Pergamon Nonlinear Analysis, Theory, Methods & Applicotiom. Vol. 28, NO. 8, pp. 1299-1332, 1997 Copyright 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00+0.00 0362-546X(95)00228-6 GLOBAL CLASSICAL SOLUTIONS FOR GENERAL QUASILINEAR HYPERBOLIC SYSTEMS WITH DECAY INITIAL DATA LI TA-TSIENI, ZHOU YI and KONG DE-XINGS TDepartment of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China; and SInstitute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China (Received 31 July 1995) Key words and phrases: Global classical solution, general quasilinear hyperbolic systems, decay initial data. 1. INTRODUCTION AND MAIN RESULTS Consider the following first order quasilinear strictly hyperbolic system au at+z4(u)~=0, whereu = (u,, . . . . uJT is the unknown vector function of (t, x) and A(u) = (ati( is an n x n matrix with suitably smooth elements au(u) (i,j = 1, . . . . n). By the definition of strict hyperbolicity, for any given u on the domain under consideration, A(u) has n distinct real eigenvalues Al(U) < A,(u) < ** * < n,(u). (l-2) Let Ii(U) = (Iii(U), . . . , ljn(U)) (resp. rj(“) = (ril(u)v ..-, ri,(U))T) be a left (resp. right) eigenvector corresponding to n,(u) (i = 1, . . ., n): li(U)A(U) = ni(U)ri(U) (resp. A(U)ri(U) = Ai(U)ri(U)), (1.3) we have detJlij(u)l z 0 (equivalently, det Irij(U)I # 0). (1.4) All n,(U), lij(u) and rij(U) (i,j = 1, . . . . n) have the same regularity as aij(U) (i,j = 1, . . . , n). Without loss of generality, we may suppose that li(U)rj(U) 3 6, (i,j = 1, . . . . n) (1.5) and r&)rj(u) = 1 (i = 1, . . . . n), (1.6) where 6, stands for the Kronecker’s symbol. In this paper we only suppose that system (1.1) is strictly hyperbolic in a neighbourhood of 2.4 = 0, namely l,(O) < A,(O) < **- < A,(O). (1.7) 1299