Abstract
In this paper, we consider the blow-up phenomenon for the -heat equation on graph with Dirichlet boundary conditions and a reaction term
where
is called the discrete weighted Laplacian operators. If
, every solution is global, and if
and under some suitable conditions, we prove that the nonnegative and nontrivial solution blows up in finite time and the blow-up rate on
-norm only depends on p. Finally, two examples are proposed to demonstrate our results.
Acknowledgements
We want to thank the referees for their comments, which helped us to improve this manuscript.
Notes
This paper is supported in part by NSF of China [grant number 11226190] and in part by the Fund of the Key Disciplines in the General Colleges and Universities of Xin Jiang Uygur Autonomous Region [grant number 2012ZDXK08].