Blow-up of solutions for the heat equations with variable source on graphs

Volume 9, Issue 4, pp 1685--1692 http://dx.doi.org/10.22436/jnsa.009.04.24

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1671 Downloads
• 2672 Views

Authors

Qiao Xin - College of Mathematics and Statistics, Yili Normal University, 835000 Yining, P. R. China. Dengming Liu - School of Mathematics and Computational Science, Hunan University of Science and Technology, 411201 Xiangtan, P. R. China.

Abstract

In this paper, we mainly consider the blow-up problem for the discrete heat equations with variable source on finite graphs \[u_t = \Delta_\omega u + u^{p(x)}\] with homogeneous Dirichlet boundary conditions and positive initial energy. We prove that the corresponding solutions blow up at a finite time with large enough initial data. Moreover, the blow-up rate is also considered.

Share and Cite

Keywords

• Blow-up
• discrete heat equation
• variable reaction
• finite graphs.

MSC

•  35B44
•  35R02
•  35B40

References

• [1] F. R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence (1997)


• [2] S. Y. Chung , Critical blow-up and global existence for discrete nonlinear p-laplacian parabolic equations, Discrete Dyn. Nat. Soc., 2014 (2014), 10 pages.


• [3] S. Y. Chung, C. A. Berenstein, \(\omega\)-harmonic functions and inverse conductivity problems on networks, SIAM J. Appl. Math., 65 (2005), 1200-1226.


• [4] Y. S. Chung, Y. S. Lee, S. Y. Chung, Extinction and positivity of the solutions of the heat eqautions with absorption on networks, J. Math. Anal. Appl., 380 (2011), 642-652.


• [5] E. B. Curtis, J. A. Morrow, Determining the resistors in a network, SIAM J. Appl. Math., 50 (1990), 918-930.


• [6] A. Elmoataz, O. Lézoray, S. Bougleux, Nonlocal discrete regularization on weighted graphs: A framework for iamge and manifold processing, IEEE Trans. Image Prcoess., 17 (2008), 1047-1060.


• [7] R. Ferreira, A. de Pablo, M. Préz-LLanos, J. D. Rossi, Critical exponents for a semi-linear parabolic equation with variable reaction, Proc. R. Soc. Edinburgh, 142 (2012), 1027-1042.


• [8] J. P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Anal., 71 (2009), 1094-1099.


• [9] V. Ta, O. Lézoray, A. Elmoataz, S. Schüpp, Graph-based tools for microscopic cellular image segmentation, Pattern Recognit., 42 (2009), 1113-1125.


• [10] X. Wu, B. Guo, W. Gao, Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Appl. Math. Lett., 26 (2013), 539-543.


• [11] Q. Xin, L. Xu, C. Mu , Blow-up for the \(\omega\)-heat equation with dirichlet boundary conditions and a reaction term on graphs, Appl. Anal., 93 (2014), 1691-1701.


• [12] W. Zhou, M. Chen, W. Liu, Critical exponent and blow-up rate for the \(\omega\)-diffusion equations on graphs with dirichlet boundary conditions, Electron. J. Differential Equations, 2014 (2014), 13 pages.