Abstract
This article deals with the porous medium equation with a more complicated source term,
subject to the homogeneous Dirichlet condition, where
![](//:0)
is a ball with radius
R,
m > 1 and the non-negative constants
![](//:0)
satisfying
![](//:0)
. We investigate how the three factors (the non-local source
![](//:0)
, the local source
![](//:0)
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
![](//:0)
. (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.
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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.