The main purpose of this paper is to study the thermal balance equations for the gas and solid interphase heat-transfer for the fast-igniting catalytic converter of automobiles: ∂ ∂tu (t, x) = −α ∂ ∂xu (t, x) + av (t, x) − au (t, x) for t > 0, 0<x<l; ∂ ∂tv (t, x) = bu (t, x) − bv (t, x) + λ exp (v (t, x)) for t > 0, 0<x<l; u (t, 0) = η for t ≥ 0; u (0, x) = u0 (x) and v (0, x) = v0 (x) for x>0. where u0, v0 are continuous functions on [0, l] with u0 (0) = η. We establish some results concerning the existence and uniqueness of the mild solutions and classical solutions of the above differential system. The asymptotical behavior of the solution is also addressed.
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