The paper deals with the initial-boundary value problem
where is the Dirac function concentrated at . It is supposed that for some and that and are positive for . The solution is said to quench if there exists some such that as . Using Green’s function the problem is transformed to an equivalent Volterra integral equation. Then the unicity of the solution and the existence of the quenching time is proved.
This sort of problems is motivated by applications to some phenomena occuring by the ignition of a combustible medium. The paper is a continuation of the papers [C. Y. Chan and H. T. Liu, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 8, 121–128 (2001; Zbl 0994.35073)] and [C. Y. Chan and H. Y. Tian, Q. Appl. Math. 61, 363–385 (2003; Zbl 1032.35105)].