×

zbMATH — the first resource for mathematics

Asymptotic behavior of the solution of the first initial-boundary value problem for equations of the Sobolev type from the viewpoint of oscillations. (English. Russian original) Zbl 1244.35009
Differ. Equ. 48, No. 2, 202-213 (2012); translation from Differ. Uravn. 48, No. 2, 196-206 (2012).
Summary: We study the large-time behavior of solutions of the first initial-boundary value problem for partial differential equations of the Sobolev type. We find conditions under which the derivatives of solutions are either oscillating or stabilize to zero.
MSC:
35B40 Asymptotic behavior of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Sobolev, S.L., Izbrannye trudy. T. 1. Uravneniya matematicheskoi fiziki. Vychislitel’naya matematika i kubaturnye formuly (Selected Works. Vol. 1. Equations of Mathematical Physics. Computational Mathematics and Cubature Formulas), Novosibirsk, 2003.
[2] Aleksandryan, R.A., Spectral Properties of Operators Generated by Systems of Differential Equations of the S.L. Sobolev Type, Tr. Mosk. Mat. Obs., 1960, no. 9, pp. 455–505.
[3] Aleksandryan, R.A., Berezanskii, Yu.M., Il’in, V.A., and Kostyuchenko, A.G., Some Problems of Spectral Theory for Partial Differential Equations, in Differents. uravneniya s chastnymi proizvodnymi: Tr. simp. posv. 60-letiyu S.L. Soboleva (Part. Diff. Eqs. Proc. Symp. Devoted to the 60th Birthday of S.L. Sobolev), Moscow, 1970, pp. 3–35.
[4] Dezin, A.A. and Maslennikova, V.N., Nonclassical Boundary Value Problems, in Differents. uravneniya s chastnymi proizvodnymi: Tr. simp. posv. 60-letiyu S.L. Soboleva (Part. Diff. Eqs. Proc. Symp. Devoted to the 60th Birthday of S.L. Sobolev), Moscow, 1970, pp. 81–95.
[5] Zelenyak, T.I. and Mikhailov, V.P., Asymptotic Behavior of Solutions of Some Boundary Value Problems of Mathematical Physics as t, in Differents. uravneniya s chastnymi proizvodnymi: Tr. simp. posv. 60-letiyu S.L. Soboleva (Part. Diff. Eqs. Proc. Symp. Devoted to the 60th Birthday of S.L. Sobolev), Moscow, 1970, pp. 96–118.
[6] Uspenskii, S.V. and Vasil’eva, E.N., Investigation at Infinity of Sobolev-Wiener Classes of Functions in Tube Domains, Tr. Mat. Inst. Steklova, 2001, vol. 231, pp. 327–335.
[7] Uspenskii, S.V. and Vasil’eva, E.N., Teoremy vlozheniya dlya sobolevskikh funktsional’nykh prostranstv. Prilozheniya k differentsial’nym uravneniyam (Embedding Theorems for Sobolev Function Spaces. Applications to Differential Equations), Moscow, 2006.
[8] Uspenskii, S.V., Vasil’eva, E.N., and Yanov, S.I., Embedding Theorems for Generalized Sobolev-Nikol’skii-Wiener Spaces and Applications to Differential Equations, Dokl. Akad. Nauk, 2000, vol. 373, no. 5, pp. 593–596.
[9] Demidenko, G.V. and Uspenskii, S.V., Uravneniya i sistemy, ne razreshennye otnositel’no starshei proizvodnoi (Equations and Systems Unsolved for the Highest Derivative), Novosibirsk: Nauchnaya Kniga, 1998.
[10] Sobolev, S.L., Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike (Some Applications of Functional Analysis in Mathematical Physics), Moscow: Nauka, 1988. · Zbl 0662.46001
[11] Denisova, T.E., Behavior of a Solution of a Mixed Sobolev Equation in the Course of Time, Vestn. Ross. Univ. Druzhby Nar., 2003, no. 1 (10), pp. 47–60.
[12] Lavrent’ev, M.A. and Shabat, B.V., Metody teorii funktsii kompleksnogo peremennogo (Methods of the Theory of Functions of a Complex Variable), Moscow: Nauka, 1987.
[13] Gabov, S.A. and Sveshnikov, A.G., Zadachi dinamiki stratifitsirovannykh zhidkostei (Problems of Dynamics of Stratified Fluids), Moscow: Nauka, 1986.
[14] Dubinskii, Yu.A., Zadacha Koshi v kompleksnoi oblasti (Cauchy Problem in Complex Domain), Moscow, 1996.
[15] Uspenskii, S.V. and Demidenko, G.V., On Mixed Boundary Value Problems for a Class of Equations Unsolved with Respect to the Highest Derivative, in Differents. uravneniya s chastnymi proizvodnymi: Tr. sem. akad. S.L. Soboleva (Partial Differential Equations. Proc. S.L. Sobolev Sem.), Novosibirsk: Inst. Mat., 1980, no. 2, pp. 92–115.
[16] Nafikov, Sh.G., Estimates for Solutions of the First Boundary Value Problem for Equations of Sobolev Type, in Teoremy vlozheniya i ikh prilozheniya k zadacham matematicheskoi fiziki. Tr. sem. akad. S.L. Soboleva (Embedding Theorems and Their Applications to Problems of Mathematical Physics. Proc. S.L. Sobolev Sem.), Novosibirsk: Inst. Mat., 1983, no. 1, pp. 90–107.
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.