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**Infinite series asymptotic expansions for decaying solutions of dissipative differential equations with non-smooth nonlinearity.**
*(English)*
Zbl 1477.34076

Summary: We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time approximations for the solution with the errors decaying exponentially at any given rates. The main idea is to shift the center of the Taylor expansions for the nonlinear term to a non-zero point. Such a point turns out to come from the non-trivial asymptotic behavior of the solution, which we prove by a new and simple method. Our result applies to different classes of non-linear equations that have not been dealt with previously.

### MSC:

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

34A36 | Discontinuous ordinary differential equations |

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\textit{D. Cao} et al., Qual. Theory Dyn. Syst. 20, No. 3, Paper No. 62, 38 p. (2021; Zbl 1477.34076)

### References:

[1] | Bibikov, Y.N.: Local Theory of Nonlinear Analytic Ordinary Differential Equations. In: Lecture Notes in Mathematics, vol. 702. Springer-Verlag, Berlin-New York (1979) · Zbl 0404.34005 |

[2] | Bruno, AD, Local Methods in Nonlinear Differential Equations (1989), Berlin: Springer Series in Soviet Mathematics. Springer-Verlag, Berlin |

[3] | Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. North-Holland Mathematical Library, vol. 57. North-Holland Publishing Co., Amsterdam (2000) |

[4] | Bruno, AD, Asymptotic behavior and expansions of solutions of an ordinary differential equation, Uspekhi Mat. Nauk, 59, 3, 31-80 (2004) |

[5] | Bruno, AD, Power-logarithmic expansions of solutions of a system of ordinary differential equations, Dokl. Akad. Nauk, 419, 3, 298-302 (2008) |

[6] | Bruno, AD, Power-exponential expansions of solutions of an ordinary differential equation, Dokl. Akad. Nauk, 444, 2, 137-142 (2012) |

[7] | Bruno, AD, On complicated expansions of solutions to ODES, Comput. Math. Math. Phys., 58, 3, 328-347 (2018) · Zbl 06909572 |

[8] | Cao, D.; Hoang, L., Asymptotic expansions in a general system of decaying functions for solutions of the Navier-Stokes equations, Ann. Mat. Pura Appl., 199, 3, 1023-1072 (2020) · Zbl 1442.35294 |

[9] | Cao, D., Hoang, L.: Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations, J. Evol. Equ. 21(2), 1179-1225 (2021) |

[10] | Cao, D.; Hoang, L., Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces, Proc. Roy. Soc. Edinburgh Sect. A, 150, 2, 569-606 (2020) · Zbl 1439.35366 |

[11] | Coddington, EA; Levinson, N., Theory of Ordinary Differential Equations (1955), New York: McGraw-Hill Book Company Inc, New York · Zbl 0064.33002 |

[12] | Cohen, PJ; Lees, M., Asymptotic decay of solutions of differential inequalities, Pac. J. Math., 11, 1235-1249 (1961) · Zbl 0171.35002 |

[13] | Foias, C.; Hoang, L.; Nicolaenko, B., On the helicity in 3D-periodic Navier-Stokes equations. I. The non-statistical case, Proc. Lond. Math. Soc., 94, 1, 53-90 (2007) · Zbl 1109.76015 |

[14] | Foias, C.; Hoang, L.; Nicolaenko, B., On the helicity in 3D-periodic Navier-Stokes equations. II. The statistical case, Comm. Math. Phys., 290, 2, 679-717 (2009) · Zbl 1184.35239 |

[15] | Foias, C.; Hoang, L.; Olson, E.; Ziane, M., On the solutions to the normal form of the Navier-Stokes equations, Indiana Univ. Math. J., 55, 2, 631-686 (2006) · Zbl 1246.76019 |

[16] | Foias, C.; Hoang, L.; Olson, E.; Ziane, M., The normal form of the Navier-Stokes equations in suitable normed spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 5, 1635-1673 (2009) · Zbl 1179.35212 |

[17] | Foias, C.; Hoang, L.; Saut, J-C, Asymptotic integration of Navier-Stokes equations with potential forces. II. an explicit Poincaré-Dulac normal form, J. Funct. Anal., 260, 10, 3007-3035 (2011) · Zbl 1232.35115 |

[18] | Foias, C.; Saut, J-C, Asymptotic behavior, as \(t\rightarrow +\infty \), of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indiana Univ. Math. J., 33, 3, 459-477 (1984) · Zbl 0565.35087 |

[19] | Foias, C.; Saut, J-C, Linearization and normal form of the Navier-Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéair, 4, 1, 1-47 (1987) · Zbl 0635.35075 |

[20] | Foias, C.; Saut, J-C, Asymptotic integration of Navier-Stokes equations with potential forces. I, Indiana Univ. Math. J., 40, 1, 305-320 (1991) · Zbl 0739.35066 |

[21] | Ghidaglia, J-M, Long time behaviour of solutions of abstract inequalities: applications to thermohydraulic and magnetohydrodynamic equations, J. Diff. Equ., 61, 2, 268-294 (1986) · Zbl 0549.35102 |

[22] | Ghidaglia, J-M, Some backward uniqueness results, Nonlinear Anal., 10, 8, 777-790 (1986) · Zbl 0622.35029 |

[23] | Hoang, L., Asymptotic expansions for the Lagrangian trajectories from solutions of the Navier-Stokes equations, Comm. Math. Phys., 383, 2, 981-995 (2021) · Zbl 1466.35286 |

[24] | Hoang, LT; Martinez, VR, Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces, J. Math. Anal. Appl., 462, 1, 84-113 (2018) · Zbl 1394.35130 |

[25] | Hoang, LT; Titi, ES, Asymptotic expansions in time for rotating incompressible viscous fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38, 109-137 (2021) · Zbl 1464.35182 |

[26] | Minea, G., Investigation of the Foias-Saut normalization in the finite-dimensional case, J. Dynam. Diff. Equ., 10, 1, 189-207 (1998) · Zbl 0970.34045 |

[27] | Shi, Y., A Foias-Saut type of expansion for dissipative wave equations, Comm. Partial Diff. Equ., 25, 11-12, 2287-2331 (2000) · Zbl 0963.35123 |

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