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Quenching for a degenerate parabolic problem due to a concentrated nonlinear source. (English) Zbl 1072.35100
The paper deals with the initial-boundary value problem $$x^q u_t - u_{xx} = a^2 \delta (x-b) f(u(x,t)) \quad \text{for} \quad 0<x<1, \quad 0<t \leq T,$$ $$u(x,0) = 0 \quad \text{for} \quad 0 \leq x \leq 1, \qquad u(0,t) = u(1,t) = 0 \quad \text{for} \quad 0< t \leq T,$$ where $\delta(x-b)$ is the Dirac function concentrated at $x=b \in (0,1)$. It is supposed that $\lim_{u \to c^-} f(u) = \infty$ for some $c>0$ and that $f(u)$ and $f'(u)$ are positive for $0 \leq u < c$. The solution $u(x,t)$ is said to quench if there exists some $t_q$ such that $\max \{ u(x,t): x\in [0,1] \} \to c^-$ as $t \to t_q$. 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 $t_q$ 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 [{\it C. Y. Chan} and {\it H. T. Liu}, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 8, 121--128 (2001; Zbl 0994.35073)] and [{\it C. Y. Chan} and {\it H. Y. Tian}, Q. Appl. Math. 61, 363--385 (2003; Zbl 1032.35105)].

35K65Parabolic equations of degenerate type
35K60Nonlinear initial value problems for linear parabolic equations
35K57Reaction-diffusion equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)