Blow-up for the ω-heat equation with Dirichlet boundary conditions and a reaction term on graphs

Abstract

In this paper, we consider the blow-up phenomenon for the -heat equation on graph with Dirichlet boundary conditions and a reaction termwhere 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.

Keywords:

• discrete heat equation
• reaction term
• blow-up
• graphs

AMS Subject Classifications:

• 35R02
• 58J35
• 35B44
• 35B50
• 35B51

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].