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Applicable Analysis
An International Journal
Volume 88, 2009 - Issue 1
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Original Articles

Asymptotic analysis to blow-up points for the porous medium equation with a weighted non-local source

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Pages 111-120 | Received 04 May 2008, Accepted 14 Oct 2008, Published online: 21 May 2009
 

Abstract

This article deals with the porous medium equation with a more complicated source term,

subject to the homogeneous Dirichlet condition, where is a ball with radius R, m > 1 and the non-negative constants satisfying . We investigate how the three factors (the non-local source , the local source and the weight function a(x)) influence the asymptotic behaviour of the solutions. It is proved that (i) when p < 1, the non-local source plays a dominating role, i.e. the blow-up set of the system is the whole domain B R , a , where . (ii) When p > m, this system presents single blow-up patterns. In other words, the local term dominates the non-local term in the blow-up profile. Moreover, the blow-up rate estimate is established with more precise coefficients determined.

AMS Subject Classifications::

Acknowledgements

This work is supported in part by NNSF of China (10771226), in part by Natural Science Foundation Project of CQ CSTC (2007BB0124) and in part by Natural Science Foundation Project of China SWU, SWU208029.

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