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

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

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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

Citation in format AMSBIB
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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
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
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
4. V. I. Bogachev, N. V. Krylov, M. Röckner, “Elliptic and parabolic equations for measures”, Russian Math. Surveys, 64:6 (2009), 973–1078
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
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
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
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
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
10. S. V. Shaposhnikov, “Regular and qualitative properties of solutions for parabolic equations for measures”, Theory Probab. Appl., 56:2 (2011), 252–279
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
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
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
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
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
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
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
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
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
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
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
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
23. Kochubei A.N., “Asymptotic Properties of Solutions of the Fractional Diffusion-Wave Equation”, Fract. Calc. Appl. Anal., 17:3 (2014), 881–896
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
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
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
27. Denisov V.N., “On Long-Time Asymptotics of Solutions of Parabolic Equations With Increasing Leading Coefficients”, Dokl. Math., 96:1 (2017), 308–311
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
29. V. N. Denisov, “Stabilization of solutions of parabolic equations with growing leading coefficients”, J. Math. Sci. (N. Y.), 244:2 (2020), 198–215
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