Geodesic
Internal layers in the one-dimensional reaction–diffusion equation with a discontinuous reactive term
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 12, pp. 2042-2048 
N. N. Nefedov; Minkang Ni. Internal layers in the one-dimensional reaction–diffusion equation with a discontinuous reactive term. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 12, pp. 2042-2048. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_12_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A singularly perturbed boundary value problem for a second-order ordinary differential equation known in applications as a stationary reaction–diffusion equation is studied. A new class of problems is considered, namely, problems with nonlinearity having discontinuities localized in some domains, which leads to the formation of sharp transition layers in these domains. The existence of solutions with an internal transition layer is proved, and their asymptotic expansion is constructed.

[1] Butuzov V. F., Vasileva A. B., Nefedov N. N., “Asimptoticheskaya teoriya kontrastnykh struktur”, Avtomatika i telemekhan., 1997, no. 7, 4–32 | MR

[2] Vasileva A. B., Butuzov V. F., Nefedov N. N., “Kontrastnye struktury v singulyarno vozmuschennykh zadachakh”, Fundamentalnaya i prikladnaya matem., 4:3 (1998), 799–851 | MR | Zbl

[3] Vasileva A. B., Butuzov V. F., Nefedov N. N., “Singulyarno vozmuschennye zadachi s pogranichnymi i vnutrennimi sloyami”, Tr. MIAN, 268, 2010, 268–283

[4] Vasileva A. B., Butuzov V. F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973 | MR

[5] Nefedov N., “Comparison principle for reaction-diffusion-advection problems with boundary and internal layers”, Lect. Notes in Comput. Sci., 823, 2013, 62–72 | DOI | MR

[6] Butuzov V. F., Levashova N. T., Melnikova A. A., “Kontrastnaya struktura tipa stupenki v singulyarno vozmuschennoi sisteme uravnenii s razlichnymi stepenyami malogo parametra”, Zh. vychisl. matem. i matem. fiz., 52:11 (2012), 1983–2003 | MR | Zbl

[7] Nefedov N. N., Recke L., Schnieder K. R., “Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations”, J. Math. Analys. Applicat., 405 (2013), 90–103 | DOI | MR | Zbl

[8] Volkov V., Nefedov N., “Asymptotic-numerical investigation of generation and motion of fronts in phase transition models”, Lect. Notes in Comput. Sci., 8236, 2013, 524–531 | DOI | MR | Zbl

[9] Franz S., Roos H.-G., “The capriciousness of numerical methods for singular perturbations”, SIAM Review, 53 (2011), 157–173 | DOI | MR | Zbl

[10] Franz S., Kopteva N., “Green's function estimates for a singularly perturbed convection-diffusion problem”, J. Different. Equations, 252 (2012), 1521–1545 | DOI | MR | Zbl

[11] Kopteva N., “Numerical analysis of a 2d singularly perturbed semilinear reaction-diffusion problem”, Lect. Notes in Comput. Sci., 5434, 2009, 80–91 | DOI | Zbl

[12] Butuzov V. F., Nefedov N. N., “Singularly perturbed boundary value problems in case of exchange of stabilities”, J. Math. Analys. Applicat., 229 (1999), 543–562 | DOI | MR | Zbl