Geodesic
Stability in terms of two measures for a class of semilinear impulsive parabolic equations
Sbornik. Mathematics, Tome 204 (2013) no. 4, pp. 485-507 Cet article a éte moissonné depuis la source Math-Net.Ru

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The problem of stability in terms of two measures is considered for semilinear impulsive parabolic equations. A new version of the comparison method is proposed, and sufficient conditions for stability in terms of two measures are obtained on this basis. An example of a hybrid impulsive system formed by a system of ordinary differential equations coupled with a partial differential equation of parabolic type is given. The efficiency of the described approaches is demonstrated. Bibliography: 24 titles.
Keywords: stability in a cone, impulsive effect, semilinear parabolic equation, semiordering, comparison principle. 
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A. I. Dvirnyj; V. I. Slyn'ko. Stability in terms of two measures for a class of semilinear impulsive parabolic equations. Sbornik. Mathematics, Tome 204 (2013) no. 4, pp. 485-507. http://geodesic.mathdoc.fr/item/SM_2013_204_4_a1/

[1] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995 | MR | Zbl

[2] M. O. Perestyuk, O. S. Chernikova, “Some modern aspects of the theory of impulsive differential equations”, Ukrainian Math. J., 60:1 (2008), 91–107 | DOI | MR | Zbl

[3] A. I. Dvirnyi, V. I. Slyn'ko, “Stability of solutions to impulsive differential equations in critical cases”, Siberian Math. J., 52:1 (2011), 54–62 | DOI | MR | Zbl

[4] A. I. Dvirnyi, V. I. Slynko, “Ob ustoichivosti reshenii nestatsionarnykh sistem differentsialnykh uravnenii s impulsnym vozdeistviem v odnom kriticheskom sluchae”, Nelineinye kolebaniya, 14:4 (2011), 445–467 | MR

[5] X. Liu, “Progress in stability of impulsive systems with applications to population growth models”, Advances in stability theory at the end of the 20th century, Stability Control Theory Methods Appl., 13, Taylor Francis, London, 2003, 321–338 | MR | Zbl

[6] N. A. Perestyuk, “K voprosu ustoichivosti polozheniya ravnovesiya impulsnykh sistem”, Godišnik Visš. Učebn. Zaved. Priložna Mat., 11:1 (1976), 145–151 | MR | Zbl

[7] A. O. Ignat'ev, O. A. Ignat'ev, A. A. Soliman, “Asymptotic stability and instability of the solutions of systems with impulse action”, Math. Notes, 80:3–4 (2006), 491–499 | DOI | DOI | MR | Zbl

[8] A. O. Ignat'ev, O. A. Ignat'ev, “Stability of solutions of systems with impulse effect”, Progress in nonlinear analysis, Nova Sci. Publ., New York, 2009, 363–389 | MR | Zbl

[9] X. Liu, D. Williams, “Asymptotic stability and instability of the solutions of systems with impulse action”, Canad. Appl. Math. Quart., 3:4 (1985), 320–340

[10] S. K. Kaul, X. Liu, “Generalized variation of parameters and stability of impulsive systems”, Nonlinear Anal., 40:1–8 (2000), 295–307 | DOI | MR | Zbl

[11] X. Liu, G. Ballinger, “Boundedness for impulsive delay differential equations and applications to population growth models”, Nonlinear Anal., 53:7–8 (2003), 1041–1062 | DOI | MR | Zbl

[12] X. Fu, X. Liu, S. Sivaloganathan, “Oscillation criteria for impulsive parabolic systems”, Appl. Anal., 79:1–2 (2001), 239–255 | DOI | MR | Zbl

[13] O. O. Struk, V. I. Tkachenko, “On impulse Lotka–Volterra systems with diffusion”, Ukrainian Math. J., 54:4 (2002), 629–646 | DOI | MR | Zbl

[14] M. Muslim, R. P. Agarwal, “Approximation of solutions to impulsive functional differential equations”, J. Appl. Math. Comput., 34:1–2 (2010), 101–112 | DOI | MR | Zbl

[15] L. H. Erbe, H. I. Freedman, X. Z. Liu, J. H. Wu, “Comparison principles for impulsive parabolic equations with applications to models of single species growth”, J. Austral. Math. Soc. Ser. B, 32:4 (1991), 382–400 | DOI | MR | Zbl

[16] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin–New York, 1981 | DOI | MR | MR | Zbl | Zbl

[17] A. A. Martynyuk, A. Yu. Obolenskii, Issledovanie ustoichivosti avtonomnykh sistem sravneniya, Preprint 78.28, Institut matematiki AN USSR, Kiev, 1976

[18] A. Yu. Obolenskii, Kriterii ustoichivosti dvizheniya nekotorykh nelineinykh sistem, Feniks, K., 2010

[19] A. I. Dvirny, V. I. Slyn'ko, “The global stability conditions for solutions of nonstationary differential equations with instant impulsive effects in pseudolinear form”, J. Math. Sci. (N. Y.), 179:2 (2011), 245–260 | DOI | MR

[20] M. A. Krasnoselskii, E. A. Lifshits, A. V. Sobolev, Pozitivnye lineinye sistemy, Nauka, M., 1985 | MR | Zbl

[21] V. M. Matrosov, L. Yu. Anapolskii, S. N. Vasilev, Metod sravneniya v matematicheskoi teorii sistem, Nauka, Novosibirsk, 1980 | Zbl

[22] S. G. Kreǐn, Linear differential equations in Banach space, Transl. Math. Monogr., 29, Providence, RI, Amer. Math. Soc., 1971 | MR | MR | Zbl | Zbl

[23] T. K. Sirazetdinov, Ustoichivost sistem s raspredelennymi parametrami, Nauka, Novosibirsk, 1987 | MR | Zbl

[24] A. A. Movchan, “The direct method of Liapunov in stability problems of elastic systems”, J. Appl. Math. Mech., 23:3 (1959), 686–700 | DOI | MR | Zbl