zbMATH — the first resource for mathematics

Moving fronts in integro-parabolic reaction-advection-diffusion equations. (English. Russian original) Zbl 1269.45008
Differ. Equ. 47, No. 9, 1318-1332 (2011); translation from Differ. Uravn. 47, No. 9, 1305-1319 (2011).
Summary: We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reaction-advection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method.

45K05 Integro-partial differential equations
45G10 Other nonlinear integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
Full Text: DOI
[1] Butuzov, V.F., Vasil’eva, A.B., and Nefedov, N.N., Contrast Structures in Singularly Perturbed Problems, Fund. Prikl. Mat., 1998, vol. 4, no. 3, pp. 799–851. · Zbl 0963.34043
[2] Pao, C.V., Nonlinear Parabolic and Elliptic Equations, New York, 1992. · Zbl 0777.35001
[3] Bates Peter W. and Zhao Guangyu, Existence, Uniqueness and Stability of the Stationary Solution to a Nonlocal Evolution Equation Arising in Population Dispersal, J. Math. Anal. Appl., 2007, vol. 332, no. 1, pp. 428–440. · Zbl 1114.35017
[4] Bates Peter W. and Chen Fengxin, Spectral Analysis of Traveling Waves for Nonlocal Evolution Equations, SIAM J. Math. Anal., 2006, vol. 38, no. 1, pp. 116–126. · Zbl 1134.35060
[5] Nefedov, N.N. and Nikitin, A.G., Development of the Asymptotic Method of Differential Inequalities for Step-Type Solutions in Singularly Perturbed Integro-Differential Equations, Zh. Vychisl. Mat. Mat. Fiz., 2001, vol. 41, no. 7, pp. 1057–1066. · Zbl 1029.45008
[6] Nefedov, N.N., Radziunas, M., Schneider, K.R., and Vasil’eva, A.B., Change of the Type of Contrast Structures in Parabolic Neumann Problems,Zh. Vychisl. Mat. Mat. Fiz., 2005, vol. 45, no. 1, pp. 41–55. · Zbl 1114.35094
[7] Fife, P.C. and Hsiao, L., The Generation and Propagation of Internal Layers, Nonlinear Anal., 1998, vol. 12, no. 1, pp. 19–41. · Zbl 0685.35055
[8] Nefedov, N.N. and Nikitin, A.G., The Asymptotic Method of Differential Inequalities for Singularly Perturbed Integrodifferential Equations, Differ. Uravn., 2000, vol. 36, no. 10, pp. 1398–1404.
[9] Nefedov, N.N., The Method of Differential Inequalities for Some Classes of Nonlinear Singularly Perturbed Problems with Internal Layers, Differ. Uravn., 1995, vol. 31, no. 7, pp. 1142–1149. · Zbl 0864.35011
[10] Nefedov, N.N., Omel’chenko, O.E., and Recke, L., Stationary Internal Layers in Integrodifferential System Reaction-Advection-Diffusion, Zh. Vychisl. Mat. Mat. Fiz., 2006, vol. 46, no. 4, pp. 623–645.
[11] Amann, H., Periodic Solutions of Semilinear Parabolic Equations, in Nonlinear Analysis: a Collection of Papers in Honor of Erich Rothe, New York, 1978, pp. 1–29.
[12] Sattinger, D.H., Monotone Methods in Elliptic and Parabolic Boundary Value Problems, Indiana Univ. Math. J., 1972, vol. 21, no. 11, pp. 979–1001. · Zbl 0223.35038
[13] Fife, P.C. and Tang, M., Comparison Principles for Reaction-Diffusion Systems: Irregular Comparison Functions and Applications to Question of Stability and Speed Propagation of Disturbances, J. Differential Equations, 1981, vol. 40, pp. 168–185. · Zbl 0431.35008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.