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

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  • 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.
    Mathematics Subject Classification: 35B33, 35B40, 35B44, 35K55, 35K57, 65M06, 65M12.


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  • [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), 886-907.doi: 10.1137/0520060.


    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), 525-537.


    A. Friedman, Blow up solutions of nonlinear parabolic equations, W, M. Ni, L. A. Peletier, J. Serrin (Eds. ), nonlinear diffusion equations and their equilibrium states, Birkhaser Verlag, Basel, 12 (1988), 301-318.doi: 10.1007/978-1-4613-9605-5_19.


    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), 109-124.


    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), 1199-1216.doi: 10.1016/j.amc.2015.07.029.


    K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad. Ser. A Math, 49 (1973), 503-505.doi: 10.3792/pja/1195519254.


    H. A. Levine, The role of critical exponents in blow up theorems, SIAM Rev, 32 (1990), 262-288.doi: 10.1137/1032046.


    P. Souplet, Finite time blow up for a nonlinear parabolic equation with a gradient term and applications, Math. Methods Appl. sci, 19 (1996), 1317-1333.doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M.


    P. Souplet and F. B. Weissler, Self-similar subsolutions and blow up for nonlinear parabolic equations, Nonlinear Analysis, Theory Methods and Applications, 30 (1997), 4637-4641.doi: 10.1016/S0362-546X(97)00258-7.

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