Zero Reaction Limit for Hyperbolic Conservation Laws with Source Terms

In this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source term For the Cauchy problem to the above equation, we prove that as →0, its solution converges to piecewise constant (±1) solution, where the two constants are the two stable local equilibria. The constants are separated by either shocks that travel with speed (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 is the unstable equilibrium. Our analytic tool is the method of generalized characteristics. Similar results for more general source term g(u), having finitely many simple zeros and satisfying ug(u)<0 for large u, are also given.