Abstract: In this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source termu_{t} + f(u)_{x} = 1/eps u ( 1-u^{2} ).For the Cauchy problem to the above equation, we prove that as $\eps\to 0$, its solution converges to piecewise constant ($\pm 1$) solution, where the two constants are the two stable local equilibrium. The constants are separated by either shocks that travel with speed $\frac12(f(1)-f(-1))$, as determined by the Rankine-Hugoniot jump condition, or a non-shock discontinuity that moves with speed $f'(0)$, where $0$ being the unstable equilibrium. Our analytic tool is the method of generalized characteristics. Similar results for more general source term ${1\over \eps}g(u)$, having
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