The paper deals with strict solutions u(x,t) = u(x1,x2,x3,t) of an equation [Formula: see text] where Du is the set of four first derivatives of u. For given initial values u(x,0) = εF(x), ut(x,0) = εG(x), the life span T(ε) is defined as the supremum of all t to which the local solution can be extended for all x. Blowup in finite time corresponds to T(ε) < ∞. Examples show that this can occur for arbitrarily small ε. On the other hand, T(ε) must at least be very large for small ε. By assuming that aik,F,G [unk] C∞, that aik(0) = 0, and that F,G have compact support, it is shown that [Formula: see text] for every N. This result had been established previously only for N < 4.
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机译：本文涉及方程[公式：参见文本]的严格解u（x，t）= u（x1，x2，x3，t），其中Du是u的四个一阶导数的集合。对于给定的初始值u（x，0）=εF（x），ut（x，0）=εG（ x ），寿命 T （ε）为定义为所有 t 的全部，本地解决方案可以扩展到所有 x 。有限时间内的爆破对应于 T （ε）<∞。示例表明，这对于任意小的ε都可能发生。另一方面，对于小的ε， T （ε）必须至少非常大。假设 aik，F，G [unk] C ∞，则 aik （0）= 0，并且 F，G 具有紧凑的支持，表明每个 N 的[公式：参见文本]。先前仅针对 N <4建立了此结果。
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