transactions of the american mathematical society Volume 297, Number 1, September 1986 THE BLOW-UP BOUNDARY FOR NONLINEAR WAVE EQUATIONS1 LUIS A. CAFFARELLI AND AVNER FRIEDMAN Abstract. Consider the Cauchy problem for a nonlinear wave equation Du = F(u) in N space dimensions, N < 3, with F superlinear and nonnegative. It is well known that, in general, the solution blows up in finite time. In this paper it is shown, under some assumptions on the Cauchy data, that the blow-up set is a space-like surface t = <j>(x)with <f>{x) continuously difierentiable. Introduction. Consider the nonlinear wave equation (0.1) Uu = u,t- Au = F(u) for x 6 R", í > 0, with the initial data (0.2) u(x,0)-/(x), ul(x,0) = g(x) for x G RN. It is well known that if F(u) is nonnegative and superlinear, then, in general, a solution cannot exist for all times. Furthermore, if T is the supremum of all times s such that a classical solution exists for all 0 < t < s, then sup |w(jc, t) | -* oo if t -* T. For details see [1-5]. In this paper we are interested in studying the blow-up set, i.e., the set T = d{u< oo} n{t > 0}. We assume that N < 3 in order to ensure that the fundamental solution of the d'Alembertian D is positive (the same assumption is made in [2, 3]). Our main result is that (0.3) T is a C1 space-like surface, that is, T is given by (0.3') T:t = <t>(x), with </> e C1 and |v<f>| < 1. The conditions on /, g and F are such that they ensure that (0.4) u > 0, du/dt >\vxu\; further, F(u) is convex and F(u) - Aup as u -» oo (A > 0, p > 1). Received by the editors April 2, 1985 and, in revised form, September 17, 1985. 1980 Mathematics Subject Classification. Primary 35L70; Secondary 35L05, 35L67. 'This work is partially supported by National Science Foundation Grants 7406375 and MCS 7915171. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 223 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use