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October  2016, 9(5): 1421-1445. doi: 10.3934/dcdss.2016057

Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation

1. 

Université de Monastir, Faculté des Sciences de Monastir, Avenue de l'Environnement - Monastir - 5000, Tunisia

2. 

Université de Sousse, École Supérieure des Sciences et de Technologie de Hammam Sousse, Rue Lamine El Abbessi 4011 Hammam Sousse, Tunisia

Received  November 2014 Revised  September 2015 Published  October 2016

This paper deals with a nonlinear parabolic equation for which a local solution in time exists and then blows up in a finite time. We consider the Chipot-Weissler equation: \begin{equation*} u_{t}=u_{x x} + u^{p}-|u_{x}|^{q},\ \ x\in (-1,1);\ t>0, \ \ p>1 \text{ and } 1 \leq q < \frac{2p}{p+1}. \end{equation*} We study the numerical approximation, we show that the numerical solution converges to the continuous one under some restriction on the initial data and the parameters $p$ and $q$. Moreover, we study the numerical blow up sets and we show that although the convergence of the numerical solution is guaranteed, the numerical blow up sets are sometimes different from that of the PDE.
Citation: Houda Hani, Moez Khenissi. Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1421-1445. doi: 10.3934/dcdss.2016057
References:
[1]

M. Chipot and F. B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term,, SIAM J. Math. Anal, 20 (1987), 886. doi: 10.1137/0520060. Google Scholar

[2]

M. Chlebik, M. Fila and P. Quittner, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term,, Dyn. Contin. Discrete Impulsive Syst. Ser. A Math. Anal., 10 (2003), 525. Google Scholar

[3]

A. Friedman, Blow up solutions of nonlinear parabolic equations,, W, 12 (1988), 301. doi: 10.1007/978-1-4613-9605-5_19. Google Scholar

[4]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac. Sci. Univ. tokyo Sect. IA Math, 13 (1966), 109. Google Scholar

[5]

H. Hani and M. Khenissi, On a finite difference scheme for blow up solutions for the Chipot-Weissler equation,, Applied Mathematics and Computation, 268 (2015), 1199. doi: 10.1016/j.amc.2015.07.029. Google Scholar

[6]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad. Ser. A Math, 49 (1973), 503. doi: 10.3792/pja/1195519254. Google Scholar

[7]

H. A. Levine, The role of critical exponents in blow up theorems,, SIAM Rev, 32 (1990), 262. doi: 10.1137/1032046. Google Scholar

[8]

P. Souplet, Finite time blow up for a nonlinear parabolic equation with a gradient term and applications,, Math. Methods Appl. sci, 19 (1996), 1317. doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M. Google Scholar

[9]

P. Souplet and F. B. Weissler, Self-similar subsolutions and blow up for nonlinear parabolic equations,, Nonlinear Analysis, 30 (1997), 4637. doi: 10.1016/S0362-546X(97)00258-7. Google Scholar

show all references

References:
[1]

M. Chipot and F. B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term,, SIAM J. Math. Anal, 20 (1987), 886. doi: 10.1137/0520060. Google Scholar

[2]

M. Chlebik, M. Fila and P. Quittner, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term,, Dyn. Contin. Discrete Impulsive Syst. Ser. A Math. Anal., 10 (2003), 525. Google Scholar

[3]

A. Friedman, Blow up solutions of nonlinear parabolic equations,, W, 12 (1988), 301. doi: 10.1007/978-1-4613-9605-5_19. Google Scholar

[4]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac. Sci. Univ. tokyo Sect. IA Math, 13 (1966), 109. Google Scholar

[5]

H. Hani and M. Khenissi, On a finite difference scheme for blow up solutions for the Chipot-Weissler equation,, Applied Mathematics and Computation, 268 (2015), 1199. doi: 10.1016/j.amc.2015.07.029. Google Scholar

[6]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad. Ser. A Math, 49 (1973), 503. doi: 10.3792/pja/1195519254. Google Scholar

[7]

H. A. Levine, The role of critical exponents in blow up theorems,, SIAM Rev, 32 (1990), 262. doi: 10.1137/1032046. Google Scholar

[8]

P. Souplet, Finite time blow up for a nonlinear parabolic equation with a gradient term and applications,, Math. Methods Appl. sci, 19 (1996), 1317. doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M. Google Scholar

[9]

P. Souplet and F. B. Weissler, Self-similar subsolutions and blow up for nonlinear parabolic equations,, Nonlinear Analysis, 30 (1997), 4637. doi: 10.1016/S0362-546X(97)00258-7. Google Scholar

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