Lower bounds for the life-span of solutions of nonlinear wave equations in three dimensions.
The paper deals with strict solutions u(x,t) = u(x(1),x(2),x(3),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) = epsilonF(x), u(t)(x,0) = epsilonG(x), the life span T(epsilon) 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(epsilon) < infinity. Examples show that this can occur for arbitrarily small epsilon. On the other hand, T(epsilon) must at least be very large for small epsilon. By assuming that a(ik),F,G [unk] C(infinity), that a(ik)(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.
John
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Courant Institute of Mathematical Sciences, New York University, New York, New York 10012.