Geodesic
A note on the domain of attraction for the stationary solution to a singularly perturbed parabolic equation
Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 3, pp. 353-358.

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

We consider a boundary-value problem for a singularly perturbed parabolic equation with an initial function independent of a perturbation parameter in the case where a degenerate stationary equation has smooth possibly intersecting roots. Before, the existence of a stable stationary solution to this problem was proved and the domain of attraction of this solution was investigated — due to exchange of stabilities, the stationary solution approaches the non-smooth (but continuous) composite root of the degenerate equation as the perturbation parameter gets smaller, and its domain of attraction contains all initial functions situated strictly on one side of the other non-smooth (but continuous) composite root of the degenerate equation. We show that if the initial function is out of the boundary of this family of initial functions near some point, the problem cannot have a solution inside the domain of the problem, i.e. this boundary is the true boundary of the attraction domain. The proof uses ideas of the nonlinear capacity method.
Keywords: small parameter, stationary solution, intersecting roots, exchange of stabilities, nonexistence, blow-up, nonlinear capacity.
Mots-clés : singular perturbation, parabolic equation, domain of attraction 
@article{MAIS_2017_24_3_a7,
     author = {M. A. Terentyev},
     title = {A note on the domain of attraction for the stationary solution to a singularly perturbed parabolic equation},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {353--358},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2017_24_3_a7/}
}
TY  - JOUR
AU  - M. A. Terentyev
TI  - A note on the domain of attraction for the stationary solution to a singularly perturbed parabolic equation
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2017
SP  - 353
EP  - 358
VL  - 24
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2017_24_3_a7/
LA  - ru
ID  - MAIS_2017_24_3_a7
ER  - 
%0 Journal Article
%A M. A. Terentyev
%T A note on the domain of attraction for the stationary solution to a singularly perturbed parabolic equation
%J Modelirovanie i analiz informacionnyh sistem
%D 2017
%P 353-358
%V 24
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2017_24_3_a7/
%G ru
%F MAIS_2017_24_3_a7
M. A. Terentyev. A note on the domain of attraction for the stationary solution to a singularly perturbed parabolic equation. Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 3, pp. 353-358. http://geodesic.mathdoc.fr/item/MAIS_2017_24_3_a7/

[1] Butuzov V. F., “On the stability and domain of attraction of asymptotically nonsmooth stationary solutions to a singularly perturbed parabolic equation”, Comp. Math. Math. Phys., 46:3 (2006), 413–424 | DOI | MR | Zbl

[2] Butuzov V. F., Nefedov N. N., “A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability”, Math. Notes, 63:3 (1998), 311–318 | DOI | DOI | MR | Zbl

[3] Karali G., Sourdis C., “Radial and bifurcating non-radial solutions for a singular perturbation problem in the case of exchange of stabilities”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 29:2 (2012), 131–170 | DOI | MR | Zbl

[4] Mitidieri E., Pokhozhaev S. I., “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234 (2001), 1–362 | MR