Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Quenching for a degenerate parabolic problem due to a concentrated nonlinear source


Authors: C. Y. Chan and X. O. Jiang
Journal: Quart. Appl. Math. 62 (2004), 553-568
MSC: Primary 35K60; Secondary 35K57, 35K65
DOI: https://doi.org/10.1090/qam/2086046
MathSciNet review: MR2086046
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Abstract: Let $ q$, $ a$, $ T$, and $ b$ be any real numbers such that $ q \ge 0$, $ a > 0$, $ T > 0$, and $ 0 < b < 1$. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at $ b$:

$\displaystyle {x^q}{u_t} - {u_{xx}} = {a^2}\delta \left( x - b \right)f\left( u\left( x, t \right) \right) in \left( 0, 1 \right) \times \left( 0, T \right],$

$\displaystyle u\left( x, 0 \right) = 0 on \left[ 0, 1 \right], u\left( 0, t \right) = u\left( 1, t \right) = 0 for \; 0 < t \le T,$

where $ \delta \left( x \right)$ is the Dirac delta function, $ f$ is a given function such that $ {\lim _{u \to {c^ - }}}f\left( u \right) = \infty $ for some positive constant $ c$, and $ f\left( u \right)$ and $ f'\left( u \right)$ are positive for $ 0 \le u < c$. It is shown that the problem has a unique continuous solution $ u$ before $ max\left\{ {u\left( x, t \right) : 0 \le x \le 1} \right\}$ reaches $ {c^ - }$, $ u$ is a strictly increasing function of $ t$ for $ 0 < x < 1$, and if $ max\left\{ {u\left( x, t \right):0 \le x \le 1} \right\}$ reaches $ {c^ - }$, then $ u$ attains the value $ c$ only at the point $ b$. The problem is shown to have a unique $ {a^*}$ such that a unique global solution $ u$ exists for $ a \le {a^*}$, and $ max\left\{ {u\left( x, t \right):0 \le x \le 1} \right\}$ reaches $ {c^ - }$ in a finite time for $ a > {a^*}$; this $ {a^*}$ is the same as that for $ q = 0$. A formula for computing $ {a^*}$ is given, and no quenching in infinite time is deduced.

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DOI: https://doi.org/10.1090/qam/2086046
Article copyright: © Copyright 2004 American Mathematical Society


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