Regular Article
Zero Reaction Limit for Hyperbolic Conservation Laws with Source Terms

DEDICATED TO PROFESSOR JACK K. HALE ON THE OCCASION OF HIS 70TH BIRTHDAY
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Abstract

In this paper we study the zero reaction limit of the hyperbolic conservation law with stiff source termtu+∂xf(u)=1εu(1−u2).For the Cauchy problem to the above equation, we prove that as ε→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 12(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 1ε g(u), having finitely many simple zeros and satisfying ug(u)<0 for large |u|, are also given.

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f1

E-mail: [email protected]

f2

E-mail: [email protected]

f3

E-mail: [email protected]

1

Research supported in part by NSF Grant DMS 9705732.

2

Research supported in part by NSF Grant DMS 9704957. Current address: Department of Mathematics, University of Wisconsin, Madison, WI 53706. E-mail: [email protected].

3

Research supported in part by a grant from the National Natural Science Foundation of China.