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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 4(364), Pages 145–212 (Mi umn1448)  

This article is cited in 29 scientific papers (total in 29 papers)

On the behaviour of solutions of parabolic equations for large values of time

V. N. Denisov

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: This paper is a survey of classical and new results on stabilization of solutions of the Cauchy problem and mixed problems for second-order linear parabolic equations. Proofs are given for some new results about exact sufficient conditions on the behaviour of lower-order coefficients of the parabolic equation; these conditions ensure stabilization of a solution of the Cauchy problem for the parabolic equation in the class of bounded or increasing initial functions.

DOI: https://doi.org/10.4213/rm1448

Full text: PDF file (615 kB)
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English version:
Russian Mathematical Surveys, 2005, 60:4, 721–790

Bibliographic databases:

UDC: 517.95
MSC: Primary 35K15, 35K20, 35B40; Secondary 35K05, 35B35

Citation: V. N. Denisov, “On the behaviour of solutions of parabolic equations for large values of time”, Uspekhi Mat. Nauk, 60:4(364) (2005), 145–212; Russian Math. Surveys, 60:4 (2005), 721–790

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. N. Denisov, “Necessary and sufficient conditions for the stabilization of the solution to the Dirichlet problem for the heat equation”, Dokl. Math., 73:2 (2006), 193–196  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. V. V. Zhikov, “Nash-Aronson estimates for solutions to some parabolic equations: Application to asymptotic diffusion problems”, Dokl. Math., 75:2 (2007), 247–251  mathnet  crossref  mathscinet  mathscinet  zmath  zmath  isi  elib  elib
    3. V. N. Denisov, “Stabilization of Solution to the Cauchy Problem for a Parabolic Equation with Lower Order Coefficients and an Exponentially Growing Initial Function”, Proc. Steklov Inst. Math., 261 (2008), 94–97  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. V. I. Bogachev, N. V. Krylov, M. Röckner, “Elliptic and parabolic equations for measures”, Russian Math. Surveys, 64:6 (2009), 973–1078  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    5. V. N. Denisov, “Stabilization of a solution to the Cauchy problem for a nondivergence parabolic equation with growing lower order coefficients”, Proc. Steklov Inst. Math., 270 (2010), 91–103  mathnet  crossref  mathscinet  zmath  isi  elib
    6. V. N. Denisov, “Sufficient conditions for stabilization of solutions of the Cauchy problem for nondivergent parabolic equations with lower-order coefficients”, Journal of Mathematical Sciences, 171:1 (2010), 46–57  mathnet  crossref  mathscinet
    7. Denisov V.N., “Necessary and sufficient stabilization conditions for the solution of the Cauchy problem for a parabolic equation with nonzero lower order coefficients”, Dokl. Math., 82:1 (2010), 578–580  crossref  mathscinet  mathscinet  zmath  isi  elib  elib
    8. Denisov V.N., “Stabilization of the solution to the Cauchy problem for a parabolic equation with nonzero lower order coefficients in classes of increasing initial functions”, Dokl. Math., 81:1 (2010), 91–93  crossref  mathscinet  zmath  isi  elib  elib
    9. V. N. Razzhevaikin, “Properties of spatial structures for reaction–nonlinear diffusion equations subject to Dirichlet conditions”, Comput. Math. Math. Phys., 51:10 (2011), 1729–1737  mathnet  crossref  mathscinet  isi
    10. S. V. Shaposhnikov, “Regular and qualitative properties of solutions for parabolic equations for measures”, Theory Probab. Appl., 56:2 (2011), 252–279  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    11. O. M. Botsenyuk, “On estimates of time decay of solutions of one magnetic field equation in an unbounded medium”, J Math Sci, 184:1 (2012), 36  crossref  mathscinet  elib
    12. A. G. Baskakov, N. S. Kaluzhina, “Beurlings theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations”, Math. Notes, 92:5 (2012), 587–605  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. V. N. Denisov, “Necessary and sufficient conditions for stabilization of a solution to the Cauchy problem for the parabolic equation with radial potential”, J Math Sci, 2012  crossref  mathscinet
    14. Denisov V.N., “Exact Stabilization Conditions for the Solutions of the Cauchy Problem for a Nondivergent Parabolic Equation with Decreasing Lower-Order Coefficients”, Dokl. Math., 86:2 (2012), 654–656  crossref  mathscinet  mathscinet  zmath  isi  elib  elib
    15. V. N. Denisov, “Stabilization of solutions of Cauchy problems for divergence-free parabolic equations with decreasing minor coefficients”, Journal of Mathematical Sciences, 201:5 (2014), 581–594  mathnet  crossref  mathscinet
    16. V. N. Denisov, “Stabilization conditions for the solution of the Cauchy problem for a parabolic equation with growing lower order coefficients”, Dokl. Math, 87:3 (2013), 348  crossref  mathscinet  zmath  isi  elib
    17. Alkhutov Yu.A., Denisov V.N., “Necessary and Sufficient Condition for the Stabilization of the Solution to the Initial-Boundary Value Problem for Second-Order Nondivergence Parabolic Equations”, Dokl. Math., 88:1 (2013), 381–384  crossref  mathscinet  zmath  isi  elib
    18. Denisov V.N., “Stabilization of the Solution of the Cauchy Problem for a Parabolic Equation in Nondivergence Form with Growing Lower Coefficients”, Differ. Equ., 49:5 (2013), 569–582  crossref  mathscinet  zmath  isi  elib
    19. Razzhevaikin V.N., “Stabilization of Solutions of the Cauchy Problem for the Reaction-Nonlinear Diffusion Equation to a Dominant Equilibrium”, Differ. Equ., 49:3 (2013), 320–325  crossref  mathscinet  zmath  isi  elib
    20. V. A. Litovchenko, I. M. Dovzhytska, “Stabilization of solutions to Shilov-type parabolic systems with nonnegative genus”, Siberian Math. J., 55:2 (2014), 276–283  mathnet  crossref  mathscinet  isi
    21. A. G. Baskakov, N. S. Kaluzhina, D. M. Polyakov, “Slowly varying on infinity semigroups of operators”, Russian Math. (Iz. VUZ), 58:7 (2014), 1–10  mathnet  crossref
    22. Yu. A. Alkhutov, V. N. Denisov, “Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero”, Trans. Moscow Math. Soc., 75 (2014), 233–258  mathnet  crossref  elib
    23. Kochubei A.N., “Asymptotic Properties of Solutions of the Fractional Diffusion-Wave Equation”, Fract. Calc. Appl. Anal., 17:3 (2014), 881–896  crossref  mathscinet  zmath  isi
    24. V. N. Denisov, “Stabilization of the solution of the Cauchy problem for a parabolic equation with bounded coefficients multiplying the solution gradient”, Dokl. Math, 91:1 (2015), 47  crossref  mathscinet  zmath  isi
    25. V. N. Denisov, “O skorosti stabilizatsii resheniya zadachi Koshi dlya parabolicheskogo uravneniya s mladshimi koeffitsientami”, Trudy Sedmoi Mezhdunarodnoi konferentsii po differentsialnym i funktsionalno-differentsialnym uravneniyam (Moskva, 22–29 avgusta, 2014). Chast 2, SMFN, 59, RUDN, M., 2016, 53–73  mathnet
    26. V. N. Denisov, “O povedenii pri bolshikh znacheniyakh vremeni reshenii parabolicheskikh nedivergentnykh uravnenii s rastuschimi starshimi koeffitsientami”, Trudy seminara po differentsialnym i funktsionalno-differentsialnym uravneniyam v RUDN pod rukovodstvom A. L. Skubachevskogo, SMFN, 62, RUDN, M., 2016, 72–84  mathnet
    27. Denisov V.N., “On Long-Time Asymptotics of Solutions of Parabolic Equations With Increasing Leading Coefficients”, Dokl. Math., 96:1 (2017), 308–311  crossref  mathscinet  zmath  isi
    28. V. N. Denisov, “O skorosti stabilizatsii resheniya zadachi Koshi dlya nedivergentnykh parabolicheskikh uravnenii s rastuschim mladshim koeffitsientom”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 4, Rossiiskii universitet druzhby narodov, M., 2017, 586–598  mathnet  crossref
    29. V. N. Denisov, “Stabilization of solutions of parabolic equations with growing leading coefficients”, J. Math. Sci. (N. Y.), 244:2 (2020), 198–215  mathnet  crossref
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