Blow-up for Semidiscretizations of some Semilinear Parabolic Equations with a Convection Term
Abstract
This paper concerns the study of the numerical approximation for the following parabolic equations with a convection term
where p > 1.
We obtain some conditions under which the solution of the semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove that the semidiscrete blow-up time converges to the real one, when the mesh size goes to zero. Finally, we give some numerical experiments to illustrate ours analysis.
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Abia L., J. C., Lopez-Marcos J. C. and Martinez J., On the blow-up time convergence of semidiscretizations of reaction-diffusion equations, Appl. Numer. Math., 26 (1998), pp.399-414.
J. Bec, K. Khanin, Burgers turbulence, Physics Reports, 447, (2007), 1-66.
J. von Below, An existence result for semilinear parabolic network equations with dynamical node conditions, Progress in partial Differential equations : elliptic and parabolic problems, Pitman Research Notes in Math. Ser. Longman Harlow Essex, 266, (1992), 274-283.
J. von Below, Parabolic network equations. Tubin0gen, 2 édition, (1994).
J. von Below, C. De Coster, A Qualitative Theory for Parabolic Problems under Dynamical Boundary Conditions, Journal of Inequalities and Applications, 5, (2000), 467-486.
J. von Below, S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, 21, (1996), 255-279.
J. von Below, G. Pincet-Mailly, Blow up for Reaction Diffusion Equations Under Dynamical Boundary Conditions, Communications in Partial Differential Equations, 28, (2003), 223-247.
J. von Below, G. Pincet-Mailly, Blow-up for some nonlinear parabolic problems with convection under dynamical boundary conditions, Discrete and Continuous Dynamical Systems, Supplement Volume,
(2007),10311041.
J. von Below, G. Pincet-Mailly, J-F. Rault, Growth order and blowup points for the parabolic Burgers’ equation under dynamical boundary conditions, Discrete contin. Dyn. Syst. Ser. S., 6(3), (2013), 825-836.
S. Chen, Global existence and blow-up of solutions for a parabolic equation with a gradient term, Proc. Amer. Math. Soc., 129, (2001), 975-981.
M. Chipot, F.B. Weissler Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20, (1989), 886-907.
M. Chlebik, M. Fila, P. Quittner, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10, (2003), 525-537.
J. Ding, Blow-up solutions for a class of nonlinear parabolic equations with Dirichlet boundary conditions, Nonlinear Anal., 52, (2003), 16451654.
J. Ding, B.-Z. Guo, Global and blow-up solutions for nonlinear Parabolic equations with a gradient term, Houston Journal of Mathematics, 37 (4), (2011), 1265-1277.
J. Ding, B.-Z. Guo, Global existence and blow-up solutions for quasilinear reaction-diffusion equations with a gradient term, Applied Mathematics Letters, 24, (2011), 936-942.
M. Fila, Remarks on blow-up for a nonlinear parabolic equation with a gradient term, Proc. Amer. Math. Soc, 111, (1991), 795-801.
U. Frisch, Turbulence: The legacy of A.N. Kolmogorov, Cambridge University Press, (1995).
O. Ladyzenskaya, V. Solonnikov, N. Uraltseva, Linear and quasilinear equations of parabolic type, Trans. of Math. Monographs, 23, (1968).
G. Pincet-Mailly, J-F. Rault, Nonlinear convection in reaction-diffusion equations under dynamical boundary
conditions, Electronic Journal of Differential Equations, 2013(10), (2013), 1-14.
J-F. Rault, Phénomène d’explosion et existence globale pour quelques problèmes paraboliques sous les conditions au bord dynamiques, PhD thesis, Université du Littoral Côte d’Opale, (2010).
J-F. Rault, A Bifurcation for a Generalized Burgers’ Equation in Dimension One, Discrete contin. Dyn. Syst. Ser. S., 5(3), (2012),683-706.
P. Souplet, Finite time blow-up for a non-linear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., 19, (1996), 1317-1333.
P. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electronic Journal of Differential Equations, 2001(20),(2001), 1-19.
P. Souplet, F. Weissler, Self-Similar Subsolutions and Blow-up for Nonlinear Parabolic equations, Journal of
Mathematical Analysis and Applications, 212, (1997), 60-74.
P. Souplet, S. Tayachi, F. B. Weissler, Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term, Indiana Univ. Math. J., 45, (1996), 655-682.
M. M. Taha, A. K. Touré and P. E. Mensah, Numerical approximation of the blow-up time for a semilinear parabolic equation with nonlinear boundary conditions, Far East journal of Mathematical Sciences (FJMS), 60(45), (2012), 125-167.
F. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation. Israel Journal of Mathematics, 38, (1981), 29-40.
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