Quenching time of some nonlinear wave equations

N’gohisse, Firmin K., and Boni, Théodore K.. "Quenching time of some nonlinear wave equations." Archivum Mathematicum 045.2 (2009): 115-124. <http://eudml.org/doc/250555>.

@article{N2009,
abstract = {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.},
author = {N’gohisse, Firmin K., Boni, Théodore K.},
journal = {Archivum Mathematicum},
keywords = {nonlinear wave equations; quenching; convergence; numerical quenching time; nonlinear wave equation; quenching; convergence; numerical quenching time},
language = {eng},
number = {2},
pages = {115-124},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Quenching time of some nonlinear wave equations},
url = {http://eudml.org/doc/250555},
volume = {045},
year = {2009},
}

TY - JOUR
AU - N’gohisse, Firmin K.
AU - Boni, Théodore K.
TI - Quenching time of some nonlinear wave equations
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 2
SP - 115
EP - 124
AB - 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.
LA - eng
KW - nonlinear wave equations; quenching; convergence; numerical quenching time; nonlinear wave equation; quenching; convergence; numerical quenching time
UR - http://eudml.org/doc/250555
ER -