| Title:
|
Quenching for semidiscretizations of a semilinear heat equation with Dirichlet and Neumann boundary conditions (English) |
| Author:
|
Nabongo, Diabate |
| Author:
|
Boni, Théodore K. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
3 |
| Year:
|
2008 |
| Pages:
|
463-475 |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper concerns the study of the numerical approximation for the following boundary value problem: $$ \cases u_t(x,t)-u_{xx}(x,t) = -u^{-p}(x,t), & 0<x<1, t>0, \ u_{x}(0,t)=0, & u(1,t)=1, t>0, \ u(x,0)=u_{0}(x)>0, & 0\leq x \leq 1, \endcases $$ where $p>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis. (English) |
| Keyword:
|
semidiscretizations |
| Keyword:
|
discretizations |
| Keyword:
|
heat equations |
| Keyword:
|
quenching |
| Keyword:
|
semidiscrete quenching time |
| Keyword:
|
convergence |
| MSC:
|
35B40 |
| MSC:
|
35K20 |
| MSC:
|
35K55 |
| MSC:
|
35K91 |
| MSC:
|
65M06 |
| idZBL:
|
Zbl 1212.35217 |
| idMR:
|
MR2490440 |
| . |
| Date available:
|
2009-05-05T17:12:24Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119736 |
| . |
| Reference:
|
[1] Abia L.M., López-Marcos J.C., Martinez J.: On the blow-up time convergence of semidiscretizations of reaction-diffusion equations.Appl. Numer. Math. 26 (1998), 399-414. MR 1612360, 10.1016/S0168-9274(97)00105-0 |
| Reference:
|
[2] Acker A., Kawohl B.: Remarks on quenching.Nonlinear Anal. 13 (1989), 53-61. Zbl 0676.35021, MR 0973368, 10.1016/0362-546X(89)90034-5 |
| Reference:
|
[3] Boni T.K.: Extinction for discretizations of some semilinear parabolic equations.C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), 795-800. Zbl 0999.35004, MR 1868956, 10.1016/S0764-4442(01)02078-X |
| Reference:
|
[4] Boni T.K.: On quenching of solutions for some semilinear parabolic equations of second order.Bull. Belg. Math. Soc. Simon Stevin 7 (2000), 73-95. Zbl 0969.35077, MR 1741748 |
| Reference:
|
[5] Fila M., Kawohl B., Levine H.A.: Quenching for quasilinear equations.Comm. Partial Differential Equations 17 (1992), 593-614. Zbl 0801.35057, MR 1163438 |
| Reference:
|
[6] Guo J.S., Hu B.: The profile near quenching time for the solution of a singular semilinear heat equation.Proc. Edinburgh Math. Soc. 40 (1997), 437-456. Zbl 0903.35007, MR 1475908 |
| Reference:
|
[7] Guo J.: On a quenching problem with Robin boundary condition.Nonlinear Anal. 17 (1991), 803-809. MR 1131490, 10.1016/0362-546X(91)90154-S |
| Reference:
|
[8] Levine H.A.: Quenching, nonquenching and beyond quenching for solutions of some parabolic equations.Annali Mat. Pura Appl. 155 (1990), 243-260. MR 1042837 |
| . |
