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(0.3) P * [ ] = 9 ? Z 9, (a „(*,*) 9,)+A( t ,* ;u ,A*)[w] «,/=! (0.4) A ( f , * ; u , / l K ) [ ] = S aaB(t.x\u.Au)A*A, 9t"a~, ^~ir~> i — 1, '",n, Au (dtu, d\u, • • • , 9Mti), a= (a0, ai, • • , a«), /J— OSo, A, • • • , ^8W) are multi-indices, and Aw= (dfdT'"dnua— (a0, ai, • • , aj). We next make following assumptions on PM. n (A-I) 2 d l ( a l j ( t , x}d}) is an elliptic operator satisfying for all r] ^W 1 u=i (0.5) Zfl, /)7,)?^CoS7??(Co>0)I u'=i »=i fliy(t, ^) =ay< (t, x) (i, ; = ! ,—, n) for all (t, *) e [0, oo) x Q,

publications of the research institute for mathematical sciences

Global existence in time and decay property of solutions of boundary value problems for semilinear hyperbolic equations of second order in the interior domain