MATHEMATICS

E. K. Isakova

ASYMPTOTICS OF THE SOLUTION OF A DIFFERENTIAL EQUATION OF PARABOLIC TYPE WITH A SMALL PARAMETER

(Presented by Academician S. L. Sobolev on 11 XII 1957)

1. Let us consider the behavior, as \(\varepsilon \to 0\) \((\varepsilon > 0)\), of the solution \(u^\varepsilon(x,t)\) of the following Cauchy problem:

\[ L_\varepsilon u^\varepsilon(x,t) = \varepsilon \sum A_{ij}(x,t)\frac{\partial^2 u^\varepsilon}{\partial x_i \partial x_j} + \sum B_j(x,t)\frac{\partial u^\varepsilon}{\partial x_i} + C(x,t)u^\varepsilon - \frac{\partial u^\varepsilon}{\partial t} =0, \tag{1} \]

\[ u^\varepsilon(x,t)\big|_{t=0}=\Psi(x), \tag{2} \]

where \(x=(x_1,\ldots,x_n)\) is a point of \(n\)-dimensional space \(E^n\); \(t\in[0,T]\); \(A_{ij}(x,t)\), \(B_i(x,t)\), \(C(x,t)\), \(i,j=1,2,\ldots,n\), are assumed to be bounded together with their derivatives up to order \(2n\) with respect to all \(x_i\), \(i=1,2,\ldots,n\); moreover, \(\partial A_{ij}/\partial t\), \(\partial B_j/\partial t\), \(i,j=1,2,\ldots,n\), exist and are bounded; at every point \((x,t)\in E^n\times[0,T]\),

\[ \sum A_{ij}(x,t)\xi_i\xi_j \ge \alpha \sum \xi_i^2,\quad \alpha>0. \]

The function \(\Psi(x)\), everywhere in \(E^n\), except for the points of a certain surface \(F(x)=0\), has bounded derivatives up to order \(2n\). On the surface \(F(x)=0\) the function \(\Psi(x)\) and all its derivatives up to order \(n\) have a discontinuity of the first kind (the behavior of derivatives of higher order on \(F(x)=0\) is of no interest to us). The surface \(F(x)=0\) has the same smoothness as the coefficients of equation (1); moreover, it is intersected by every straight line parallel to the \(x_1\)-axis, and in exactly one point.

Denote by \(v(x,t)\) the solution of problem (1)—(2) for \(\varepsilon=0\) (we shall call it problem \((1^0)\)—\((2^0)\)), and by \(l(x,t)\) the characteristic of equation \((1^0)\) passing through the point \((x,t)\). Then along each characteristic of equation \((1^0)\) passing through the surface \(F(x)=0\) (we shall call it a characteristic of discontinuity), the solution \(v(x,t)\) will be discontinuous. (We do not consider the convergence of the solution \(u^\varepsilon(x,t)\), as \(\varepsilon\to0\), of problem (1)—(2), and also of other problems to be considered below, to the solution \(v(x,t)\) of problem \((1^0)\)—\((2^0)\) at all points of continuity of the function \(v(x,t)\), since it was proved in \((^1)\) even for solutions of nonlinear parabolic equations.)

Definition. The principal, with respect to \(\varepsilon\) as \(\varepsilon\to0\), part of the difference \(u^\varepsilon(x,t)-v(x,t)\) will be called the “inner parabolic boundary layer.”

Let us find the inner parabolic boundary layer and the asymptotics with respect to \(\varepsilon\) for \(u^\varepsilon(x,t)\) in a neighborhood of any characteristic of discontinuity.

Replacing \(x\) and \(t\), respectively, by \(y\) and \(\tau\), where \(\tau\) is the length of the arc of the characteristic \(l(x,t)\) between the points \((y,0)\) and \((x,t)\), and \(y=(y_1,\ldots,y_n)\) are the coordinates of the point of intersection of \(l(x,t)\) with the plane \(t=0\), we obtain

\[ L_\varepsilon u^\varepsilon(y,\tau) \equiv \varepsilon \sum a_{ij}(y,\tau)\frac{\partial^2 u^\varepsilon}{\partial y_i \partial y_j} + c(y,\tau)u^\varepsilon - \frac{\partial u^\varepsilon}{\partial \tau} =0, \tag{1'} \]

\[ u^\varepsilon(y,\tau)\big|_{\tau=0}=\psi(y);\qquad y=(y_1,\ldots,y_n)\in E^n,\quad \tau\in[0,T_1]. \tag{2'} \]

In this case the surface \(F(x)=0\) passes into the surface \(f(y)=0\), on which \(\psi(y)\) is discontinuous together with all its derivatives up to order \(n\). In a neighborhood of \(f(y)=0\) introduce local coordinates \(\varphi\) and \(\rho\): \(\varphi=(\varphi_1,\ldots,\varphi_{n-1})\) are coordinates on the surface \(f(y)=0\), and \(\rho\) is the length of the normal at the corresponding point \((\varphi_1,\ldots,\varphi_{n-1})\). Then \((1')\)—\((2')\) passes into

\[ L_\varepsilon u^\varepsilon(\rho,\varphi,\tau)\equiv \overline L_\varepsilon u^\varepsilon+\sigma(\sqrt{\varepsilon},u^\varepsilon),\qquad u^\varepsilon(\rho,\varphi,\tau)\big|_{\tau=0}=\widetilde\psi(\rho,\varphi), \]

where

\[ \overline L_\varepsilon u^\varepsilon\equiv \varepsilon \widetilde a(\rho,\varphi,\tau)\frac{\partial^2 u^\varepsilon}{\partial \rho^2} +\widetilde c(\rho,\varphi,\tau)u^\varepsilon-\frac{\partial u^\varepsilon}{\partial \tau}, \qquad \sigma(\sqrt{\varepsilon},u^\varepsilon)=O(\sqrt{\varepsilon}). \]

Theorem 1. For the solution \(u^\varepsilon\) of problem (1)—(2), in a neighborhood of every characteristic of discontinuity the representation

\[ u^\varepsilon=z_0^\varepsilon+z_1^\varepsilon+\cdots+z_n^\varepsilon+O(\sqrt{\varepsilon^{\,n+1}}), \]

is valid, where \(z_k^\varepsilon,\ k=0,1,\ldots,n,\) are determined recursively:

\[ \overline L_\varepsilon z_0^\varepsilon=0,\qquad z_0^\varepsilon\big|_{\tau=0}=\widetilde\psi(\rho,\varphi),\qquad \overline L_\varepsilon z_k^\varepsilon=-\sigma(\sqrt{\varepsilon},z_{k-1}^\varepsilon),\qquad z_k^\varepsilon\big|_{\tau=0}=0, \]

\[ k=1,2,\ldots,n, \]

and \(z_0^\varepsilon\) has the asymptotics in \(\varepsilon\) found in (2), while \(z_k^\varepsilon=O(\sqrt{\varepsilon^{\,k}})\).

1. We proceed to the study of the behavior, as \(\varepsilon\to0\), of the solution \(u^\varepsilon(x,t)\) of equation (1), satisfying the conditions

\[ u^\varepsilon(x,t)\big|_{t=0}=\Psi_1(x);\qquad u^\varepsilon(x,t)\big|_{x_1=0}=\Psi_2(x_2,\ldots,x_n,t), \tag{3} \]

where \((x,t)\in D\ (0\le x_1<\infty,\ 0\le t\le T)\); \(\Psi_1(x)\) and \(\Psi_1(x_2,\ldots,x_n,t)\) are bounded and have bounded derivatives with respect to all \(x_i,\ i=1,2,\ldots,n,\) up to order \(2n\), and \(\Psi_1(0,x_2,\ldots,x_n)=\Psi_2(x_2,\ldots,x_n,0)\), and \(B_1(x,t)<0\). Then the solution \(v(x,t)\) of problem (1)—(3) for \(\varepsilon=0\) will be continuous, while its first derivatives on characteristics issuing from the \((n-1)\)-dimensional plane \(t=0,\ x_1=0\) (characteristics of discontinuity) may have discontinuities of the first kind. In this case, for the derivatives of the solution \(u^\varepsilon\) in a neighborhood of such characteristics, one can observe the phenomenon of an inner parabolic boundary layer. We shall find the asymptotics in \(\varepsilon\), in a neighborhood of every characteristic of discontinuity, for the solution \(u^\varepsilon\) and its first derivatives.

Under the change of \(x\) and \(t\) to \(y\) and \(\tau\), the plane \(x_1=0\) passes into a certain surface \(\Gamma\), which intersects \(y_1=0\) in the \((n-1)\)-dimensional plane \(\tau=0,\ y_1=0\), the domain \(D\) into the domain \(D'\), and condition (3) into the condition

\[ u^\varepsilon(y,\tau)\big|_{\tau=0}=\psi_1(y);\qquad u^\varepsilon(y,\tau)\big|_{\Gamma}=\psi_2. \tag{3'} \]

In this case the role of the surface of discontinuity will be played by \(\tau=0,\ y_1=0\), and

\[ \overline L_\varepsilon u^\varepsilon =\varepsilon a_{11}(y,\tau)u^\varepsilon_{y_1y_1} +c(y,\tau)u^\varepsilon-u^\varepsilon_\tau. \]

If \(z_0^\varepsilon\) is the solution of \(\overline L_\varepsilon z_0^\varepsilon=0\) under the condition \((3')\) (where \(y_2,y_3,\ldots,y_n\) are regarded as parameters), and \(z_k^\varepsilon\) are the solutions of

\[ L_\varepsilon z_k^\varepsilon=-\sigma(\sqrt{\varepsilon},z_{k-1}^\varepsilon),\qquad k=1,2,\ldots,n, \]

under homogeneous conditions \((3')\), then

\[ u^\varepsilon(y,\tau)=z_0^\varepsilon+z_1^\varepsilon+\cdots+z_n^\varepsilon+O(\sqrt{\varepsilon^{\,n+1}}), \tag{4} \]

where \(z_k^\varepsilon=O(\sqrt{\varepsilon^{\,k}}),\ k=1,2,\ldots,n\).

It remains to study the behavior of \(z_0^\varepsilon\) as \(\varepsilon\to0\), i.e., the behavior as \(\varepsilon\to0\) of the solution of problem \((1')\)—\((3')\) for \(n=1\) (we shall denote the solution of this problem by \(w^\varepsilon(y,\tau)\)). We shall assume the coefficients of equation \((1')\) to be given for all \(y\) and \(\tau>0\), and therefore \(v(y,\tau)\) may be regarded as defined for all \(y\) and \(\tau>0\). Denote by \(w_0^\varepsilon(y,\tau)\) the solution of \((1')\), satis-

satisfying the condition $\left. w_0^\varepsilon(y,\tau)\right|_{\tau=0}=\left. v(y,\tau)\right|_{\tau=0}$; then, as $\varepsilon\to0$, $w_0^\varepsilon(y,\tau)$ and $w^\varepsilon(y,\tau)$ converge in the domain $D'$ to the same limiting solution $v(y,\tau)$, and it can be shown that $w_0^\varepsilon(y,\tau)-w^\varepsilon(y,\tau)=O(\varepsilon)$ for $(y,\tau)\in D'$. Define $w_k^\varepsilon$, $k=1,2,\ldots,n$, as solutions of equation $(1')$ under the conditions $\left. w_k^\varepsilon\right|_{\tau=0}=\left. v_k\right|_{\tau=0}$, where $v_0(y,\tau)=v(y,\tau)$, while $v_k(y,\tau)$, $k=1,2,\ldots,n$, are solutions of $(1')$ for $\varepsilon=0$, satisfying respectively the conditions
$\left. v_k(y,\tau)\right|_{\substack{\tau=0\\ y_1\ge 0}}=0$,
$\left. v_k(y,\tau)\right|_{\Gamma}=\left. (v_{k-1}-w_{k-1}^\varepsilon)\right|_{\Gamma}$; then $w_k^\varepsilon(y,\tau)=O(\varepsilon^k)$, $k=1,2,\ldots,n$.

Theorem 2. For the solution $w^\varepsilon(y,\tau)$ and its derivative $\partial w^\varepsilon(y,\tau)/\partial y$ in a neighborhood of $y=0$, the following representations hold

\[ w^\varepsilon(y,\tau)=w_0^\varepsilon(y,\tau)+w_1^\varepsilon(y,\tau)+\ldots+w_n^\varepsilon(y,\tau)+O(\varepsilon^{n+1}) \quad \text{for } \tau\in[0,T_1], \]

\[ \frac{\partial w^\varepsilon(y,\tau)}{\partial y} = \frac{\partial w_0(y,\tau)}{\partial y} + \frac{\partial w_1^\varepsilon(y,\tau)}{\partial y} +\ldots+ \frac{\partial w_n^\varepsilon(y,\tau)}{\partial y} + O\!\left(\varepsilon^n\sqrt{\frac{\varepsilon}{\tau}}\right) \]

\[ \text{for } 0<\tau\le T_1, \]

where the solutions $w_k^\varepsilon(y,\tau)=O(\varepsilon^k)$ were found above, and moreover

\[ \frac{\partial w_k^\varepsilon(y,\tau)}{\partial y} = O\!\left(\frac{\varepsilon^k}{\sqrt{\varepsilon\tau}}\right), \quad k=1,2,\ldots,n, \]

while $\partial w_0/\partial y$ has the asymptotics in $\varepsilon$ found in $(2)$.

In the general case, when $n>1$, the following theorem is valid.

Theorem 3. For the solution $u^\varepsilon$ of problem $(1')$—$(3')$ (and hence also of problem $(1)$—$(3)$), in a neighborhood of every characteristic of discontinuity there is a representation of the form $(4)$, and $z_0^\varepsilon(y,\tau)$ satisfies Theorem 2.

3. Analogous results, with some modifications, can be obtained in studying the behavior, as $\varepsilon\to0$, of the solution $u^\varepsilon(x,t)$ of equation $(1)$ satisfying the conditions

\[ \left. u^\varepsilon(x,t)\right|_{t=0}=\Psi_1(x); \qquad \left. u^\varepsilon(x,t)\right|_{S}=\Psi_2, \quad x\in\Omega,\quad t\in[0,T], \tag{5} \]

where $\Omega$ is some domain in the space $x_1,\ldots,x_n$; $S$ is its boundary. For simplicity we restrict ourselves to the case $n=1$; then $(5)$ takes the form

\[ \left. u^\varepsilon(x,t)\right|_{t=0}=\Psi_1(x); \qquad \left. u^\varepsilon(x,t)\right|_{x=0}=\Psi_2(t); \tag{6} \]

\[ \left. u^\varepsilon(x,t)\right|_{x=1}=\Psi_3(t), \tag{7} \]

where $x\in[0,1]$, $t\in[0,T]$; $\Psi_1(x)$ and $\Psi_2(t)$ are $n$ times continuously differentiable and bounded together with all their derivatives; $\Psi_3(t)$ is continuous and bounded; $\Psi_3(0)=\Psi_1(1)$; $\Psi_1(0)=\Psi_2(0)=0$; the coefficient $B_1(x,t)<0$. Let $v(x,t)$ be the solution of $(1)$—$(6)$ for $\varepsilon=0$. Then, along the characteristic $l(0,0)$, the first derivatives of the solution $v(x,t)$ may have a discontinuity of the first kind. In this case, for the derivatives of the solution $u^\varepsilon(x,t)$ in a neighborhood of $l(0,0)$, the phenomenon of an internal parabolic boundary layer is observed. Moreover, since $v(x,t)$ does not satisfy $(7)$, in a neighborhood of $x=1$ for the solution $u^\varepsilon(x,t)$ there is also another phenomenon—the phenomenon of an external boundary layer $(^3)$.

Let us find the asymptotics in $\varepsilon$ for the solution $u^\varepsilon(x,t)$ and its first derivatives in a neighborhood of $l(0,0)$. We shall assume $\Psi_1(x)$ and $\Psi_2(t)$ to be given, respectively, for all $x\ge0$ and $t\ge0$. Denote by $\bar u^\varepsilon(x,t)$ the solution of $(1)$—$(6)$ for $x\ge0$, $t\in[0,T]$; then, as $\varepsilon\to0$, $\bar u^\varepsilon(x,t)$ and $u^\varepsilon(x,t)$ converge to $v(x,t)$ for $0\le x<1$. It turns out that one can construct a function

$w^\varepsilon(x,t)$, which cancels the discrepancy in the boundary conditions at $x=1$ between $\overline{u}^{\,\varepsilon}(x,t)$ and $u^\varepsilon(x,t)$ (i.e., $(\overline{u}^{\,\varepsilon}(x,t)-u^\varepsilon(x,t))|_{x=1}$), and
\[ u^\varepsilon(x,t)=\overline{u}^{\,\varepsilon}(x,t)+w^\varepsilon(x,t)+O(\varepsilon^{n+1}) \]
for $x\in[0,1]$, $t\in[0,T]$. The function $w^\varepsilon(x,t)$ has the form of an external boundary layer and is constructed by the method developed by L. A. Lyusternik and M. I. Vishik$^{3}$. In this case the corresponding asymptotics for the first derivatives of the solution $u^\varepsilon(x,t)$ in a neighborhood of $l(0,0)$, outside certain neighborhoods of the origin and of the point of intersection of $l(0,0)$ with $x=1$, is obtained by differentiating only $\overline{u}^{\,\varepsilon}(x,t)$.

In the case $n>1$, analogous arguments are carried out.

Moscow State University
named after M. V. Lomonosov

Received
9 XII 1957

REFERENCES

$^{1}$ O. A. Oleinik, Uspekhi Mat. Nauk, 10, no. 3 (65), 229 (1955).
$^{2}$ E. K. Isakova, DAN, 117, no. 6 (1957).
$^{3}$ M. I. Vishik, L. A. Lyusternik, Uspekhi Mat. Nauk, 12, 5, 77 (1957).