Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source


Authors: C. Y. Chan and H. Y. Tian
Journal: Quart. Appl. Math. 61 (2003), 363-385
MSC: Primary 35K55; Secondary 35B45, 35K20, 35K65
DOI: https://doi.org/10.1090/qam/1976376
MathSciNet review: MR1976376
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Abstract: Let $ q$ be a nonnegative real number, and $ T$ be a positive real number. This article studies the following degenerate semilinear parabolic first initial-boundary value problem:

$\displaystyle {x^q}{u_t}\left( x, t \right) - {u_{xx}}\left( x, t \right) = {a^... ...b \right)f\left( u\left( x, t \right) \right) \qquad for 0 < x < 1, 0 < t \le T$

,

$\displaystyle u\left( x, 0 \right) = \psi \left( x \right) \qquad for 0 \le x \le 1$

,

$\displaystyle u\left( 0, t \right) = u\left( 1, t \right) = 0 \qquad for 0 < t \le T$

, where $ \delta \left( x \right)$ is the Dirac delta function, and $ f$ and $ \psi $ are given functions. It is shown that the problem has a unique solution before a blow-up occurs, $ u$ blows up in a finite time, and the blow-up set consists of the single point $ b$. A lower bound and an upper bound of the blow-up time are also given. To illustrate our main results, an example is given. A computational method is also given to determine the finite blow-up time.

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DOI: https://doi.org/10.1090/qam/1976376
Article copyright: © Copyright 2003 American Mathematical Society


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