| Title:
|
Quenching time of some nonlinear wave equations (English) |
| Author:
|
N’gohisse, Firmin K. |
| Author:
|
Boni, Théodore K. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
45 |
| Issue:
|
2 |
| Year:
|
2009 |
| Pages:
|
115-124 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we consider the following initial-boundary value problem
\[ {\left\rbrace \begin{array}{ll} u_{tt}(x,t)=\varepsilon Lu(x,t)+f\big (u(x,t)\big )\quad \mbox {in}\quad \Omega \times (0,T)\,,\\ u(x,t)=0 \quad \mbox {on}\quad \partial \Omega \times (0,T)\,, \\ u(x,0)=0 \quad \mbox {in}\quad \Omega \,, \\ u_t(x,0)=0 \quad \mbox {in}\quad \Omega \,, \end{array}\right.}\]
where $\Omega $ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega $, $L$ is an elliptic operator, $\varepsilon $ is a positive parameter, $f(s)$ is a positive, increasing, convex function for $s\in (-\infty ,b)$, $\lim _{s\rightarrow b}f(s)=\infty $ and $\int _0^b\frac{ds}{f(s)}<\infty $ with $b=\operatorname{const}>0$. Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation
\[ {\left\rbrace \begin{array}{ll} \alpha ^{\prime \prime }(t)=f(\alpha (t))\,,&\quad t>0\,, \\ \alpha (0)=0\,,\quad \alpha ^{\prime }(0)=0\,, \end{array}\right.}\]
as $\varepsilon $ goes to zero. We also show that the above result remains valid if the domain $\Omega $ is large enough and its size is taken as parameter. Finally, we give some numerical results to illustrate our analysis. (English) |
| Keyword:
|
nonlinear wave equations |
| Keyword:
|
quenching |
| Keyword:
|
convergence |
| Keyword:
|
numerical quenching time |
| MSC:
|
35B40 |
| MSC:
|
35B50 |
| MSC:
|
35L20 |
| MSC:
|
35L70 |
| MSC:
|
65M06 |
| idZBL:
|
Zbl 1212.35016 |
| idMR:
|
MR2591668 |
| . |
| Date available:
|
2009-06-25T18:16:41Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128294 |
| . |
| Reference:
|
[1] Abia, L. M., López-Marcos, J. C., Martínez, 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] Boni, T. K.: On quenching of solution for some semilinear parabolic equation of second order.Bull. Belg. Math. Soc. 5 (2000), 73–95. MR 1741748 |
| Reference:
|
[3] Boni, T. K.: Extinction for discretizations of some semilinear parabolic equations.C. R. Acad. Sci. Paris Sér. I Math. 333 (8) (2001), 795–800. Zbl 0999.35004, MR 1868956, 10.1016/S0764-4442(01)02078-X |
| Reference:
|
[4] Chang, H., Levine, H. A.: The quenching of solutions of semilinear hyperbolic equations.SIAM J. Math. Anal. 12 (1982), 893–903. MR 0635242, 10.1137/0512075 |
| Reference:
|
[5] Friedman, A., Lacey, A. A.: The blow-up time for solutions of nonlinear heat equations with small diffusion.SIAM J. Math. Anal. 18 (1987), 711–721. Zbl 0643.35013, MR 0883563, 10.1137/0518054 |
| Reference:
|
[6] Glassey, R. T.: Blow-up theorems for nonlinear wave equations.Math. Z. 132 (1973), 183–203. Zbl 0247.35083, MR 0340799, 10.1007/BF01213863 |
| Reference:
|
[7] Kaplan, S.: On the growth of solutions of quasi-linear parabolic equations.Comm. Pure Appl. Math. 16 (1963), 305–330. Zbl 0156.33503, MR 0160044, 10.1002/cpa.3160160307 |
| Reference:
|
[8] Keller, J. B.: On solutions of nonlinear wave equations.Comm. Pure Appl. Math. 10 (1957), 523–530. Zbl 0090.31802, MR 0096889, 10.1002/cpa.3160100404 |
| Reference:
|
[9] Levine, H. A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form $\rho u_{tt}=-Av+F(u)$.Trans. Amer. Math. Soc. 192 (1974), 1–21. MR 0344697 |
| Reference:
|
[10] Levine, H. A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations.SIAM J. Math. Anal. 5 (1974), 138–146. Zbl 0243.35069, MR 0399682, 10.1137/0505015 |
| Reference:
|
[11] Levine, H. A.: The quenching solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions.SIAM J. Math. Anal. 14 (1983), 1139–1153. MR 0718814, 10.1137/0514088 |
| Reference:
|
[12] Levine, H. A.: The phenomenon of quenching: A survey.Proc. VIth International Conference on Trends in the Theory and Practice of Nonlinear Analysis (Lakshmitanthan, V., ed.), Elservier Nork-Holland, New York, 1985. Zbl 0581.35037, MR 0817500 |
| Reference:
|
[13] Levine, H. A., Smiley, M. W.: The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions.J. Math. Anal. Appl. 103 (1984), 409–427. MR 0718814 |
| Reference:
|
[14] Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations.Prentice Hall, Englewood Cliffs, NJ, 1967. MR 0219861 |
| Reference:
|
[15] Rammaha, M. A.: On the quenching of solutions of the wave equation with a nonlinear boundary condition.J. Reine Angew. Math. 407 (1990), 1–18. Zbl 0698.35088, MR 1048525 |
| Reference:
|
[16] Reed, M.: Abstract nonlinear wave equations.Lecture Notes in Math., vol. 507, Springer-Verlag Berlin, New-York, 1976. Zbl 0319.35060, MR 0605679 |
| Reference:
|
[17] Sattinger, D. H.: On global solution of nonlinear hyperbolic equations.Arch. Rational Mech. Anal. 30 (1968), 148–172. Zbl 0159.39102, MR 0227616, 10.1007/BF00250942 |
| Reference:
|
[18] Smith, R. A.: On a hyperbolic quenching problem in several dimensions.SIAM J. Math. Anal. 20 (1989), 1081–1094. Zbl 0687.35056, MR 1009347, 10.1137/0520072 |
| Reference:
|
[19] Walter, W.: Differential-und Integral-Ungleichungen.Springer, Berlin, 1954. |
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