THE ASYMPTOTICS OF A SOLUTION OF THE MULTIDIMENSIONAL HEAT EQUATION WITH UNBOUNDED INITIAL DATA1

For the multidimensional heat equation, the long-time asymptotic approximation of the solution of the Cauchy problem is obtained in the case when the initial function grows at infinity and contains logarithms in its asymptotics. In addition to natural applications to processes of heat conduction and diffusion, the investigation of the asymptotic behavior of the solution of the problem under consideration is of interest for the asymptotic analysis of equations of parabolic type. The auxiliary parameter method plays a decisive role in the investigation.


Introduction
In 1822, J. Fourier published his most fundamental work [4], where the heat conduction equation was presented and analyzed. This event provided a strong impetus for later researches in the fields of partial differential equations and trigonometric series. The famous equation has been further successfully used for effective descriptions of molecular diffusion, stochastic motion, the capillary conduction of liquids in porous media, and even for the analysis of social economic data. Already Fourier himself pointed out the universality of this mathematical model sine qua non in his eminent book as follows: "Il est facile de juger combien ces recherches intéressent les sciences physiques et l'économie civile, et quelle peutêtre leur influence sur les progrès des arts qui exigent l'emploi et la distribution du feu." 2 Fourier's preliminary theoretical studying of heat phenomena and some vivid particulars of his elaborations in early 1800s are expressively reflected in the prefatory part of [4]. The historical survey [10] supplied with appropriate general and specialized references depicts many significant details of the subsequent life of the heat equation during the XIX and XX centuries.
Since the literature about the heat equation, in particular, and parabolic equations, in general, is immense, it is impossible in this introduction to give a complete picture of available results, and the bibliography below is of course by no means exhaustive. Here, we mention that existence and uniqueness theorems were obtained for a wide class of parabolic equations and systems [6,15,18,19]; some results for unbounded solutions were presented in [11,13]. As for the long-time behavior of solutions, we see that their stabilization, certain estimates, and the leading terms of asymptotics were mainly considered [2,8,12,17]. Complete asymptotic expansions of solutions into infinite series in inverse integer powers of the time variable were earlier obtained by Friedman in [5] and [6,Ch. 6] for bounded space-domains.
In the present paper, the long-time asymptotics of the solution of the Cauchy problem for the multidimensional heat equation is obtained for a locally Lebesgue integrable initial function Λ : R m → R of polynomial growth.
As is well known [18], in the class of smooth functions of moderate growth for t > 0, there exists a unique solution of problem (1.1)-(1.2) and it can be written in the form of the Poisson integral 3 It should be noted that the investigation of the asymptotic behavior of the function u(x, t), in addition to possible natural applications to the modeling of physical processes of heat conduction and diffusion, may be of interest for the asymptotic analysis of solutions of nonlinear parabolic equations by the matching method [9,21] as well as for the theory of invariants [7] and some issues of matrix geometry [14].
Below, a complete asymptotic expansion of the solution u(x, t) of problem (1.1)-(1.2) is found as |x| + t → +∞ under the following suppositions: where p is a positive integer and Λ n,j (x ′ ) are Lebesgue integrable functions of x ′ = (x 2 , . . . , x m ); for simplicity, we also suppose that supp Λ ⊂ (x 1 , . . . , x m ) : x 1 > 0, |x 2 | + . . . + |x m | < x ν 1 , ν > 0, supp Λ n,j ⊂ (x 2 , . . . , x m ) : |x 2 | + . . . + |x m | < r n , r n > 0. (1.6) Although Λ is a function of several variables, the asymptotic series (1.5) must be understood here in the usual sense of Poincaré [16, § 1] due to the second condition (1.6), that is for any integer N 1. It should be also said that the appearance of asymptotic series of form (1.5) is typical for the matching method [9]. The main difficulty of the calculation of the asymptotic expansion of integral (1.3) is exactly due to condition (1.5) and the "smearing" of the integrand exponent as t → +∞; if we formally put t = +∞, then we generally get the divergence of the integral. Thus, the asymptotic limit under consideration is diametrically opposite to the well-known case of the integrals of Laplace's type with the sharpening exponent and a suitable computational technique suggested by Danilin in [1] is therefore complementary to the standard Laplace method. This technique is called the auxiliary parameter method.

Applying the auxiliary parameter method
To obtain the asymptotic behavior of integral (1.3) as the space-time variables (x, t) independently tend to infinity, we apply a scheme similar to that used in [20] for the solution of the heat the dots denote the integrand in formula (1.3) together with the factor (4πt) −m/2 , the number β is an arbitrary parameter, and ds ′ = ds 2 . . . ds m . Under conditions (1.4) and (1.5), the asymptotics of the integrals U 0 (x, t) and U 1 (x, t) can be computed by using the expansions of the kernel exponent and the initial function Λ, respectively.

Asymptotics of U 1 (x, t)
In the integral U 1 (x, t), we make the change s 1 = 2z √ t and put Next, using condition (1.5), for any integer N p + 1, we obtain (hereinafter we often omit the arguments of σ and µ) by formula (1.7). Then, for N p + 1, we have Changing the order of summation, we find To handle the integral with respect to z, it is convenient to consider first the following set of independent variables: The obvious inequalities therefore, on account of the first definition (2.3), we obtain For 0 n p, we have Since by (2.7) µ → +0 as σ → +∞ for (x, t) ∈ T α , it follows that where the finite sum over s with b ′ s being some constants and n s , r s , l s being some nonnegative integers depends naturally on N . For n > p, we have where P r (η 1 ) are some polynomials of degree r, (2.9) and the sum in the square brackets is a partial sum of the Maclaurin series for the function exp(2zη 1 − z 2 ) in variable z with H r (η 1 ) being the Hermite polynomials of degree r. This implies the equality with b ′′ s being some constants and n s , r s , l s being some nonnegative integers. From formula (2.9) we easily conclude that the function Ψ n−p (z, η 1 ) has no singularities as z → 0; therefore, the last two integrals in (2.10) converge and relation (2.10) itself thus becomes b ′′′ s are some constants, n s , r s , l s are some nonnegative integers, and γ is defined in (2.7). Using the second condition (1.6) and Maclaurin's expansion for the exponent in the integrand of (2.4) in s ′ t −1/2 , for any natural N * 1, we obtain where Q (n,j) l (η ′ ) are some lth degree polynomials in η ′ = 2 −1 t −1/2 x ′ whose coefficients depend on n and j. Substituting expressions (2.8), (2.11), and (2.13) into formula (2.4) and taking into account that σ −(γ+α/2β)N = O(σ −N ), since γ + α/2β = α/β − 1 > 1, we find that as σ → +∞, where, according to formulas (2.9) and (2.12), the coefficients S n,l (η) are some smooth functions of polynomial growth for 0 n p and of superexponential decreasing for n > p, a ′ s t ks η ns µ rs ln ls µ (2.15) is a finite sum with η ns = η n 1,s 1 . . . η nm,s m , a ′ s being some real constants, k s being half-integer numbers, and n j,s , r s , l s being some nonnegative integers. Because of the factor exp(−|η| 2 ), the estimate of the remainder in formula (2.14) remains true for the values of the independent variables from the set since for (x, t) ∈ X α there hold the following inequalities: Now, let us pass to the evaluation of the integral From the obvious inequality |x| 2 [σ(x, t)] 2/β and inequality (2.6) we conclude that for |s| σ and (x, t) ∈ T α , where 1 k m. Then, using conditions (1.6), (1.7) and estimates (2.18), we represent the integral U 0 (x, t) in the following form: as σ → +∞ with any N 1. Because of the factor exp(−|η| 2 ), the estimate of the remainder holds also true on the set X α defined by (2.16). Expanding the parenthesis in the above formula for U 0 (x, t) and changing the order of summation, we obtain as σ → +∞, where a k,l = a k 1 ,...,km,l 1 ,l 2,2 ,...,l 2,m are some constants, η k = η k 1 1 . . . η km m , and (s ′ ) l 2 = s l 2,2 2 . . . s l 2,m m . Keeping in mind the asymptotic condition (1.7), we transform the multiple integral appeared above as follows: with c l 1 ,l 2 ,j and c * l 1 ,l 2 ,i,j being some constants, where the finite sum over i, j depends naturally on a sufficiently large N * ; here we used the elementary relation From formulas (2.3), inequality (2.6), the uniform estimate t −m/2 η n exp −|η| 2 = O σ −αm/2β , and the previous asymptotic expression for U 0 (x, t), it follows that as σ → +∞, where δ is defined in (2.18), Π n,j (η) are some polynomials of degree n, and the finite sum V 0,N (µ, η, t) = exp −|η| 2 s: r 2 s +l 2 s =0 a ′′ s t ks η ns µ rs ln ls µ, (2.20) with a ′′ s being some constants, is obtained similarly to expression (2.15).
By virtue of the arbitrariness of the value β, from formulas (2.21) and (2.23) with β = β 1 and β = β 2 such that all numbers 2β 1 r 1 , . . . , 2β 1 r L(N ) , 2β 2 r 1 , . . . , 2β 2 r L(N ) are pairwise distinct, we obtain the following asymptotic relation with r 2 s + l 2 s = 0: a ′′′ s t ks ln k ′ s t η ns ε −2β 1 rs ln ls ε 2β 1 − ε −2β 2 rs ln ls ε 2β 2 = O ε 2(α−1−β 1 )N −2β 1 (p+1) as ε → +0. Consequently, taking into account the finiteness of the sum in the left-hand side, we have to conclude about every particular term in the left-hand side that either its order is not greater than the estimate in the right-hand side or the corresponding coefficient a ′′′ s is equal to zero. Thus, we arrive at the following statement with β = β 1 .

Asymptotics of the solution
Immediately from Lemmas 1 and 2, we obtain our main result.