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Lower Bounds for the Life-Span of Solutions of Nonlinear Wave Equations in Three Dimensions

Fritz John
Proceedings of the National Academy of Sciences of the United States of America
Vol. 79, No. 12, [Part 2: Physical Sciences] (Jun. 15, 1982), pp. 3933-3934
Stable URL: http://www.jstor.org/stable/12566
Page Count: 2
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Abstract

The paper deals with strict solutions u(x,t) = u(x1, x2, x3, t) of an equation □ u=utt-Δ u=∑i,k=1 3aik(Du)uxixk , where Du is the set of four first derivatives of u. For given initial values u(x,0)=ε F(x), ut(x,0)=ε G(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 ∈ C∞, that aik(0)=0, and that F, G have compact support, it is shown that $\underset \varepsilon \rightarrow 0\to{\lim}\varepsilon ^{N}T(\varepsilon)=\infty $ for every N. This result had been established previously only for N < 4.

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