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Classification of singularities for blowing up solutions in higher dimensions


Author: J. J. L. Velázquez
Journal: Trans. Amer. Math. Soc. 338 (1993), 441-464
MSC: Primary 35K60; Secondary 35A20, 35B05
DOI: https://doi.org/10.1090/S0002-9947-1993-1134760-2
MathSciNet review: 1134760
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Abstract: Consider the Cauchy problem (P)

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{u_t} - \Delta u = {u^p}} \hfil... ...text{when}}\;x \in {\mathbb{R}^N},} \hfill & {} \hfill \\ \end{array} } \right.$

where $ p > 1$, and $ {u_0}(x)$ is a continuous, nonnegative and bounded function. It is known that, under fairly general assumptions on $ {u_0}(x)$, the unique solution of $ ({\text{P}})$, $ u(x,t)$, blows up in a finite time, by which we mean that

$\displaystyle \mathop {\lim \sup }\limits_{t \uparrow T} \left( {\mathop {\sup }\limits_{x \in {\mathbb{R}^N}} \;u(x,t)} \right) = + \infty .$

In this paper we shall assume that $ u(x,t)$ blows up at $ x = 0$, $ t = T < + \infty $ , and derive the possible asymptotic behaviours of $ u(x,t)$ as $ (x,t) \to (0,T)$, under general assumptions on the blow-up rate.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1134760-2
Keywords: Semilinear diffusion equations, asymptotic behaviour, classification of singularities, blow-up
Article copyright: © Copyright 1993 American Mathematical Society

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