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Semilinear Parabolic Equations (semilinear + parabolic_equation)

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Mathematics and Statistics100%

Selected Abstracts

On the numerical computation of blowing-up solutions for semilinear parabolic equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001
D. Fayyad
Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end, with a finite unknown ,blow-up' time Tb have been studied in a previous work. Specifically, for , a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ,mass control' property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve-\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon-u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end, \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow-up time Tb and the blow-up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end, with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end. Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end, we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end, defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end. In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end, the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end, verifies, $\|(u-u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1-\alpha}$\nopagenumbers\end, \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end. The two special cases of a single-point blow-up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p-1})$\nopagenumbers\end. The estimate $|T_\eta-T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Asymptotic behaviour of solutions of quasilinear evolutionary partial differential equations of parabolic type on unbounded spatial intervals

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2006
Poul
Abstract We study the long-time behaviour of solutions to a quasilinear parabolic problem on a half-line. The main result lies in showing the existence of a positive solution that converges to the travelling wave of solution to the stationary problem on the whole line. The main tools used here are the zero number theory and the concentration compactness principle. This result is a generalization of a result know for semilinear parabolic equations. Copyright © 2006 John Wiley & Sons, Ltd. [source]

On the numerical computation of blowing-up solutions for semilinear parabolic equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001
D. Fayyad
Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end, with a finite unknown ,blow-up' time Tb have been studied in a previous work. Specifically, for , a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ,mass control' property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve-\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon-u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end, \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow-up time Tb and the blow-up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end, with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end. Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end, we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end, defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end. In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end, the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end, verifies, $\|(u-u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1-\alpha}$\nopagenumbers\end, \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end. The two special cases of a single-point blow-up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p-1})$\nopagenumbers\end. The estimate $|T_\eta-T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd. [source]

The probability approach to numerical solution of nonlinear parabolic equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2002
G. N. Milstein
Abstract A number of new layer methods for solving semilinear parabolic equations and reaction-diffusion systems is derived by using probabilistic representations of their solutions. These methods exploit the ideas of weak sense numerical integration of stochastic differential equations. In spite of the probabilistic nature these methods are nevertheless deterministic. A convergence theorem is proved. Some numerical tests are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 490,522, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10020 [source]