-\Delta u=F(u,u',u'')\quad (x\in {\mathbb{R}}^ 3,t>0),\quad u=\epsilon u_ 0(x),\quad u_ t=\epsilon u_ 1(x)\quad at\quad t=0,u_ 0,u_ 1\in C_ 0^{\infty}, \] is studied. Here F is a smooth function which is ”quadratic” at the origin; u’ denotes the first derivatives of u, u” the second derivatives. It is shown that if F satisfies any one of three mild conditions (e.g., if F is independent of u, or if F is in conservation form), then for \(\epsilon\) sufficiently small, the life span of such a solution u is at least of the order \(\exp (B\epsilon^{-1})\) for some constant B.
This result had been proved earlier by the second author in the radial case. The main tools are new weighted estimates for the classical inhomogeneous wave equation [see the second author, ibid. 37, 269-288 (1984; Zbl 0583.35068)] which in turn were inspired by a paper of the first author [ibid. 36, 1-35 (1983; Zbl 0487.35065)]. The lower bound for the life-span is sharp, as will be shown in a paper of John [”Improved estimates for blow-up for solutions of strictly hyperbolic equations in 3 space dimensions”, to appear].

MSC:

35L70 Second-order nonlinear hyperbolic equations
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References:

[1] John, Comm. Pure Appl. Math. 36 pp 1– (1983)
[2] Klainerman, Comm. Pure Appl. Math. 36 pp 325– (1983)
[3] Improved estimates for blow-up for solutions of strictly hyperbolic equations in 3-space dimensions, in preparation.
[4] Klainerman, Comm. Pure Appl. Math. 37 pp 269– (1984)
[5] Klainerman, Comm. Pure Appl. Math. 33 pp 43– (1980)
[6] Nirenberg, Ann. Scuola Norm. Sup. Pisa (3) 13 pp 115– (1959)
[7] Moser, Ann. Scuola Norm. Sup. Pisa (3) 20 pp 265– (1966)
[8] John, Manuscripta Math. 28 pp 235– (1979)
[9] John, Comm. Pure Appl. Math. 29 pp 649– (1976)
[10] Partial Differential Equations, Holt, Rheinhart and Winston, New York, 1969.
[11] Klainerman, Comm. Pure Appl. Math. 34 pp 481– (1981)
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