We prove an inequality (Lemmas 1.1') which has been applied by one of the authors and by J. Moser in their papers in this issue. The inequality expresses that a function, which in every subcube C of a cube C, can be approximated in the L1 mean by a constant ac with an error independent of C, differs then also in the L" mean from a, in C by an error of the same order of magnitude. More precisely, the measure of the set of points in C, where the function differs from a, by more than an amount u decreases exponentially as a increases.I n Section 2 we apply Lemma 1' to derive a result of Weiss and Zygmund [3], and in Section 3 we present an extension of Lemma 1'. LEMMA 1. Let ~( x ) be a n integrable function defined in a finite cube C, in n-dimensional space; x = (xl, --+, XJ. Assume that there is a constant K such that for every parallel subcube C , and some constant a,, the inequality holds. Here dx denotes element of volume and m ( C ) i s the Lebesgue measure of C. Then, if p(u) is the measure of the set of points where /u-aco~ > 0, we have (2) p(u) 5 Be"+m(C,) for u > 0, where B, b are constants depending only on n.

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Solutions u(x, t) of the inequality 0 u > A I u I for x e R3, t > 0 are considered, where 3 is the d'Alembertian, and A,p are constants with A > 0,1 < p < 1 + V . It is shown that the support of u is compact and contained in the cone 0 < t < to -Ix -xO-, if the "initial data" u(x, 0), ut(x, 0) have their support in the ball I x -x < to. In particular, "global" solutions of 3 u = A ju I P with initial data of compact support vanish identically. On the other hand, for A > 0, p > 1 + X"_2, global solutions of 3 u = A ju I P exist, if the initial data are of compact support and "sufficiently" small. This paper is concerned with global existence of solutions u(x, t) = u(x1, x2, X3, t) of a nonlinear wave equation of the form Ou = 0(u) [1] or of an inequality of the form [2], of class C2 in the half-space x e R3, t > 0, for which f E C3, g E C2 for x e R.3 "Blow-up" consists in nonexistence of a global solution for given fg,k. In that case, instead of global solutions there may still exist local solutions u defined for x E R3 and sufficiently small t. We denote by uO(x, t) the solution of the linear wave equation 0u0= 0 [4]

The waves considered here are solutions of a first-order strictly hyperbolic system of differential equations, written in the form where u = u ( x , t ) is a vector with n components u l , * -* , u, depending on two scalar independent variables x, t, and u = u ( u ) is an n-th order square matrix. 'The question to be discussed is the formation of singularities of a solution u of (1) corresponding to initial data I t will be shown that if the system (1) is "genuinely nonlinear" in a sense to be defined below, and if the,initial data are "sufficiently small" (but not identically 0), the first derivatives of u will become infinite for certain (x, t ) with t > 0.The result is well known for n = 1 and n = 2 (see the papers by Lax and by Glimm and Lax [I], [2], [lo]). In many cases such a singular behavior can be identified physically with the formation of a shock, and considerable interest attaches to the study of the subsequent behavior of the solution. I n the present paper we shall be content just to reach the onset of singular behavior, without attempting to define and to follow a solution for all times. For that we would need a physical interpretation of our system to guide us in formulating proper shock conditions. An indication of the behavior to be expected for a solution u of our system is furnished by the case n = 1, when u and u ( u ) are scalars and the problem ( l ) , (2) can be solved explicitly (see Lax [l]). Here u is constant along the characteristic curves