## Journal of Applied Mathematics

### Numerical Blow-Up Time for a Semilinear Parabolic Equation with Nonlinear Boundary Conditions

#### Abstract

We obtain some conditions under which the positive solution for semidiscretizations of the semilinear equation ${u}_{t}={u}_{xx}-a(x,t)f(u),\text{\,}\text{\,}0 \lt x \lt 1, \text{\,}\text{\,}t\in (0,T)$, with boundary conditions ${u}_{x}(0,t)=0$, ${u}_{x}(1,t)=b(t)g(u(1,t))$, blows up in a finite time and estimate its semidiscrete blow-up time. We also establish the convergence of the semidiscrete blow-up time and obtain some results about numerical blow-up rate and set. Finally, we get an analogous result taking a discrete form of the above problem and give some computational results to illustrate some points of our analysis.

#### Article information

Source
J. Appl. Math., Volume 2008 (2008), Article ID 753518, 29 pages.

Dates
First available in Project Euclid: 2 March 2010

https://projecteuclid.org/euclid.jam/1267538643

Digital Object Identifier
doi:10.1155/2008/753518

Zentralblatt MATH identifier
1179.35168

#### Citation

Assalé, Louis A.; Boni, Théodore K.; Nabongo, Diabate. Numerical Blow-Up Time for a Semilinear Parabolic Equation with Nonlinear Boundary Conditions. J. Appl. Math. 2008 (2008), Article ID 753518, 29 pages. doi:10.1155/2008/753518. https://projecteuclid.org/euclid.jam/1267538643

#### References

• T. K. Boni, Sur l'explosion et le comportement asymptotique de la solution d'une équation parabolique semi-linéaire du second ordre,'' Comptes Rendus de l'Académie des Sciences. Série I, vol. 326, no. 3, pp. 317--322, 1998.
• T. K. Boni, On blow-up and asymptotic behavior of solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions,'' Commentationes Mathematicae Universitatis Carolinae, vol. 40, no. 3, pp. 457--475, 1999.
• M. Chipot, M. Fila, and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions,'' Acta Mathematica Universitatis Comenianae, vol. 60, no. 1, pp. 35--103, 1991.
• V. A. Galaktionov and J. L. Vázquez, The problem of blow-up in nonlinear parabolic equations,'' Discrete and Continuous Dynamical Systems. Series A, vol. 8, no. 2, pp. 399--433, 2002.
• P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, Basel, Switzerland, 2007.
• A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Problems for Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, Germany, 1995.
• L. M. Abia, J. C. López-Marcos, and J. Martínez, On the blow-up time convergence of semidiscretizations of reaction-diffusion equations,'' Applied Numerical Mathematics, vol. 26, no. 4, pp. 399--414, 1998.
• T. Nakagawa, Blowing up of a finite difference solution to ${u}_{t}={u}_{xx}+{u}^{2}$,'' Applied Mathematics and Optimization, vol. 2, no. 4, pp. 337--350, 1975.
• T. K. Boni, Extinction for discretizations of some semilinear parabolic equations,'' Comptes Rendus de l'Académie des Sciences. Série I, vol. 333, no. 8, pp. 795--800, 2001.
• G. Acosta, J. Fernández Bonder, P. Groisman, and J. D. Rossi, Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions,'' M2AN, vol. 36, no. 1, pp. 55--68, 2002.
• G. Acosta, J. Fernández Bonder, P. Groisman, and J. D. Rossi, Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions,'' Discrete and Continuous Dynamical Systems. Series B, vol. 2, no. 2, pp. 279--294, 2002.
• C. Brändle, P. Groisman, and J. D. Rossi, Fully discrete adaptive methods for a blow-up problem,'' Mathematical Models & Methods in Applied Sciences, vol. 14, no. 10, pp. 1425--1450, 2004.
• C. Brändle, F. Quirós, and J. D. Rossi, An adaptive numerical method to handle blow-up in a parabolic system,'' Numerische Mathematik, vol. 102, no. 1, pp. 39--59, 2005.
• R. G. Duran, J. I. Etcheverry, and J. D. Rossi, Numerical approximation of a parabolic problem with a nonlinear boundary condition,'' Discrete and Continuous Dynamical Systems, vol. 4, no. 3, pp. 497--506, 1998.
• J. Fernández Bonder and J. D. Rossi, Blow-up vs. spurious steady solutions,'' Proceedings of the American Mathematical Society, vol. 129, no. 1, pp. 139--144, 2001.
• A. de Pablo, M. Pérez-Llanos, and R. Ferreira, Numerical blow-up for the $p$-Laplacian equation with a nonlinear source,'' in Proceedings of the 11th International Conference on Differential Equations (Equadiff '05), pp. 363--367, Bratislava, Slovakia, July 2005.
• M. N. Le Roux, Semidiscretization in time of nonlinear parabolic equations with blowup of the solution,'' SIAM Journal on Numerical Analysis, vol. 31, no. 1, pp. 170--195, 1994.
• M. N. Le Roux, Semi-discretization in time of a fast diffusion equation,'' Journal of Mathematical Analysis and Applications, vol. 137, no. 2, pp. 354--370, 1989.
• D. Nabongo and T. K. Boni, Numerical blow-up and asymptotic behavior for a semilinear parabolic equation with a nonlilnear boundary condition,'' Albanian Journal of Mathematics, vol. 2, no. 2, pp. 111--124, 2008.
• D. Nabongo and T. K. Boni, Numerical blow-up solutions of localized semilinear parabolic equations,'' Applied Mathematical Sciences, vol. 2, no. 21--24, pp. 1145--1160, 2008.
• F. K. N'gohissé and T. K. Boni, Numerical blow-up solution for some semilinear heat equation,'' Electronic Transactions on Numerical Analysis, vol. 30, pp. 247--257, 2008.