Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form

We study the Cauchy problem for quasi-linear strictly hyperbolic systems of diagonal form with weakly linearly degenerate characteristics. The global existence and uniqueness of C-1 solution with initial data phi in C-b(1)(R) boolean AND W-1,W-1(R) are proved when the L-1(R) norm of phi or the C-0(R) norm of phi' is small. We also give some examples to show that if the initial data are not in W-1,W-1(R) or not small, then the solution to the Cauchy problem may blow up in a finite time. (C) 2003 Elsevier Ltd. All rights reserved.