By Katz N.M.

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25) TJ a) 0 n (1 8~ 12 +~a·z~~ ~ at j,l=l -Re~ ~ j,l=l J ax. ax J l a·1 au µ·au_ Jax l 1 at 18 12 n +2~h·µ ~ ~ J J j=l at 40 2. EXISTENCE OF A SOLUTION FOR A HYPERBOLIC EQUATION . ;':-... J 1 j,l=l J ax . at In the above, we set n n M = L ajlµjµi, N = Lhjµj. 25) ~ (ry + I~~ 1 2N) 2 - 2£ 112 M 112 \~~I+ ryL. Regarding the right-hand side of the above as a quadratic expres- . 25) is also non-negative. 29) ry 2 + 2Nry - M ~ 0. But, the left-hand side is the characteristic equation Po(t, x, ry, µ) of P.

On the other hand, a2 ( at 2 - a2 ) :2::: ax2 u(t,p) = 2 j=l (a2u at2 - J :2::: a2u ax2 2 j=l a2u) az2 (t,p) = J and so u(t,p)--+ o, au at (t,p)--+ 'l/;(p) = 'l/;(x) Ifwe define u(t,x) E C 2 ((0,oo) x JR2 ) by u(t,x) = u(t,x,O), (t---+ 0). o, 26 1. :: ax2 u = o, j=l -----+ J 0, au at (t, x) -----+ '1/J(x) (t --t 0). Now, we write u(t,x) = M[~](t,x,O) = - 1- 47rt { }l(y,z)-(x,O)l=t ~(y,z)dSy,z· We can express the part of the sphere of radius t and center at (x,O) with z > 0 as {(x1 + s1, x2 + s2, Jt 2 - si - s§ ); si + s§ < t 2}.

0. 47) M[g](t, x) = { g(x + tf,)dSe, 4t7r Js2 (t > 0), where g is a function defined on ~ 3 . It follows immediately from the definition that if g E cm(~ 3 ), then M[g] E cm((O, oo) x ~ 3 ). Therefore, if cp E C 3 (~ 2 ), then M[cp] E C 3 ((0,oo) x ~ 3 ). 48) - A) M[cp] = 0, { M[cp](t,x)------+ 0, :tM[cp](t,x)------+ cp(x). The convergence, in the above, is generalized and uniform in ~ 3 . Now, since M[cp] E C 3 ((0, oo) x ~ 3 ), the following holds: EP2 - A ) ot a M[cp] = 8ta ( 8t2 a2 - A ) M[cp] = O.