Geodesic
On the stability and domain of attraction of a stationary nonsmooth limit solution of a singularly perturbed parabolic equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 3, pp. 433-444 
V. F. Butuzov. On the stability and domain of attraction of a stationary nonsmooth limit solution of a singularly perturbed parabolic equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 3, pp. 433-444. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_3_a7/
@article{ZVMMF_2006_46_3_a7,
     author = {V. F. Butuzov},
     title = {On the stability and domain of attraction of a~stationary nonsmooth limit solution of a~singularly perturbed parabolic equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {433--444},
     year = {2006},
     volume = {46},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_3_a7/}
}
TY  - JOUR
AU  - V. F. Butuzov
TI  - On the stability and domain of attraction of a stationary nonsmooth limit solution of a singularly perturbed parabolic equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2006
SP  - 433
EP  - 444
VL  - 46
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_3_a7/
LA  - ru
ID  - ZVMMF_2006_46_3_a7
ER  - 
%0 Journal Article
%A V. F. Butuzov
%T On the stability and domain of attraction of a stationary nonsmooth limit solution of a singularly perturbed parabolic equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2006
%P 433-444
%V 46
%N 3
%U http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_3_a7/
%G ru
%F ZVMMF_2006_46_3_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

A stationary solution to the singularly perturbed parabolic equation $-u_t+\varepsilon^2u_{xx}-f(u,x)=0$ with Neumann boundary conditions is considered. The limit of the solution as $\varepsilon\to0$ is a nonsmooth solution to the reduced equation $f(u,x)=0$ that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found.

[1] Butuzov V. F., Nefedov H. H., “Singulyarno vozmuschennaya kraevaya zadacha dlya uravneniya vtorogo poryadka v sluchae smeny ustoichivosti”, Matem. zametki, 63:3 (1998), 354–362 | MR | Zbl

[2] Gudkov V. V., Klokov Yu. A., Lepin A. Ya., Ponomarev V. D., Dvukhtochechnye kraevye zadachi dlya obyknovennykh differentsialnykh uravnenii, Zinatne, Riga, 1973 | MR | Zbl

[3] Butuzov V. F., Smurov I., “Initial boundary value problem for a singularly perturbed parabolic equation in case of exchange of stability”, J. Math. Analys. and Appl., 234 (1999), 183–192 | DOI | MR | Zbl

[4] Butuzov V. F., Gromova E. A., “O kraevoi zadache dlya sistemy bystrogo i medlennogo uravnenii vtorogo poryadka v sluchae peresecheniya kornei vyrozhdennogo uravneniya”, Zh. vychisl. matem. i matem. fiz., 41:8 (2001), 1165–1179 | MR | Zbl

[5] Butuzov V. F., Nefedov H. H., Shnaider K. P., “Singulyarno vozmuschennye zadachi v sluchae smeny ustoichivosti”, Differents. ur-niya. Singulyarnye vozmuscheniya, Itogi nauki i tekhn. Ser. Sovrem. matem. i ee prilozh. Tematich. obzory., 109, VINITI, M., 2002

[6] Pao C. V., Nonlinear parabolic and elliptic equations, Plenum Press, New York, London, 1992 | MR