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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the blow up of $ u\sb t$ at quenching


Authors: Keng Deng and Howard A. Levine
Journal: Proc. Amer. Math. Soc. 106 (1989), 1049-1056
MSC: Primary 35B40; Secondary 35K55
MathSciNet review: 969520
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Abstract: Let $ \Omega $ be a bounded convex domain in $ {{\mathbf{R}}^n}$ with smooth boundary. We consider the problems $ \left( C \right):{u_t} = \Delta u + \varphi \left( u \right)$ in $ \Omega \times \left( {0,T} \right)$, while $ u = 0$ on $ \partial \Omega \times \left( {0,T} \right)$ and $ u\left( {x,0} \right) = {u_0}\left( x \right)$. Here $ \varphi \left( u \right):\left( { - \infty ,A} \right) \to \left( {0,\infty } \right)\left( {A > 0} \right)$ satisfies $ \varphi '\left( u \right) \geq 0,\varphi ''\left( u \right) \geq 0$, and $ {\lim _{u \to {A^ - }}}\varphi \left( u \right) = + \infty $, while $ {u_0}$ satisfies $ \Delta {u_0}\left( x \right) + \varphi \left( {{u_0}\left( x \right)} \right) \geq 0$. We show that if $ u$ quenches (reaches $ A$ in finite time), then the quenching points are in a compact subset of $ \Omega $ and $ {u_t}$ blows up. We also extend the result to the third boundary value problem.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0969520-0
PII: S 0002-9939(1989)0969520-0
Article copyright: © Copyright 1989 American Mathematical Society