Abstract: In this paper the author considers the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with ``slow'' decay initial data. By constructing an example, the author first illustrates that the classical solution to this kind of Cauchy problem may blow up in a finite time, even if the system is weakly linearly degenerate. The author then gives some lower bounds of the life-span of classical solutions in the case that system is weakly linearly degenerate. These estimates imply that, when the system is weakly linearly degenerate, the classical solution exists almost globally in time. Finally, the author proves that Theorems 1.1 and 1.2 in  are still valid for this kind of initial data.
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