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It is shown that solutions of hyperbolic equations in cylindrical space time domains which vanish on the lateral boundary of the cylinder must have arbitrarily large zeros in the interior of the cylinder. In case the coefficients of the equation are time independent the solutions will have arbitrarily large zeros on any interior line of the cylinder. This paper concerns the oscillations of solutions of hyperbolic equations. More specifically, we shall consider solutions of the equation a a (1) utt =Z 2 aI (k(X, t) -U C(X, t)U in some space-time cylinder Q x (0, xc), where we assume that the operator on the right is uniformly elliptic: (2) Z' ajk(X, t)$j$k >_ , 12 (e > O) j.k and that c(x, t)>0 in Q x (0, xc). We shall further assume that the coefficients aj are in C'(5 x (0, cx)) and that their derivatives are uniformly bounded in _ x (0, cc); while c(x, t) will be assumed continuous in nx (0, oc). We will first want to consider classical solutions of (1) with well-defined boundary values on an x (0, cc) and we will take this to mean a function u(x, t) in C2(! X (0, cc)) which satisfies (1) in Qlx (0, cx). Concerning Q we assume it to be a bounded domain in En with a& a sufficiently smooth n-I dimensional manifold having a well-defined exterior normal at every point. This being understood our first result is THEOREM 1. Let u(x, t) be a solution of (1) in Q) x (0, xo) wt-hich vtanishes on an x (O, cx). Then for any r>0, u must haLe a zero in Q x [ir, cc). REMARK. A result of this type has been obtained by K. Kreith [3] for the case of one space variable xl with the operator on the right of (1) having the form a2Ix2K(Xl, t). His method extends to second order elliptic operators in higher dimensions provided that the principal part Received by the editors October 12, 1972 and, in revised form, January 19, 1973. AMS (MOS) subject class ifcations (1970). Primary 35B05.

Oscillation theorems for solutions of hyperbolic equations