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2014 Blow-Up Solutions and Global Solutions to Discrete p-Laplacian Parabolic Equations
Soon-Yeong Chung, Min-Jun Choi
Abstr. Appl. Anal. 2014: 1-11 (2014). DOI: 10.1155/2014/351675


We discuss the conditions under which blow-up occurs for the solutions of discrete p-Laplacian parabolic equations on networks S with boundary S as follows: ut(x,t)=Δp,ωu(x,t)+λ|u(x,t)|q-1u(x,t), (x,t)S×(0,+); u(x,t)=0, (x,t)S×(0,+); u(x,0)=u00, xS¯, where p>1, q>0, λ>0, and the initial data u0 is nontrivial on S. The main theorem states that the solution u to the above equation satisfies the following: (i) if 0<p-1<q and q>1, then the solution blows up in a finite time, provided u¯0>ω0/λ1/q-p+1, where ω0:=maxxSyS¯ω(x,y) and u¯0:=maxxS u0(x); (ii) if 0<q1, then the nonnegative solution is global; (iii) if 1<q<p-1, 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.


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Soon-Yeong Chung. Min-Jun Choi. "Blow-Up Solutions and Global Solutions to Discrete p-Laplacian Parabolic Equations." Abstr. Appl. Anal. 2014 1 - 11, 2014.


Published: 2014
First available in Project Euclid: 27 February 2015

zbMATH: 07022208
MathSciNet: MR3285158
Digital Object Identifier: 10.1155/2014/351675

Rights: Copyright © 2014 Hindawi


Vol.2014 • 2014
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