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Abstract and Applied Analysis
Volume 2014, Article ID 351675, 11 pages
http://dx.doi.org/10.1155/2014/351675
Research Article

Blow-Up Solutions and Global Solutions to Discrete -Laplacian Parabolic Equations

1Department of Mathematics and Program of Integrated Biotechnology, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received 21 August 2014; Accepted 16 October 2014; Published 24 November 2014

Academic Editor: Chengming Huang

Copyright © 2014 Soon-Yeong Chung and Min-Jun Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss the conditions under which blow-up occurs for the solutions of discrete -Laplacian parabolic equations on networks with boundary as follows: , ; , ; , where , , , and the initial data is nontrivial on . The main theorem states that the solution to the above equation satisfies the following: (i) if and , then the solution blows up in a finite time, provided , where and ; (ii) if , then the nonnegative solution is global; (iii) if , then the solution is global. In order to prove the main theorem, we first derive the comparison principles for the solution of the equation above, which play an important role throughout this paper. Moreover, when the solution blows up, we give an estimate for the blow-up time and also provide the blow-up rate. Finally, we give some numerical illustrations which exploit the main results.