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Constructing the asymptotics of a solution of the heat equation from the known asymptotics of the initial function in three-dimensional space S. V. Zakharov N. N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
Russian version:
Matematicheskii Sbornik, 2024, Volume 215, Number 1, DOI: https://doi.org/10.4213/sm9890 
Bibliographic databases:
    § 1. IntroductionThe subject of our study here is the solution of the Cauchy problem for the heat equation in three-dimensional space, which is a fundamental model in mathematical physics:
$$
\begin{equation}
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x_1^2} +\frac{\partial^2 u}{\partial x_2^2}+\frac{\partial^2 u}{\partial x_3^2}, \qquad t>0,
\end{equation}
\tag{1.1}
$$
$$
\begin{equation}
u(x,0)=\Lambda (x), \qquad x=(x_1,x_2,x_3) \in \mathbb{R}^3.
\end{equation}
\tag{1.2}
$$
As is well known, for a wide class of initial data this solution can be represented as an integral convolution1
$$
\begin{equation}
u(x,t)=\frac{1}{8(\pi t)^{3/2}}\int_{\mathbb{R}^3} \Lambda(s) \exp\biggl(-\frac{|s-x|^2}{4t} \biggr)\,ds,
\end{equation}
\tag{1.3}
$$
which was in fact already known to Fourier (see [1], Ch. IX, § 392). He deduced (1.1) in his fundamental treatise and stressed there the universal nature of heat phenomena in the physical world: “La chaleur pénètre, comme la gravité, toutes les substances de l’univers, ses rayons occupent toutes les parties de l’espace”. The sine qua non status of the heat equation has fully been confirmed by the subsequent history (see [2]). Recall that if the initial function $\Lambda\colon \mathbb{R}^3 \to \mathbb{R}$ does not increase too rapidly (for example, has a power growth), then $u(x,t)$ is the unique solution (see [3] and [4]). The main aim of our investigation is to construct an asymptotic approximation as $t\to +\infty$ to the solution of the Cauchy problem (1.1), (1.2) for initial functions satisfying certain special conditions Apart from direct applications to models of physical processes of heat transfer and diffusion, this problem is of independent theoretical interest for the asymptotic analysis of solutions of parabolic-type equations because the need in such information arises in a natural way in applications of the matching method (see [5]) to initial-boundary value problems. For example, in the Cauchy problem with large initial gradient for a quasilinear parabolic equation (see [6], § 2), to construct a uniform approximation to a solution even in a small neighbourhood of a singular point we have to use the asymptotic behaviour at infinity of the solution of a linear heat equation with respect to the inner variables, which was considered separately in [7]. Generally speaking, the investigations of the asymptotic behaviour of solutions of evolution partial differential equation at large times has always been — and is still a topical problem in mathematical physics, which is a subject of a great number of papers, including ones on parabolic-type equations: see the surveys [8] and [9], which also give information on results describing conditions under which the Cauchy problem for quite general equations and systems of parabolic type is well posed, and — with a considerable margin — ensuring the existence and uniqueness of solutions, which also covers the special conditions that we state in what follows. In particular, authors considered the behaviour as $t\to +\infty$ of solutions of equations supplemented by various boundary conditions: the Cauchy problem, whose solution was considered in [10] and [11]; the first, second and third boundary value problems considered in [12]–[15]; mixed problems considered in [16]–[18]. There are results on the stabilization of solutions whose initial functions have a power growth (see [19] and [20]). On the other hand convergence theorems do not give one a full picture of the behaviour of solutions, and the asymptotic analysis of solutions is often limited to finding the leading approximation. The first full asymptotic expansions as $t\to +\infty$ of solutions of parabolic equations in infinite series were perhaps due to Friedman (see [21] and [22], Ch. 6), who obtained them for domains bounded in the space variables. For nondecreasing and even increasing initial data full asymptotic formulae for solutions in spaces of various dimension were constructed in [7], [23] and [24]. The main problem with analytic investigation of solutions with nondecreasing initial data in spatially unbounded domains is related to the singular behaviour of such solutions. In fact, if we set formally $t=+\infty$ in (1.3), then we obtain in general a divergent integral because the initial function is only assumed to be locally integrable. So the second important aim of our paper is to explain in details, using the example of a solution of the Cauchy problem expressed as a convolution by formula (1.3), how the method of introduction of an auxiliary parameter works, which allows one to find with arbitrary accuracy the structure of asymptotic approximations to integrals depending singularly on small parameters. § 2. Choice of an initial functionIn this paper we construct an asymptotic formula as $t \to +\infty$ for the solution $u(x,t)$ of the Cauchy problem (1.1), (1.2) under the assumption that the initial function $\Lambda$ is locally Lebesgue integrable and, for all $N \geqslant 1$, satisfies the asymptotic relation
$$
\begin{equation}
\Lambda(x_1, x_2, x_3)=\sum_{n=0}^{N-1} x_1^{-n} \Lambda_{2,n}(x_2) \Lambda_{3,n}(x_3) +O(x_1^{-N}), \qquad x_1 \to +\infty,
\end{equation}
\tag{2.1}
$$
in the sense of Poincaré (see [25], § 1, or [26], Ch. I, § 1.3), where $\Lambda_{2,n}\colon \mathbb{R} \to \mathbb{R}$ and $\Lambda_{3,n}\colon \mathbb{R} \to \mathbb{R}$ are locally integrable functions and the estimate of the remainder term on the right-hand side of (2.1) is independent of $(x_2,x_3)\in \mathbb{R}^2$. We assume that for all $n\geqslant 0$ the functions $\Lambda_{2,n}$ tend to zero at a superpower rate, and the functions $\Lambda_{3,n}$, satisfy the following asymptotic relations for all $N \geqslant 1$:
$$
\begin{equation}
\Lambda_{3,n}(x_3)=\sum_{m=0}^{N-1} x_3^{-m} \Lambda_{3,n,m}^\pm +O(|x_3|^{-N}), \qquad x_3 \to \pm \infty.
\end{equation}
\tag{2.2}
$$
In addition, assume that
$$
\begin{equation}
\Lambda (x_1, x_2, x_3)=O(1), \qquad |x| \to \infty,
\end{equation}
\tag{2.3}
$$
which, in combination with the local integrability of $\Lambda$, certainly ensures the convergence of the integral (1.3), and let and for all $n\geqslant 0$ let
$$
\begin{equation}
\int_{0}^{1} s_1^{n} \Lambda (s_1, s_2, s_3)\,ds_1 =\sum_{m=0}^{N-1}s_3^{-m} \Phi_{n,m}^\pm (s_2) +O(|s_3|^{-N}), \qquad s_3 \to \pm \infty,
\end{equation}
\tag{2.5}
$$
where $N \geqslant 1$, and the integrable functions $\Phi_{n,m}^\pm$ tend to zero at infinity at a superpower rate. Our choice of the assumptions (2.1)–(2.4) is motivated by Hilbert’s thesis of the importance of special setups of problems2 on the one hand and by reasons of simplicity on the other; the meaning of (2.5) becomes clear in the calculations that follow. In physical interpretations these assumptions correspond to the initial distribution of heat localized in a neighbourhood of the half-plane
$$
\begin{equation*}
\bigl\{ (x_1, x_2, x_3)\colon x_1>0,\, x_2=0,\, -\infty<x_3<+\infty \bigr\}
\end{equation*}
\notag
$$
in the space of independent variables. It is clear that, as the heat equation is linear, combining (2.1)–(2.4) with rearrangements of the space variables we can extend the class of initial data for which we can readily extract asymptotic expressions for the solution directly from our results in this paper. § 3. Using the auxiliary parameter methodIn the investigation of optimal control problems Danilin developed an original method of the introduction of an auxiliary parameter in order to analyze integrals displaying singular behaviour (see [27] and [28]). Following the general idea of this method (see [29], Ch. 7, § 30) and taking assumption (2.4) on the initial data into account we represent the solution (1.3) as a sum of two integrals, over a bounded layer and over a half-space: where
$$
\begin{equation*}
U_0(x,t)=\frac{1}{8(\pi t)^{3/2}} \int_{0}^{\sigma(x,t)}\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Lambda(s) \exp\biggl(-\frac{|s-x|^2}{4t} \biggr) ds_2 \, ds_3 \, ds_1
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
U_1(x,t)=\frac{1}{8(\pi t)^{3/2}} \int_{\sigma(x,t)}^{+\infty} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Lambda(s) \exp\biggl(-\frac{|s-x|^2}{4t} \biggr) ds_2 \, ds_3 \, ds_1;
\end{equation*}
\notag
$$
here
$$
\begin{equation}
\sigma (x,t)=(x_1^2 +x_2^2 +x_3^2 +t)^{\beta/2}, \qquad 0<\beta<1,
\end{equation}
\tag{3.2}
$$
integration from $0$ to $\sigma (x,t)$ in $U_0(x,t)$ and from $\sigma (x,t)$ to $+\infty$ in $U_1(x,t)$ is with respect to $s_1$, and $\beta$ is an arbitrary parameter. In itself, the computational component of the method in the case of the decomposition (3.1) of the convolution (1.3) can be described as follows. To calculate the asymptotic behaviour of the integral $U_1(x,t)$ over a half-space, first we use the asymptotics of the initial function and then regularize the singularities arising by subtracting the Taylor polynomial for the exponential heat kernel. Conversely, to find the asymptotic behaviour of the integral $U_0(x,t)$ over a bounded layer we use Taylor’s formula for the heat kernel first and then subtract an asymptotic expression for the initial function. The formulae obtained in this way contain ‘virtual’ sums, which are expressions involving an arbitrary auxiliary parameter $\beta$. The proof that the total sum of this expressions is sufficiently small as $t\to +\infty$ is the final step of the calculations and, at the same time, the conceptual component of the method. On the other hand, if in splitting the convolution integral we fix some partition of the interval of integration $[0,+\infty)$, then we obtain many fictitious expressions, of which we can get rid only by calculating them explicitly, which is quite difficult when we look for a complete asymptotic approximation in the form of an infinite series. 3.1. An asymptotic expression for $U_1(x,t)$The precise meaning of the asymptotic expressions below will be explained separately each time; usually they should be understood in the sense of Erdélyi (see [26], Ch. II, § 2.1) with respect to the asymptotic sequences $\{ (|x|^2 +t)^{-\varkappa n} \}_{n=1}^{\infty}$ for various values of $\varkappa>0$ as $|x|^2 +t \to +\infty$. In the expression for $U_1(x,t)$ it is convenient to make the change $s_1=2 z \sqrt{t}$ of the variable of integration. Now we introduce the natural self-similar variables3
$$
\begin{equation}
\mu=\frac{\sigma}{2\sqrt{t}}\quad\text{and} \quad \eta_1=\frac{x_1}{2\sqrt{t}}
\end{equation}
\tag{3.3}
$$
and using the asymptotic condition (2.1), for each positive integer $N$ we obtain the formula
$$
\begin{equation}
\begin{aligned} \, \notag U_1(x,t)&=\frac{1}{4\pi^{3/2} t} \int_{\mu}^{+\infty} \exp(-(\eta_1-z)^2) \notag \\ &\qquad\qquad\times\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Lambda(2z\sqrt{t},s_2,s_3) E(s_2,s_3,x_2,x_3,t)\,ds_2\,ds_3\,dz \notag \\ \notag &=\frac{1}{\sqrt{\pi}} \sum_{n=0}^{N-1} 2^{-n} t^{-n/2}\int_{\mu}^{+\infty} z^{-n} \exp(-(z-\eta_1)^2)\,dz \\ &\qquad\qquad\times \frac{1}{4\pi t} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Lambda_{2,n}(s_2) \Lambda_{3,n}(s_3) E(s_2,s_3,x_2,x_3,t)\,ds_2\,ds_3 \notag \\ &\qquad +R_N (x,t,\mu), \end{aligned}
\end{equation}
\tag{3.4}
$$
where
$$
\begin{equation}
E(s_2,s_3,x_2,x_3,t)= \exp\biggl(-\frac{(x_2-s_2)^2 +(x_3-s_3)^2}{4t} \biggr),
\end{equation}
\tag{3.5}
$$
and the remainder term has the estimate
$$
\begin{equation*}
| R_N (x,t,\mu) | \leqslant \mathrm{const}\cdot t^{-N/2} \int_{\mu}^{+\infty} z^{-N} \exp(-(z-\eta_1)^2)\,dz,
\end{equation*}
\notag
$$
which is refined below by (3.15). In what follows we use the fact that, as $\sigma \to +\infty$, in the unbounded domain
$$
\begin{equation}
T_{\alpha}=\bigl\{ (x,t)\colon x\in\mathbb{R}^3,\, t>(x_1^2 +x_2^2 +x_3^2)^{\alpha/2} \bigr\}, \qquad 1 +\beta<\alpha<2,
\end{equation}
\tag{3.6}
$$
we have the following inequalities, which are easy to verify:
$$
\begin{equation}
t>2^{-\alpha/2} \sigma^{\alpha/\beta}, \qquad \mu<2^{-1+\beta/2} t^{\beta/\alpha -1/2}\quad\text{and} \quad \mu<2^{-1+\alpha/4} \sigma^{-\gamma},
\end{equation}
\tag{3.7}
$$
where by definition Using the last inequality in (3.7), for the integral $\displaystyle\int_{\mu}^{+\infty}$ on the right-hand side of (3.4), for $n=0$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\mu}^{+\infty} \exp(-(z-\eta_1)^2)\,dz =\int_{0}^{+\infty} \exp(-(z-\eta_1)^2)\,dz-\int_{0}^{\mu} \exp(-(z-\eta_1)^2)\,dz \\ \notag &\qquad=\int_{-\eta_1}^{+\infty} \exp(-z^2)\,dz-\exp(-\eta_1^2) \int_{0}^{\mu} \exp(2 z \eta_1-z^2) \,dz \\ &\qquad=\int_{-\eta_1}^{+\infty} \exp(-z^2)\,dz + \exp(-\eta_1^2) \sum_{r=1}^{N-1} H_{r}(\eta_1) \frac{\mu^{r}}{r!} +O(\sigma^{-\gamma N}) \end{aligned}
\end{equation}
\tag{3.9}
$$
as $\sigma \to +\infty$, where $H_{r}(\eta_1)$ is the Hermite polynomial of degree $r$, because the expression $\exp (2z\eta_1-z^2)$ is precisely the generating function for Hermite polynomials (see [30], Ch. IV, § 4.9). For $n \geqslant 1$ we must regularize the integrand to isolate the singularity $z^{-n}$ as $z=\mu \to 0$. To do this, using the Taylor expansion of the exponential function, we write the identity
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mu}^{+\infty} z^{-n} \exp(-(z-\eta_1)^2)\,dz = \int_{1}^{+\infty} z^{-n} \exp(- (z-\eta_1)^2) \,dz \\ &\qquad +\int_{\mu}^{1} \Psi_{n}(z,\eta_1)\,dz +\exp(-\eta_1^2) \sum_{r=0}^{n-1} \frac{H_{r}(\eta_1)}{r!} \int_{\mu}^{1} z^{r-n}\,dz, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\Psi_{n}(z,\eta_1)=z^{-n} \biggl[ \exp(-(z-\eta_1)^2)-\exp(-\eta_1^2) \sum_{r=0}^{n-1} \frac{H_{r}(\eta_1)}{r!} z^{r} \biggr],
\end{equation}
\tag{3.10}
$$
and the sum with respect to $r$ in square brackets is a partial Taylor sum for the function $\exp (2 z \eta_1-z^2)$ as $z\to 0$. Hence
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\mu}^{+\infty} z^{-n} \exp(-(z-\eta_1)^2)\, dz = \int_{1}^{+\infty} z^{-n} \exp(-(z-\eta_1)^2) \,dz \\ \notag &-\exp(-\eta_1^2) \frac{H_{n-1}(\eta_1)}{(n-1)!} \log\mu + \exp(-\eta_1^2) \sum_{r=0}^{n-2}\frac{H_{r}(\eta_1)}{r!\, (r-n +1)} \\ &-\exp(-\eta_1^2) \sum_{r=0}^{n-2} \frac{H_{r}(\eta_1)}{r!\, (r-n +1)} \mu^{r-n +1} \,{+}\int_{0}^{1}\Psi_{n}(z,\eta_1)\,dz \,{-} \int_{0}^{\mu}\Psi_{n}(z,\eta_1)\,dz, \end{aligned}
\end{equation}
\tag{3.11}
$$
where the sums with respect to $r$ are set equal to zero for $n=1$. It follows from the definition (3.10) that the function $\Psi_n(z,\eta_1)$ has no singularities as $z \to 0$, and it is also easy to see that $\Psi_n(z,\eta_1)$ is an analytic function, which can be represented by a convergent series:
$$
\begin{equation*}
\Psi_{n}(z,\eta_1)=\exp(-\eta_1^2) \sum_{r=n}^{\infty} \frac{H_{r}(\eta_1)}{r!} z^{r-n} =\exp(-\eta_1^2) \sum_{m=0}^{\infty} \frac{H_{n+m}(\eta_1)}{(n+m)!} z^{m}, \qquad z \to 0.
\end{equation*}
\notag
$$
Then for each $N\geqslant 2$ we have the asymptotic relations
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{0}^{\mu}\Psi_{n}(z,\eta_1)\,dz =\sum_{r=1}^{N-1}\frac{\partial^{r-1} \Psi_{n}(0,\eta_1)}{\partial z^{r-1}} \frac{\mu^{r}}{r!}+O(\sigma^{-\gamma N}) \\ &\qquad=\exp(-\eta_1^2) \sum_{r=1}^{N-1} \frac{H_{n+r-1}(\eta_1)}{r (n +r-1)!} \mu^{r}+O(\sigma^{-\gamma N}), \qquad \sigma \to +\infty, \end{aligned}
\end{equation}
\tag{3.12}
$$
where the estimate for the remainder term follows from (3.7) and (3.8). For what follows we need to know the order of growth as $\eta_1 \to \infty$ of the first and fifth terms on the right-hand side of (3.11), which are independent of $\mu$. Proof. From (3.10) we obtain an estimate for the second integral: as $\eta_1 \to \infty$, because the expression in square brackets, according to the definition of $\Psi_n(z,\eta_1)$, eliminates the nonintegrable singularity $z^{-n}$ as $z\to 0$, and the function $\exp (-\eta_1^2/2)$ certainly suppresses the growth of the integral $\displaystyle\int_{0}^{1} z^{-n} [\dots]\, dz$.
We apply integration by parts to the first integral as $\eta_1 \to -\infty$:
$$
\begin{equation*}
\begin{aligned} \, \int_{1}^{+\infty} z^{-n} \exp(-(z-\eta_1)^2)\,dz &=\frac{\exp(-\eta_1^2)}{2\eta_1} \int_{1}^{+\infty} z^{-n} \exp(-z^2)\, d\exp(2\eta_1 z) \\ &=O\biggl( \frac{\exp(-(\eta_1 -1)^2)}{1+|\eta_1|} \biggr),\qquad \eta_1 \to -\infty. \end{aligned}
\end{equation*}
\notag
$$
As $\eta_1 \to +\infty$, we make the substitution $ z=\eta_1 w$ and use a Laplace-type asymptotic formula: which shows that the estimate in the statement of the lemma is sharp. The proof is complete. Now it is convenient to work with the class of smooth functions with powerlike growth
$$
\begin{equation}
\mathscr{B}_{m,n}=\bigl\{ f\in C^{\infty}(\mathbb{R}^m)\colon |f(x)| \leqslant M_f (1 +|x|^n) \ \forall\, x\in\mathbb{R}^m,\, M_f>0 \bigr\}.
\end{equation}
\tag{3.13}
$$
It follows from (3.11), (3.12) and Lemma 1 that
$$
\begin{equation}
\begin{aligned} \, &\int_{\mu}^{+\infty} z^{-n} \exp(-(z-\eta_1)^2)\,dz \nonumber \\ &\qquad=J_{n}(\eta_1)-\exp(-\eta_1^2) v_n (\eta_1, \mu)+O(\sigma^{-\gamma N}), \qquad \sigma \to +\infty, \end{aligned}
\end{equation}
\tag{3.14}
$$
where $J_{n} \in \mathscr{B}_{1,n-2}$,
$$
\begin{equation*}
\begin{aligned} \, J_{n}(\eta_1) &=\exp(-\eta_1^2) \sum_{r=0}^{n-2}\frac{H_{r}(\eta_1)}{r!\, (r-n +1)} \\ &\qquad+\int_{1}^{+\infty} z^{-n} \exp(-(z-\eta_1)^2)\,dz+\int_{0}^{1}\Psi_{n}(z,\eta_1)\,dz, \end{aligned}
\end{equation*}
\notag
$$
the sum with respect to $r$ is set equal to zero for $n=1$, and
$$
\begin{equation*}
v_n (\eta_1, \mu)=\frac{H_{n-1}(\eta_1)}{(n-1)!} \log\mu +\sum_{r=-n +1}^{N-1} \frac{H_{n+r-1}(\eta_1)}{r (n +r-1)!} \mu^{r}, \qquad r \neq 0.
\end{equation*}
\notag
$$
Using formula (3.14) and the second inequality in (3.7), we obtain a new estimate for the remainder in (3.4)
$$
\begin{equation}
R_N (x,t,\mu)=O(\sigma^{-\alpha N/ 2\beta}), \qquad \sigma \to +\infty.
\end{equation}
\tag{3.15}
$$
Next we consider the expression
$$
\begin{equation*}
\begin{aligned} \, & \frac{1}{4\pi t} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Lambda_{2,n}(s_2) \Lambda_{3,n}(s_3) E(s_2,s_3,x_2,x_3,t)\,ds_2\,ds_3 \\ &\qquad =\frac{1}{4\pi t}\int_{-\infty}^{+\infty} \Lambda_{2,n}(s_2) \exp\biggl(-\frac{(s_2-x_2)^2}{4t} \biggr)\,ds_2 \\ &\qquad\qquad\times \int_{-\infty}^{+\infty} \Lambda_{3,n}(s_3) \exp\biggl(-\frac{(s_3-x_3)^2}{4t} \biggr)\,ds_3 \end{aligned}
\end{equation*}
\notag
$$
on the right-hand side of (3.4). Bearing in mind that the functions $\Lambda_{2,n}(s_2)$ decay at a superpower rate, for all $n \geqslant 0$ and $N \geqslant 1$ we have (for instance, by [7])
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{2\sqrt{\pi t}} \int_{-\infty}^{+\infty}\Lambda_{2,n}(s_2) \exp\biggl(-\frac{(s_2-x_2)^2}{4t} \biggr)\,ds_2 \\ &\qquad =\exp (-\eta_2^2) \frac{1}{2\sqrt{\pi t}} \int_{-\infty}^{+\infty} \Lambda_{2,n}(s_2)\, ds_2 \notag \\ &\qquad\qquad +\exp (-\eta_2^2) \sum_{m=1}^{N} t^{-m/2} q_{2,n,m-1}(\eta_2) +O \biggl( \frac{(1 +|\eta_2|^N) \exp (-\eta_2^2)}{1 +t^{(N+1)/2}} \biggr), \end{aligned}
\end{equation}
\tag{3.16}
$$
where the $q_{2,n,m-1}(\eta)$ are some polynomials of $\eta_2=2^{-1} t^{-1/2} x_2$ of degree $m-1$, with coefficients depending on $n$. From condition (2.2) we obtain (by the same paper [7])
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{2\sqrt{\pi t}} \int_{-\infty}^{+\infty}\Lambda_{3,n}(s_3) \exp\biggl(-\frac{(s_3-x_3)^2}{4t} \biggr)\,ds_3 \\ \notag &\quad\ =\frac{1}{\sqrt{\pi}} \biggl[ \Lambda^{-}_{3,n,0}\int_{\eta_3}^{+\infty} \exp (-z^2)\,dz + \Lambda^{+}_{3,n,0}\int_{- \eta_3}^{+\infty} \exp (-z^2)\,dz \biggr] \\ &\quad\ \qquad +\sum_{j=1}^{N}t^{-j/2} \bigl( q_{3,n,j,0}(\eta_3) +q_{3,n,j,1}(\eta_3) \exp (-\eta_3^2) \log t \bigr) \!+\!O\biggl(\! \frac{(1 +|\eta_3|^N) \log t}{1 +t^{(N+1)/2}} \!\biggr), \end{aligned}
\end{equation}
\tag{3.17}
$$
where $\eta_3=2^{-1} t^{-1/2} x_3$ and $q_{3,n,j,l} \in \mathscr{B}_{1,j-1}$. Using asymptotic relations (3.16) and (3.17), for the second (double) integral on the right-hand side of (3.4) we finally obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{4\pi t} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \Lambda_{2,n}(s_2) \Lambda_{3,n}(s_3) \exp\biggl(-\frac{(s_2-x_2)^2 +(s_3-x_3)^2}{4t} \biggr)\,ds_2\,ds_3 \\ \notag &\qquad =\exp(-\eta_2^2) \frac{ \Lambda^{*}_{2,n}}{2 \pi \sqrt{t}} \biggl[ \Lambda^{-}_{3,n,0}\int_{\eta_3}^{+\infty} \exp (-z^2)\,dz + \Lambda^{+}_{3,n,0}\int_{- \eta_3}^{+\infty} \exp (-z^2)\,dz \biggr] \\ \notag &\qquad\qquad +\sum_{j=2}^{N} t^{-j/2} \exp (-\eta_2^2) \bigl[ Q_{n,j,0}( \eta_2, \eta_3) +Q_{n,j,1}( \eta_2, \eta_3) \exp (-\eta_3^2) \log t \bigr] \\ &\qquad\qquad +O\biggl( \frac{\exp(-\eta_2^2) (1 +|\eta_2|^N +|\eta_3|^N) \log t}{1 + t^{(N+1)/2}} \biggr), \qquad t \to +\infty, \quad \forall\, N \geqslant 1, \end{aligned}
\end{equation}
\tag{3.18}
$$
where $Q_{n,j,0}\in \mathscr{B}_{2,j-1}$, $Q_{n,j,1}\in \mathscr{B}_{2,0}$,
$$
\begin{equation}
\Lambda^{*}_{2,n}=\int_{-\infty}^{+\infty} \Lambda_{2,n}(s_2)\, ds_2\quad\text{and} \quad \Lambda^{\pm}_{3,n,0}=\lim_{x_3 \to \pm\infty}\Lambda_{3,n}(x_3).
\end{equation}
\tag{3.19}
$$
Substituting (3.9), (3.14) and (3.18) into (3.4) and taking (3.15) into account, we find the asymptotics of $U_1(x,t)$. It is very important that in the terms containing the logarithm $\log\mu$ we do not transform the expressions
$$
\begin{equation*}
\frac{1}{4\pi t} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \Lambda_{2,n}(s_2) \Lambda_{3,n}(s_3)E(s_2,s_3,x_2,x_3,t)\,ds_2\,ds_3
\end{equation*}
\notag
$$
using formula (3.18); furthermore, taking the definition (3.3) of $\mu$ into account we write $\log\mu=\log\sigma-2^{-1} \log{t}-\log{2}$. We can state our result in the following form. Lemma 2. For all $N \geqslant 2$ 3.2. The asymptotics of $U_0(x,t)$Note that by the definitions (3.2) and (3.6), for $(x,t) \in T_{\alpha}$ it follows from the inequalities $0\leqslant s_1 \leqslant \sigma$ and (3.7) that
$$
\begin{equation}
x_1^2<\sigma^{2/\beta}, \qquad \frac{|x_1 s_1|}{t}<2 \sigma^{-\delta}\quad\text{and} \quad \frac{s_1^2}{t}<2 \sigma^{2-\alpha/\beta},
\end{equation}
\tag{3.23}
$$
where
$$
\begin{equation}
\delta=\frac{\alpha-1}{\beta}-1>0\quad\text{and} \quad \frac{\alpha}{\beta}-2>\delta.
\end{equation}
\tag{3.24}
$$
Using inequalities (3.23), (3.24) and Taylor’s formula we can represent the integral expression $U_0(x,t)$ in the following form:
$$
\begin{equation*}
\begin{aligned} \, &U_0(x,t)=\frac{\exp(-\eta_1^2)}{8(\pi t)^{3/2}} \int_{0}^{\sigma} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \Lambda(s_1,s_2,s_3) \\ &\qquad \times\sum_{m=0}^{N-1} \frac{1}{m!} \biggl(\frac{\eta_1 s_1}{\sqrt{t}}-\frac{s_1^2}{4t}\biggr)^m\,ds_2\,ds_3\,ds_1 +O(\sigma^{-\delta N}), \qquad \sigma \to +\infty, \quad \forall\, N \geqslant 1, \end{aligned}
\end{equation*}
\notag
$$
where $E(s_2,s_3,x_2,x_3,t)$ is the exponential function (3.5). Multiplying out the power expression $(\dots)^m$, as $\sigma \to +\infty$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &U_0(x,t)=\frac{\exp(-\eta_1^2)}{8(\pi t)^{3/2}} \sum_{m=0}^{N-1} \sum_{k=0}^{m} \frac{(-1)^k}{4^k k!\, (m-k)!} t^{-(m+k)/2} \eta_1^{m-k} \\ &\qquad \times \int_{0}^{\sigma} s_1^{m+k}\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \Lambda(s_1,s_2,s_3)\,ds_2\,ds_3\,ds_1 +O(\sigma^{-\delta N}). \end{aligned}
\end{equation*}
\notag
$$
Introducing the new index of summation $n=m +k$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &U_0(x,t)=\frac{\exp(-\eta_1^2)}{8(\pi t)^{3/2}} \sum_{n=0}^{N-1} t^{-n/2} \frac{H_n(\eta_1)}{2^n n!} \\ &\qquad \times \int_{0}^{\sigma} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} s_1^{n} E(s_2,s_3,x_2,x_3,t) \Lambda (s_1,s_2,s_3)\,ds_2\,ds_3\,ds_1 +O(\sigma^{-\delta N}), \end{aligned}
\end{equation}
\tag{3.25}
$$
where we again see Hermite polynomials in the form
$$
\begin{equation*}
H_n (\eta_1)=2^n n! \sum_{k=0}^{[n/2]} \frac{(-1)^k \eta_1^{n-2k}}{4^k k! \,(n-2k)!}
\end{equation*}
\notag
$$
(see [30], Ch. IV, § 4.9) and we have used that, according to (3.7) and (3.24), we have the inequality $\sigma^{\delta}<2^{\beta\delta/2} t^{1/2}$, so that $t^{-N/2}=O (\sigma^{-\delta N})$ as $t\to +\infty$. To compensate for the increasing factor $s_1^n$ in the integrand in (3.25), it is convenient to perform regularization by subtracting from $\Lambda$ a partial sum of its asymptotic expression. To do this, using the asymptotic conditions (2.1) and (2.2) for the initial data, we transform the integral on the right-hand side of (3.25) as follows:
$$
\begin{equation*}
\begin{aligned} \, & \int_{0}^{\sigma}\int_{ -\infty}^{+\infty}\int_{ -\infty}^{+\infty} s_1^{n} E(s_2,s_3,x_2,x_3,t) \Lambda (s_1, s_2, s_3)\,ds_2\,ds_3\,ds_1 \\ &\ =\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \Phi_n (s_2, s_3) \,ds_2\,ds_3 \\ &\ \qquad +\int_{1}^{\sigma}\int_{ -\infty}^{+\infty}\int_{ -\infty}^{+\infty} s_1^{n} E(s_2,s_3,x_2,x_3,t) \bigl[ \Lambda(s_1, s_2, s_3) - \Lambda_{2,0}(s_2)\Lambda_{3,0}(s_3) \\ &\ \qquad- \dots-s_1^{-n-1} \Lambda_{2,n+1}(s_2)\Lambda_{3,n+1}(s_3) \bigr]\,ds_2\,ds_3\,ds_1 \\ &\ \qquad+\int_{1}^{\sigma} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \\ &\ \qquad\qquad\qquad\times \bigl[ s_1^{n} \Lambda_{2,0}(s_2)\Lambda_{3,0}(s_3) +\dots +s_1^{-1} \Lambda_{2,n+1}(s_2)\Lambda_{3,n+1}(s_3)\bigr]\,ds_2\,ds_3\,ds_1. \end{aligned}
\end{equation*}
\notag
$$
Bearing in mind that in the second term, using regularization as $s_1 \to +\infty$, for the integrand we have
$$
\begin{equation*}
s_1^{n} \bigl[ \Lambda(s_1, s_2, s_3) - \Lambda_{2,0}(s_2)\Lambda_{3,0}(s_3)-\dots - s_1^{-n-1} \Lambda_{2,n+1}(s_2)\Lambda_{3,n+1}(s_3) \bigr]=O( s_1^{-2}),
\end{equation*}
\notag
$$
and the integral $\displaystyle\int_{1}^{\sigma} \dots\, ds_1$ can explicitly be calculated, as $\sigma \to +\infty$, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_{0}^{\sigma}\int_{ -\infty}^{+\infty}\int_{ -\infty}^{+\infty} s_1^{n} E(s_2,s_3,x_2,x_3,t) \Lambda (s_1, s_2, s_3)\,ds_2\,ds_3\,ds_1 \\ &\qquad=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \Phi_n (s_2, s_3) \,ds_2\,ds_3 \\ &\qquad+\sum_{m=-N+1}^{n-1} \frac{\sigma^{m} -1}{m} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \\ &\qquad\qquad\qquad\times \Lambda_{2,n-m+1}(s_2)\Lambda_{3,n-m+1} (s_3)\,ds_2\,ds_3 \\ &\qquad+\log\sigma \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \Lambda_{2,n+1}(s_2) \Lambda_{3,n+1} (s_3)\,ds_2\,ds_3 +O(\sigma^{-N}), \end{aligned}
\end{equation*}
\notag
$$
where the function $\Phi_n (s_2, s_3)={\displaystyle\int_{0}^{1} s_1^{n} \Lambda (s_1, s_2, s_3)\,ds_1}$ satisfies (2.5) by assumption, and the sum with respect to $m$ contains no term with $m=0$, but contains an expression with $\log\sigma$ instead. Applying (3.18) to all terms in the previous formula, with the exception of the term containing $\log\sigma$, and using the asymptotic formula
$$
\begin{equation*}
\begin{aligned} \, & \frac{1}{4\pi t} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \Phi_n (s_2, s_3) \,ds_2\,ds_3 \\ &\qquad =\exp (-\eta_2^2) \sum_{j=1}^{N-1} t^{-j/2} \bigl[ \widehat{S}_{n,j,0} (\eta_2,\eta_3) +t^{-1/2} \log t\widehat{S}_{n,j,1} (\eta_2,\eta_3) \exp (-\eta_3^2) \bigr] \\ &\qquad\qquad +O \bigl( (\eta_2^2 +\eta_3^2 +t)^{-\rho N} \bigr), \qquad \eta_2^2 +\eta_3^2 +t \to +\infty, \quad \widehat{S}_{n,j,l} \in \mathscr{B}_{2,j}, \quad \rho>0, \end{aligned}
\end{equation*}
\notag
$$
which was found in [23], Theorem 1, as $\sigma \to +\infty$, we obtain
$$
\begin{equation*}
\begin{aligned} \, & \frac{1}{4\pi t} \int_{0}^{\sigma} \int_{ -\infty}^{+\infty} \int_{ -\infty}^{+\infty} s_1^{n} E(s_2,s_3,x_2,x_3,t) \Lambda (s_1, s_2, s_3)\,ds_2\,ds_3\,ds_1 \\ &\qquad =t^{-1/2} \exp (-\eta_2^2) \bigl[ D_{n,1,0} (\eta_3) +D_{n,1,1} (\eta_3) \log t \bigr] \\ &\qquad\qquad +\exp (-\eta_2^2) \sum_{j=2}^{N-1} t^{-j/2} \bigl[D_{n,j,0} (\eta_2,\eta_3) +D_{n,j,1} (\eta_2,\eta_3) \exp (-\eta_3^2) \log t \bigr] \\ &\qquad\qquad +\frac{\log\sigma}{4\pi t} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} E(s_2,s_3,x_2,x_3,t) \Lambda_{2,n+1}(s_2) \Lambda_{3,n+1} (s_3)\,ds_2\,ds_3 \\ &\qquad\qquad +\sum_{k=-N+1}^{n-1} \!\!\mu^{k}\sum_{j=1}^{N} t^{-(k+j)/2} \bigl[ D'_{n,j,0} (\eta_2,\eta_3) +D'_{n,j,1} (\eta_2,\eta_3) \log t \bigr] \!+\!O(\sigma^{-\rho N}), \end{aligned}
\end{equation*}
\notag
$$
where $D_{n,j,l}, D'_{n,j,l} \in \mathscr{B}_{2,j}$, and the sum with respect to $k$ contains no term with index $k=0$. Using the explicit expression, presented in [31], § 2, for the coefficient of the leading approximation we obtain
$$
\begin{equation*}
\begin{aligned} \, D_{n,1,0} (\eta_3) &=\frac{1}{2\pi} \biggl[ \int_{-\infty}^{+\infty} \Phi^{-}_{n,0}(s_2)\,ds_2 \int_{\eta_3}^{+\infty} \exp (-z^2)\,dz \\ &\qquad +\int_{-\infty}^{+\infty} \Phi^{+}_{n,0}(s_2)\,ds_2 \int_{- \eta_3}^{+\infty} \exp (-z^2)\,dz \biggr], \\ D_{n,1,1} (\eta_3) &=\frac{\Lambda^{*}_{2,n+1}}{4\pi} \biggl[ \Lambda^{-}_{3,n+1,0}\int_{\eta_3}^{+\infty}\exp (-z^2)\,dz +\Lambda^{+}_{3,n+1,0}\int_{- \eta_3}^{+\infty}\exp (-z^2)\,dz \biggr]. \end{aligned}
\end{equation*}
\notag
$$
As a result, we arrive at the following statement. Lemma 3. The following asymptotic formula holds for all $N \geqslant 2$ as $\sigma \to +\infty$: § 4. Estimating ‘virtual’ sumsNoticing that, according to (3.2), (3.6) and (3.24), we have $\alpha/(2\beta)>\delta$, from Lemmas 2 and 3 we derive the outcome of the computational part of our work, an asymptotic relation for the solution of the problem under consideration:
$$
\begin{equation*}
u(x,t)=A_N(\eta,t) +\widetilde{W}_N(\eta,t,\mu) +O (\sigma^{-\delta N}), \qquad \sigma \to +\infty,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, A_N(\eta,t) \equiv t^{-1/2} \Omega^{(0)}_{1,0} (\eta) +\sum_{n=2}^{N} t^{-n/2} \bigl[ {S}_{n,0} (\eta) +{S}_{n,1} (\eta) \log t \bigr], \\ {S}_{n,l} (\eta) \equiv \Omega^{(0)}_{n,l} (\eta) +\Omega^{(1)}_{n,l} (\eta)\quad\text{and} \quad \widetilde{W}_N (\eta,t,\mu) \equiv V_{0,N}(\eta,t,\mu) +V_{1,N}(\eta,t,\mu). \end{gathered}
\end{equation*}
\notag
$$
It remains to make the final step in the use of the method of auxiliary parameter, to find an estimate for the sum of ‘virtual expressions’ involving the quantity $\mu$, which depends on an arbitrary parameter $\beta$. Noting that when we add (3.22) and (3.27), the terms containing the factor $\log\sigma$ annihilate, we find that
$$
\begin{equation*}
\widetilde{W}_N(\eta,t,\mu)=\sum_{r=-N+1}^{N-1} \mu^{r} \sum_{m \in \mathbb{Z}} t^{m/2} \bigl[ \widetilde{v}_{r,m,1} (\eta) +\widetilde{v}_{r,m,1} (\eta) \log t \bigr].
\end{equation*}
\notag
$$
Substituting $\mu=2^{-1} t^{-1/2} \sigma$ into the previous formulae we obtain
$$
\begin{equation}
u(x,t)=A_N(\eta,t) +W_N(\eta,t,\sigma) +O(\sigma^{-\delta N}), \qquad \sigma \to +\infty,
\end{equation}
\tag{4.1}
$$
where
$$
\begin{equation}
W_N(\eta,t,\sigma)=\sum_{r=-N+1}^{N-1}\sigma^{r} w_{N,r} (\eta,t),
\end{equation}
\tag{4.2}
$$
where the coefficients
$$
\begin{equation*}
w_{N,r} (\eta,t)=\sum_{m \in \mathbb{Z}} t^{m/2} \bigl[ v_{r,m,1} (\eta) +v_{r,m,1} (\eta) \log t \bigr]
\end{equation*}
\notag
$$
are finite sums and the sum (4.2) contains no term with index $r=0$. Before we turn directly to showing that the sum (4.2) is small, we present a very simple example giving one readily an insight in the question. Given a function $F$ and some $N>1$, assume that for all $\beta \in (0,1)$ we have
$$
\begin{equation*}
F(t)=t^{\beta} a(t) +O(t^{-\beta N}), \qquad t \to +\infty.
\end{equation*}
\notag
$$
Since $\beta$ is arbitrary, we have
$$
\begin{equation*}
F(t)=t^{\beta-\varepsilon} a(t) +O(t^{- (\beta-\varepsilon) N}), \qquad t \to +\infty,
\end{equation*}
\notag
$$
for an arbitrary $\varepsilon>0$ such that $0<\beta - \varepsilon<1$. It follows from these asymptotic relations that
$$
\begin{equation*}
t^{\beta} a(t)=O(t^{- (\beta-\varepsilon) N}), \qquad t \to +\infty.
\end{equation*}
\notag
$$
Thus, we obtain the estimate where $h \in (0,1)$ can be taken arbitrarily close to $1$. Now the idea to introduce an arbitrary parameter becomes clear: this enables us to get rid of the ‘virtual’ expressions arising by estimating them. In fact, if we construct as asymptotic formula for some integral by splitting it, some expressions produced by different intervals of integration annihilate, and in simple cases we can trace this by explicit calculations; however, it is intuitively clear after all that the asymptotics of a solution cannot depend on arbitrary quantities introduced artificially. Now we state and prove a result on an estimate for the sum $W_N (\eta,t,\sigma)$. Lemma 4. There exists $\varkappa>0$ such that for all $N \geqslant 2$ the sum (4.2) has the estimate Proof. Setting $F_N(\eta,t,\sigma)=u(x,t)-A_N(\eta,t)$, from (4.1) we obtain
$$
\begin{equation*}
F_N(\eta,t,\sigma)=W_N(\eta,t,\sigma) +O(\sigma^{-\delta N}), \qquad \sigma \to +\infty,
\end{equation*}
\notag
$$
where $W_N(\eta,t,\sigma)$ is the sum (4.2). Let $\theta=(|x|^2 +t)^{-1/2}$; then
$$
\begin{equation*}
F_N(\eta,t,\sigma)=\sum_{r=-N+1}^{2N-1} \theta^{-r \beta} w_{N,r} (\eta,t) +O(\theta^{\delta N \beta}), \qquad \theta \to +0.
\end{equation*}
\notag
$$
Since $\beta$ is an arbitrary parameter satisfying $0<\beta<1$, and the sum with respect to $r$ is finite, we can find $\varepsilon>0$ such that
$$
\begin{equation*}
F_N(\eta,t,\sigma)=\sum_{r=-N+1}^{2N-1} \theta^{-r (\beta-\varepsilon)} w_{N,r} (\eta,t) +O(\theta^{\delta N (\beta- \varepsilon)}), \qquad \theta \to +0,
\end{equation*}
\notag
$$
where $0<\beta - \varepsilon<1$ and all powers $\theta^{-r \beta}$ and $\theta^{-r (\beta-\varepsilon)}$ are distinct. Starting with the greatest power $\theta^{-(2N-1)\beta}$, we obtain in turn
$$
\begin{equation*}
\theta^{-r \beta} w_{N,r} (\eta,t)=O(\theta^{\delta N (\beta-\varepsilon)}), \qquad \theta \to +0,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\theta^{-r (\beta-\varepsilon)} w_{N,r} (\eta,t)=O(\theta^{\delta N (\beta- \varepsilon)}), \qquad \theta \to +0.
\end{equation*}
\notag
$$
Now, recalling that the sums with respect to $r$ contain no terms with $r=0$, we conclude that
$$
\begin{equation*}
W_N(\eta,t,\sigma)=O(\theta^{\delta N (\beta-\varepsilon)}), \qquad \theta \to +0.
\end{equation*}
\notag
$$
Hence (4.3) follows once we set $\varkappa=\delta (\beta-\varepsilon) / \beta$.
The proof is complete. § 5. Asymptotic formula for the solution and its applicationsCollecting the above results, now we can formulate the final conclusion of our work. Theorem. Let $\Lambda\colon \mathbb{R}^3 \to \mathbb{R}$ be a locally integrable function such that relations (2.1)–(2.5) hold. Then, as $t \to +\infty$, an asymptotic approximation to the solution of the heat equation (1.1) with initial condition $u(x,0)=\Lambda (x)$ in the unbounded domain $\{ (x,t)\colon t>|x|^{\alpha},\,1<\alpha<2\}$ has the following form: Proof. Substituting (3.20) and (3.26) into the right-hand side of the representation for $u(x,t)$ as a sum (3.1), from formulae (3.9), (3.18) and (3.21), providing an explicit expression for the leading approximation, and from the notation (3.2), (3.3) and (3.13), we obtain the ansatz from (5.1). Using (4.1) and the estimate (4.3) we obtain the following relation for the difference between the solution $u(x,t)$ and the asymptotic ansatz $A_N(\eta,t)$: where $h=\min \{ \delta \beta/2,\,\delta (\beta-\varepsilon) /2 \}>0$. In this way we arrive at the assertion of the theorem. Remark 1. The approximation formula (5.1) describes the asymptotic behaviour of the solution $u(x,t)$ in the sense of Erdélyi (see [26], Ch. II, § 2.1) with respect to the gauge sequence $\{ (|x|^2 +t)^{- h n}\}_{n=1}^{\infty}$, where the number $H$ depends as follows on the domain $T_{\alpha}$: Remark 2. If necessary, from the above calculations and formulae (3.4), (3.9), (3.10), (3.11), and others, in which the coefficients of powers of the time variable have an explicit functional form, after overcoming some technical difficulties we can also extract an explicit, although rather awkward expression for the functions $S_{n,l}$ which determine higher asymptotic approximations to the solution $u(x,t)$. Now we explain how our result in this paper can be applied to the investigation of the following Cauchy problem for the vector Burgers equation
$$
\begin{equation}
\frac{\partial \mathbf{v}}{\partial \vartheta} +(\mathbf{v}\nabla_{\xi}) \mathbf{v} =\varepsilon\triangle_{\xi} \mathbf{v}, \qquad \vartheta>0,
\end{equation}
\tag{5.2}
$$
with the initial condition
$$
\begin{equation}
\mathbf{v}({\xi},0,\varepsilon,\rho)= \rho \nabla_{\xi} \Psi \biggl( \frac{\xi}{\rho}\biggr), \qquad \xi=(\xi_1, \xi_2, \xi_3)\in\mathbb{R}^3,
\end{equation}
\tag{5.3}
$$
where $\mathbf{v}=(v_1, v_2, v_3)$ is a potential vector field, $\varepsilon$ and $\rho$ are independent small positive parameters and $\Psi (x)=\Psi (x_1,x_2,x_3)$ is a continuously differentiable function with bounded partial derivatives. If in (5.2) and (5.3) we make the natural change of variables (see [31], § 2, where another notation is used)
$$
\begin{equation}
\xi=\rho x, \qquad \vartheta=\frac{\rho^2 t}{\varepsilon},
\end{equation}
\tag{5.4}
$$
then for the components ${u_l}(x,t,\varepsilon,\rho)={v_l}(\xi,\theta,\varepsilon,\rho)$ we obtain the equation
$$
\begin{equation*}
\frac{\partial u_l}{\partial t}-\triangle_x u_l =- \frac{\rho}{\varepsilon} \sum_{m=1}^{3} u_m \,\frac{\partial u_l}{\partial x_m}.
\end{equation*}
\notag
$$
Then, as $\rho/ \varepsilon \to 0$, in the leading approximation we obtain the following initial problem:
$$
\begin{equation*}
\frac{\partial u_l}{\partial t}=\triangle_x u_l, \qquad u_l (x,0)=\frac{\partial \Psi (x)}{\partial x_l}.
\end{equation*}
\notag
$$
Now, to construct an asymptotic formula for the solution of (5.2), (5.3) by means of the matching method (see [5]) we must investigate the behaviour of the solution as $|x|+t \to +\infty$, because small values of the outer variables (5.4) correspond to large values of the inner (stretched) variables $(x,t)$. Summarizing, we can say that, apart from the obvious possible use in linear models for the physical processes of heat transfer, diffusion, and other phenomena, the knowledge of the asymptotic behaviour of the solution (1.3) and the analytic techniques of calculations presented above (using which one can in principle obtain an approximation of any prescribed accuracy to the solution, including in the case of the heat equations with another number of independent space variables, for a wide class of various asymptotic conditions imposed on the initial data) are also of interest for the asymptotic analysis of solutions of nonlinear equations and parabolic-type systems by means of the matching method. 
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\Bibitem{Zak24} |
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