Geodesic
Constructing the asymptotics of a solution of the heat equation from the known asymptotics of the initial function in three-dimensional space
Sbornik. Mathematics, Tome 215 (2024) no. 1, pp. 101-118
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An asymptotic approximation, as time increases without limit, is constructed to the solution of the Cauchy problem for the heat equation in three-dimensional space. The locally integrable initial function, which does not necessarily tend to zero at infinity, is assumed to have powerlike asymptotics. The method of introduction of an auxiliary parameter, which also involves the regularization of singularities in integrals, plays the central role in the research. The asymptotic expression for the solution is shown to have the form of a series in negative half-integer powers of the time variable, with coefficients depending on self-similar variables and the logarithm of time; the leading term is found explicitly. Using the example of the Cauchy problem for the vector Burgers equation, it is shown that to perform an asymptotic analysis of the solution by the matching method one needs to construct an asymptotic approximation to a solution of the heat equation. Bibliography: 31 titles.
Keywords: heat equation, Cauchy problem, asymptotic formula, auxiliary parameter method, regularization of singularities. 
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S. V. Zakharov. Constructing the asymptotics of a solution of the heat equation from the known asymptotics of the initial function in three-dimensional space. Sbornik. Mathematics, Tome 215 (2024) no. 1, pp. 101-118. http://geodesic.mathdoc.fr/item/SM_2024_215_1_a5/

[1] J. Fourier, Théorie analytique de la chaleur, Firmin Didot, Père et Fils, Paris, 1822, xxii+639 pp. | MR | Zbl

[2] T. N. Narasimhan, “Fourier's heat conduction equation: history, influence, and connections”, Rev. Geophys., 37:1 (1999), 151–172 | DOI

[3] O. A. Ladyzhenskaya, “On the uniqueness of the solution of the Cauchy problem for a linear parabolic equation”, Mat. Sb., 27(69):2 (1950), 175–184 (Russian) | MR | Zbl

[4] A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, “Linear equations of the second order of parabolic type”, Russian Math. Surveys, 17:3 (1962), 1–143 | DOI | MR | Zbl

[5] A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, Transl. Math. Monogr., 102, Amer. Math. Soc., Providence, RI, 1992, x+281 pp. | DOI | MR | Zbl

[6] S. V. Zakharov, “Cauchy problem for a quasilinear parabolic equation with a large initial gradient and low viscosity”, Comput. Math. Math. Phys., 50:4 (2010), 665–672 | DOI | MR | Zbl

[7] S. V. Zakharov, “Heat distribution in an infinite rod”, Math. Notes, 80:3 (2006), 366–371 | DOI | MR | Zbl

[8] V. N. Denisov, “On the behaviour of solutions of parabolic equations for large values of time”, Russian Math. Surveys, 60:4 (2005), 721–790 | DOI | MR | Zbl

[9] V. N. Denisov, “On the large-time behaviour of solutions of parabolic equations”, Partial differential equations, Sovrem. Mat. Fundam. Napravl., 66, no. 1, RUDN University, Moscow, 2020, 1–155 (Russian) | DOI | MR

[10] A. M. Il'in, “The behaviour of the solution of the Cauchy problem for a parabolic equation under unlimited growth of time”, Uspekhi Mat. Nauk, 16:2(98) (1961), 115–121 (Russian) | MR | Zbl

[11] V. N. Denisov, “On stabilization of the Poisson integral and Tikhonov–Stieltjes means: two-sided estimate”, Dokl. Math., 103:1 (2021), 32–34 | DOI | MR | Zbl

[12] F. Kh. Mukminov, “The behavior, for $t\to \infty$, of solutions of the first mixed problem for the heat-transfer equation in regions unbounded in the space variables”, Differ. Equ., 15 (1980), 1444–1453 | MR | Zbl

[13] F. Kh. Mukminov, “On uniform stabilization of solutions of the first mixed problem for a parabolic equation”, Math. USSR-Sb., 71:2 (1992), 331–353 | DOI | MR | Zbl

[14] A. V. Lezhnev, “On the behavior, for large time values, of nonnegative solutions of the second mixed problem for a parabolic equation”, Math. USSR-Sb., 57:1 (1987), 195–209 | DOI | MR | Zbl

[15] V. I. Ushakov, “The behavior for $t\to \infty$ of solutions of the third mixed problem for second-order parabolic equations”, Differ. Equ., 15 (1979), 212–219 | MR | Zbl

[16] Yu. N. Čeremnyh, “The behavior of solutions of boundary value problems for second order parabolic equations as $t$ grows without bound”, Math. USSR-Sb., 4:2 (1968), 219–232 | DOI | MR | Zbl

[17] V. V. Zhikov, “Asymptotic problems connected with the heat equation in perforated domains”, Math. USSR-Sb., 71:1 (1992), 125–147 | DOI | MR | Zbl

[18] A. M. Il'in and R. Z. Has'minskiĭ, “Asymptotic behavior of solutions of parabolic equations and an ergodic property of nonhomogeneous diffusion processes”, Amer. Math. Soc. Transl. Ser. 2, 49, Amer. Math. Soc., Providence, RI, 1966, 241–268 | DOI | MR | Zbl

[19] V. N. Denisov, “Stabilization of a Poisson integral in the class of functions with power growth rate”, Differ. Equ., 21 (1985), 24–31 | MR | Zbl

[20] V. N. Denisov, “Stabilization of the solution to the Cauchy problem for a parabolic equation with lowest order coefficients and an increasing initial function”, Dokl. Math., 70:1 (2004), 571–573 | MR | Zbl

[21] A. Friedman, “Asymptotic behavior of solutions of parabolic equations of any order”, Acta Math., 106:1–2 (1961), 1–43 | DOI | MR | Zbl

[22] A. Friedman, Partial differential equations of parabolic type, Englewood Cliffs, NJ, Prentice-Hall, Inc., 1964, xiv+347 pp. | MR | Zbl

[23] S. V. Zakharov, “Asymptotic calculation of the heat distribution in a plane”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 243–249 | DOI | MR | Zbl

[24] S. V. Zakharov, “The asymptotics of a solution of the multidimensional heat equation with unbounded initial data”, Ural Math. J., 7:1 (2021), 168–177 | DOI | MR | Zbl

[25] H. Poincaré, “Sur les intégrales irrégulières. Des équations linéaires”, Acta Math., 8:1 (1886), 295–344 | DOI | MR | Zbl

[26] A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956, vi+108 pp. | MR | Zbl

[27] A. R. Danilin, “Asymptotic behaviour of bounded controls for a singular elliptic problem in a domain with a small cavity”, Sb. Math., 189:11 (1998), 1611–1642 | DOI | MR | Zbl

[28] A. R. Danilin, “Asymptotic behavior of the optimal cost functional for a rapidly stabilizing indirect control in the singular case”, Comput. Math. Math. Phys., 46:12 (2006), 2068–2079 | DOI | MR

[29] A. M. Il'in and A. R. Danilin, Asymptotic methods in analysis, Fizmatlit, Moscow, 2009, 248 pp. (Russian) | Zbl

[30] N. N. Lebedev, Special functions and their applications, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965, xii+308 pp. | MR | Zbl

[31] S. V. Zakharov, “Asymptotic solution of the multidimensional Burgers equation near a singularity”, Theoret. and Math. Phys., 196:1 (2018), 976–982 | DOI | MR | Zbl