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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 \left(x-b\right)f\left(u\left(x,t\right)\right)\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{1.em}{0ex}}0
$u\left(x,0\right)=0\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{1.em}{0ex}}0\le x\le 1,\phantom{\rule{2.em}{0ex}}u\left(0,t\right)=u\left(1,t\right)=0\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{1.em}{0ex}}0

where $\delta \left(x-b\right)$ is the Dirac function concentrated at $x=b\in \left(0,1\right)$. It is supposed that ${lim}_{u\to {c}^{-}}f\left(u\right)=\infty$ for some $c>0$ and that $f\left(u\right)$ and ${f}^{\text{'}}\left(u\right)$ are positive for $0\le u. The solution $u\left(x,t\right)$ is said to quench if there exists some ${t}_{q}$ such that $max\left\{u\left(x,t\right):x\in \left[0,1\right]\right\}\to {c}^{-}$ 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 [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)].

##### MSC:
 35K65 Parabolic equations of degenerate type 35K60 Nonlinear initial value problems for linear parabolic equations 35K57 Reaction-diffusion equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)