MATHEMATICS

M. I. Vishik and Corresponding Member of the Academy of Sciences of the USSR L. A. Lyusternik

ON THE ASYMPTOTICS OF SOLUTIONS OF BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR DIFFERENTIAL EQUATIONS

The method of constructing asymptotics with respect to a small parameter $\varepsilon$ for solutions of boundary-value problems for linear differential equations $(^{1,2})$ also carries over to certain classes of nonlinear differential equations. We shall illustrate this with the example of the ordinary equation

\[ L_{\varepsilon}y \equiv \varepsilon y''+\varphi(x,y)y' - \psi(x,y)=0,\qquad y(0)=A,\quad y(1)=B. \tag{1} \]

The asymptotics of solutions of this problem in powers of the parameter $A$ was studied by Vazov $(^3)$. Consider the limiting equation

\[ L_0 w \equiv \varphi(x,w)w' - \psi(x,w)=0. \tag{2} \]

Suppose that some domain $D$ is covered by limiting curves $w=w(x)$ (i.e., by solutions of (2)). Then in $D$ the quantities
$w'(x)=\psi(x,y)/\varphi(x,y)=p(x,y)$ $(y=w(x))$ and
$w''(x)=p'_x+p'_y p=q(x,y)$ are functions of $(x,y)$.

Consider the case when in $D$

\[ \varphi(x,y)\ge \gamma>0, \tag{3} \]

which ensures, for solutions of (1), the appearance of a boundary layer in a neighborhood of $x=0$. We shall call a curve $y=u(x)$ a curve cutting from above (from below) for solutions $\widetilde y_{\varepsilon}(x)$ of equation (1) if, when $\widetilde y_{\varepsilon}(x)\le u(x)$ ($\widetilde y_{\varepsilon}(x)\ge u(x)$), the line $y=\widetilde y_{\varepsilon}(x)$ cannot have, inside the strip $0<x<1$, contact with $y=u(x)$. For this it is sufficient that

\[ L_{\varepsilon}u \equiv \varepsilon u''+\varphi(x,u)\,[u'-p(x,u)]<0\quad (>0) \]

for $0<x<1$. (For example, a segment $y=\mathrm{const}$ on which $\psi>0$ will cut from above, or a limiting curve $y=w(x)$ on which $q<0$.)

Suppose there exists in the strip $0\le x\le 1$ a domain satisfying the following conditions (we shall call it an $M$-domain): 1) it is covered by a field of limiting curves $y=w(x)$ connecting points of the lines $x=0$, $x=1$; 2) it is bounded by the segments $[A_0,A_1]$ and $[B_0,B_1]$ of the lines $x=0$ and $x=1$ and by the curves $r_1$ and $r_2$, cutting from above and from below; 3) the segment $[B_0,B_1]$ of the line $x=1$ contains a segment $[\overline B_0,\overline B_1]$ such that every limiting curve $y=w(x)$ issuing from a point $[1,B]$, $\overline B_0\le B\le \overline B_1$, passes, for $0\le x\le 1$, entirely inside this $M$-domain. It is easy to see that problem (1), for
$A_0<A<A_1$, $\overline B_0<B<\overline B_1$, and sufficiently small $\varepsilon>0$, has a solution
$y=\widetilde y_{\varepsilon}(x)$ passing inside the $M$-domain.

In what follows, unless otherwise stated, we shall assume that an $M$-domain exists and that the indicated inequalities are satisfied for the initial values $A$ and $B$, and also that (3) is fulfilled. Note that every solution (1) lying in the $M$-domain is bounded: $|\widetilde y_{\varepsilon}(x)|\le C$; hence it is easy to derive
$|\widetilde y'_{\varepsilon}(x)|\le C_1/\varepsilon$. In studying the solution $y=\widetilde y_{\varepsilon}(x)$ it is convenient to use-

the function \(z(x)=\widetilde y_\varepsilon'(x)-p(x,\widetilde y_\varepsilon(x))\). It satisfies the equation

\[ \varepsilon z'=-\varphi_1(x,\widetilde y_\varepsilon)z-\varepsilon q(x,\widetilde y_\varepsilon), \qquad \varphi_1=\varphi+\varepsilon p_y'. \tag{4} \]

Obviously, for sufficiently small \(\varepsilon\), \(\varphi_1>\gamma_1>0\). Therefore, solving the Cauchy problem for (4) with initial conditions at \(x=0\), we find that \(z(x)\) is the sum of an exponentially decreasing term of boundary-layer type and a term of order \(\varepsilon\). Hence it follows (for sufficiently small \(\varepsilon\)) that the solution \(\widetilde y_\varepsilon(x)\) of problem (1), for \(x>x_0\), where \(x_0=O(\varepsilon|\ln\varepsilon|)\), falls into the \(\varepsilon\)-neighborhood of the limiting curve \(y=w(x)\) \((w(1)=B)\). For \(0<x<x_0\) the difference \(v(x)=\widetilde y_\varepsilon(x)-w(x)\) is a function of boundary-layer type, with \(v(x_0)=O(\varepsilon)\), \(v'(x_0)=O(\varepsilon)\). For \(0<x<x_0\), \(v'(x)=O(1/\varepsilon)\), \(\varepsilon v''=O(1)\). Neglecting quantities of order \(O(1)\), one may write for the principal part \(v_0\) of this difference \(v\) the equation

\[ \varepsilon v_0''+\varphi(v_0+a)v_0'=0 \qquad \bigl(v_0(0)=A-a,\ a=w(0);\ \varphi(y)=\varphi(0,y)\bigr). \]

This equation is easily solved by quadratures and, as may be verified, for \(\varphi\geq\gamma>0\),

\[ v_0(x)=O(1)\exp(-\gamma x/\varepsilon),\qquad v_0'(x)=O(1/\varepsilon)\exp(-\gamma x/\varepsilon). \]

Theorem. If (3) holds in \(\overline M\) and \(\varphi(x,y)\) and \(\psi(x,y)\) have the corresponding smoothness, then the following asymptotic representations hold for the solutions \(\widetilde y_\varepsilon(x)\) of problem (1) (where \(A_0<A<A_1,\ B_0<B<B_1\)), lying in \(M\):

\[ \widetilde y_\varepsilon(x)=w_0(x)+v_0(x)+\widetilde R_0(x), \qquad \widetilde R_0(x)=O(\varepsilon|\ln\varepsilon|), \tag{5} \]

\[ \widetilde y_\varepsilon(x)= \left[w_0(x)+\sum_{s=1}^{n}\varepsilon^s w_s(x)\right] + \left[v_0(x)+\sum_{s=1}^{n+1}\varepsilon^s v_s\right] +R_n(x), \]

\[ R_n(x)=O(\varepsilon^{n+1}). \tag{6} \]

We present a scheme of the proof of formula (6). After separating out the principal terms \(w_0(x)+v_0(x)\) of the asymptotics, we are able to linearize the equations determining the higher terms of this asymptotics. The construction of the asymptotics (6) is analogous to the process described in \((1,2)\) for the linear case. We require that

\[ L_\varepsilon\overline w_n=O(\varepsilon^{n+1}),\qquad \overline w_n=\sum_{0}^{n}\varepsilon^s w_s,\qquad w_0(1)=\widetilde y(1),\quad w_s(1)=0\ \text{for }s\geq1. \tag{7} \]

Expanding at the point \((x,w_0(x))\) the functions \(\varphi(x,\overline w_n)\) and \(\psi(x,\overline w_n)\) in powers of \(\varepsilon\) and equating in (7) the terms with identical powers of \(\varepsilon\), we obtain:

\[ \varphi(x,w_0)w_0'-\psi(x,w_0)=0,\qquad w_0(1)=B; \]

\[ \varphi(x,w_0)w_k' + \bigl[\varphi_y'(x,w_0)w_0'-\psi_y'(x,w_0)\bigr]w_k = \Phi_k-w_{k-1}'', \qquad w_k(0)=0; \tag{8} \]

\(\Phi_k\) is a function of \(w_0,w_1,\ldots,w_{k-1},w_0',\ldots,w_{k-1}'\). Thus the \(w_k\) are successively determined by solving the linear equations (8); \(w_k,w_k',w_k''\) are functions bounded on \([0,1]\). To find the asymptotics of the boundary layer \(\overline v_n=v_0+\varepsilon v_1+\cdots+\varepsilon^{n+1}v_{n+1}\), we proceed from the equation

\[ L_\varepsilon(\overline v_n+\overline w_n)-L_\varepsilon(\overline w_n)=O(\varepsilon^{n+1}), \qquad (\overline v_n+\overline w_n)\big|_{x=0}=A, \tag{9} \]

from which, in view of (7), it follows that \(L_\varepsilon(\overline v_n+\overline w_n)=O(\varepsilon^{n+1})\). Introduce, as in \((1,2)\), the variable \(t=x/\varepsilon\); in this variable

\[ \varepsilon L_\varepsilon u\equiv u''(t)+\varphi(\varepsilon t,u)u_t'-\varepsilon\psi(\varepsilon t,u). \]

Let us expand the function found, \(\overline{w}_n=\sum_{s}^{n}\varepsilon^s w_s\), in a series in powers of \(x=\varepsilon t\); recalling that \(w(0)=a\), and grouping the terms according to powers of \(\varepsilon\), we obtain:

\[ \overline{w}_n(x)=\overline{w}_n(\varepsilon t)=a+\sum_{s}^{n}\varepsilon^s p_s(t)+O(\varepsilon^{n+1}), \tag{10} \]

where \(p_s(t)\) are polynomials in \(t\). Substituting this expression for \(\overline{w}_n\) into (9) and expanding the coefficients \(\varphi(\varepsilon t,\overline{w}_n)\) and \(\psi(\varepsilon t,\overline{w}_n)\) in powers of \(\varepsilon\), we successively obtain

\[ v_0''(t)+\varphi(a+v_0)v_0'(t)=0,\qquad v_0\big|_{t=0}=A-w_0(0)=A-a; \tag{11} \]

\[ v_k''(t)+\varphi(a+v_0)v_k'(t)+\varphi_y'(a+v_0)v_0'v_k(t)=\Psi_k \quad (k=0,1,\ldots,n+1); \tag{12} \]

\[ v_k(0)=-w_k(0)\quad \text{for } 1\le k\le n;\qquad v_{n+1}(0)=0, \]

where \(\Psi_k\) is a function of \(v_0,v_1,\ldots,v_{k-1}\), and also of \(p_s(t)\), i.e. of already found functions. Here we seek \(v_k\) as functions of boundary-layer type \((v|_{\infty}=0)\), which replaces the second boundary condition. It is proved by induction that all functions \(v_k\) \((k=0,1,\ldots,n+1)\) are functions of boundary-layer type.

To estimate \(R_n(x)\) in formula (6), note that, in view of (7), (9), denoting \(\widetilde{y}_\varepsilon=\widetilde{y}\), \(\widetilde{y}_1=\widetilde{y}-R_n(=\overline{v}_n+\overline{w}_n)\), we have:

\[ L_\varepsilon \widetilde{y}-L_\varepsilon \widetilde{y}_1=-L_\varepsilon \widetilde{y}_1=O(\varepsilon^{n+1}), \tag{13} \]

i.e.

\[ \varepsilon R_n''+\bigl[\varphi(x,\widetilde{y})\widetilde{y}'-\varphi(x,\widetilde{y}_1)\widetilde{y}_1'\bigr] -\bigl[\psi(x,\widetilde{y})-\psi(x,\widetilde{y}_1)\bigr] =O(\varepsilon^{n+1}). \tag{14} \]

If \(z_1=\widetilde{y}_1'-p(x,\widetilde{y}_1)\), we obtain for \(z_1\) an equation differing from (4) by the addition of \(O(\varepsilon^{n+1})\) to the right-hand side. Denoting \(\delta z=z-z_1\), we have:

\[ R_n'=\widetilde{y}'-\widetilde{y}_1'=\overline{p}_y R_n+\delta z;\qquad \overline{p}_y=p_y(x,\widetilde{y}_1+\theta R_n),\quad 0<\theta<1. \tag{15} \]

Further, solving equation (4) and the corresponding equation for \(z_1\), and noting that both \(z(0)\) and \(z_1(0)\) will be of order \(1/\varepsilon\), we obtain

\[ \delta z(x)=O(1/\varepsilon)\exp[-\gamma_1 x/\varepsilon]+O(\varepsilon)R_n(\theta x)+O(\varepsilon^{n+1}). \tag{16} \]

Considering (15) as a linear equation with respect to \(R_n\), solving it under the condition \(R_n(1)=0\), and using (16), we obtain:

\[ R_n(x)=O(1)\exp[-\gamma_1 x/\varepsilon]+O(\varepsilon)R_n(\xi)+O(\varepsilon^{n+1}), \tag{17} \]

where \(\xi\) is the point at which \(|R_n(x)|\) attains its maximum. If \(\xi\ge \varepsilon^{1-k}\), \(0<k<1\), then from (17) it follows that

\[ R_n(\xi)=O(1)\exp(-\gamma_1\varepsilon^{-k})+O(\varepsilon^{n+1}). \tag{18} \]

Let \(0<\xi<\varepsilon^{1-k}\). Then, integrating equation (14) from \(\varepsilon^{1-k}\) to \(\xi\), we obtain, since \(R_n'(\xi)=0\):

\[ -\varepsilon R_n'(\varepsilon^{1-k}) +\int_{\widetilde{y}(\varepsilon^{1-k})}^{\widetilde{y}(\xi)} \varphi(y)\,dy -\int_{\widetilde{y}_1(\varepsilon^{1-k})}^{\widetilde{y}_1(\xi)} \varphi(y)\,dy +\int_{\varepsilon^{1-k}}^{\xi}\Phi\,dx +O(\varepsilon^{n+1})=0, \tag{19} \]

where, as is easily verified, \(\Phi=O(1)R_n+O(x)R_n'\), and

\[ \int_{\varepsilon^{1-k}}^{\xi}\Phi\,dx =O(\varepsilon^{1-k})\,|R_n(\xi)|. \]

Further, \(\varepsilon R'_n(\varepsilon^{1-k})\), by virtue of (15) and (16), is equal to \(O(\varepsilon)|R_n(\xi)|+O(1)\exp[-\gamma_1\varepsilon^{-k}]+O(\varepsilon^{n+1})\). The difference of the integrals in (19) reduces to the integrals of \(\varphi(y)\) over the interval \((\tilde y_1(\varepsilon^{1-k}), \tilde y(\varepsilon^{1-k}))\), of length \(|R_n(\varepsilon^{1-k})|=O(1)\exp(-\gamma_1\varepsilon^{-k})+O(\varepsilon)|R_n(\xi)|+O(\varepsilon^{n+1})\), and over the interval \((\tilde y_1(\xi), \tilde y(\xi))\), of length \(|R_n(\xi)|\). We note that the integral over the second interval, which we denote by \(P\), exceeds \(\gamma|R_n(\xi)|\) in absolute value. The remaining terms in (19) give an expression of the form

\[ O(1)\exp(-\gamma_1\varepsilon^{-k})+O(\varepsilon^{1-k})|R_n(\xi)|+O(\varepsilon^{n+1}). \tag{20} \]

Hence, from (19), using the inequality \(|P|\ge \gamma |R_n(\xi)|,\ \gamma>0\), we obtain

\[ |R_n(\xi)|=O(\varepsilon^{n+1})+O(1)\exp(-\gamma_1\varepsilon^{-k}). \tag{21} \]

Since the second term in (21) is of higher order in comparison with the first, we obtain \(|R_n(\xi)|=O(\varepsilon^{n+1})\). The theorem is proved. Analogously one proves:

If in the \(M\)-domain, under the fulfillment of the conditions of the theorem, there exist two solutions \(\tilde{\tilde y}(x)\) and \(\tilde y(x)\) of problem (1), then

\[ \tilde{\tilde y}(x)-\tilde y(x)=O(\exp(-\gamma_1\varepsilon^{-k})), \]

where \(k\) is any fixed number between 0 and 1, i.e. uniqueness always holds up to a quantity exponentially small with respect to \(\varepsilon\).

Sufficient conditions for uniqueness in the \(M\)-domain will be the simultaneous fulfillment of the inequalities:

\[ \varphi>\gamma>0,\qquad p_y>0,\qquad (A-a)\varphi'_y\ge 0. \tag{22} \]

Remark. The constructions given above also carry over to equations of a more general form, for example \(\varepsilon y''+f(x,y,y')=0\), under restrictions corresponding to those indicated above. It should be noted that in this case the boundary layer may have a weaker character of variation.

Remarks on quasilinear partial differential equations. As was shown in \((^1,{}^2)\), for the linear case the construction of the boundary layer reduces to the solution of an ordinary equation in the direction transverse to the boundary. Constructions of the same type also carry over to some classes of quasilinear elliptic partial differential equations. For example, for the equation \(\varepsilon^2\Delta u-\psi(\rho,\varphi,u)=0\) under the conditions \(u|_{\rho=0}=f(\varphi)\) (\(\rho=0\) is the equation of the boundary \(\Gamma\)), \(\psi(\rho,\varphi,0)=0\), \(\psi'_u>\gamma^2>0\), the solution of the limiting equation (for \(\varepsilon=0\)) will be \(w\equiv 0\), while for the boundary layer in the first approximation we obtain the ordinary equation

\[ \varepsilon^2 A(\varphi)\frac{\partial^2 v}{\partial \rho^2}-\psi(0,\varphi,v)=0,\qquad v|_{\rho=0}=f(\varphi), \]

which is analogous to (1). For the following approximations one obtains linear equations and an expansion of type (6) holds. In the same way one can obtain an asymptotic expansion of the form (6), for example, for quasilinear elliptic equations \(L_\varepsilon u=h\) with a small parameter in the highest derivatives, if: 1) for arbitrarily small \(\varepsilon\) there exists and is unique a smooth solution \(u_\varepsilon(x,y)\) of the boundary-value problem for \(L_\varepsilon u=h\), which depends continuously (uniformly with respect to \(\varepsilon\)) on \(h\) (classes of such equations are easy to indicate, relying on the work of S. N. Bernstein \((^4)\)); 2) the solution \(w\) of the limiting equation (for \(\varepsilon=0\)) is sufficiently smooth; 3) the construction of the boundary layer reduces, for example, to an ordinary equation of the form (1).

Received
10 V 1958

REFERENCES

\(^1\) M. I. Vishik, L. A. Lyusternik, DAN, 113, No. 4 (1957).
\(^2\) M. I. Vishik, L. A. Lyusternik, Uspekhi Mat. Nauk, 12, issue 5, 3 (1957).
\(^3\) W. Wasow, Commun. Pure Appl. Math., 9, No. 1, 93 (1956).
\(^4\) S. N. Bernstein, Math. Ann., 69 (1910).