Abstract
Full Text
MATHEMATICS
SU YU-CHEN
ASYMPTOTICS OF SOLUTIONS OF CERTAIN DEGENERATING QUASILINEAR HYPERBOLIC EQUATIONS OF SECOND ORDER
(Presented by Academician I. G. Petrovskii on 25 XI 1960)
In the present note an asymptotic expansion with respect to the small parameter \(\varepsilon\) is constructed for the solutions of the Cauchy problem and of a mixed problem for a quasilinear partial differential equation of hyperbolic type with small parameter \(\varepsilon\), and for a boundary-value problem in the case of an equation of elliptic type. For quasilinear ordinary differential equations with a small parameter, the Cauchy problem and the boundary-value problem were studied in work \((^1)\). In the present note we use the methods of \((^{1,2})\).
- The Cauchy problem. In the domain \(G:\{-\infty<x<+\infty,\ t\geqslant 0\}\) we consider the following problem, which we call problem \(A_\varepsilon\):
\[ L_\varepsilon u_\varepsilon \equiv \varepsilon \left(\partial^2 u_\varepsilon/\partial x^2-\partial^2 u_\varepsilon/\partial t^2\right) -\varphi(x,t,u_\varepsilon)\,\partial u_\varepsilon/\partial t+\psi(x,t,u_\varepsilon)=0; \tag{1} \]
\[ u_\varepsilon\big|_{t=0}=\alpha(x),\qquad \partial u_\varepsilon/\partial t\big|_{t=0}=\beta(x). \tag{2} \]
The degenerate problem \(A_0\) \((\varepsilon=0)\) consists in solving the equation
\[ L_0 w \equiv -\varphi(x,t,w)\,\partial w/\partial t+\psi(x,t,w)=0 \tag{3} \]
under the condition
\[ w\big|_{t=0}=\alpha(x). \tag{4} \]
We assume that \(\varphi(x,t,u)\geqslant a>0\) for all points
\((x,t,u)\in G\times\{-\infty<u<+\infty\}\). With the aid of the first iterative process we construct a sequence \(w_0,w_1,\ldots,w_n\) so that \(L_\varepsilon \bar w_n=O(\varepsilon^{n+1})\), where
\[ \bar w_n=\sum_{i=0}^{n}\varepsilon^i w_i. \]
Expanding \(\varphi(x,t,\bar w_n)\), \(\psi(x,t,\bar w_n)\) at the point \((x,t,w_0)\) in powers of \(\varepsilon\) and equating to zero the terms with like powers, we obtain
\[ -\varphi_0(x,t,w_0)\,\partial w_0/\partial t+\psi_0(x,t,w_0)=0,\qquad w_0\big|_{t=0}=\alpha(x); \tag{5} \]
\[ \varphi_0(x,t,w_0)\,\partial w_k/\partial t+ \left(\varphi_1\,\partial w_0/\partial t-\psi_1\right)w_k +\partial^2 w_{k-1}/\partial t^2 -\partial^2 w_{k-1}/\partial x^2 +U_k(x,t,w)=0 \quad (k=1,2,\ldots,n), \tag{6} \]
where \(U_k(x,t,w)\) depends only on \(w_i,\ \partial w_i/\partial t\ (i<k)\); \(\varphi_0,\psi_0,\varphi_1,\psi_1\) are the coefficients of the expansions of the functions \(\varphi,\psi\). The initial conditions for \(w_k\) will be determined below.
To find the asymptotics of the boundary layer
\[ \bar v_n=\sum_{i=0}^{n+1}\varepsilon^i v_i \]
we introduce a new variable \(y=t/\varepsilon\). We construct functions \(v_i\) \((i=0,1,\ldots,n+1)\) so that for the function \(\bar v_n\) we have
\[ L_\varepsilon(\bar w_n+\bar v_n)-L_\varepsilon \bar w_n=O(\varepsilon^{n+1}). \tag{7} \]
Then, in view of \(L_\varepsilon \bar w_n=O(\varepsilon^{n+1})\), we obtain that
\[ L_\varepsilon(\bar w_n+\bar v_n)=O(\varepsilon^{n+1}). \tag{8} \]
We rewrite equation (1) in the new variable
\[ \varepsilon L_\varepsilon u = \varepsilon^2\left(\frac{\partial^2 u}{\partial x^2} -\frac{1}{\varepsilon^2}\frac{\partial^2 u}{\partial y^2}\right) -\varphi(x,\varepsilon y,u)\frac{\partial u}{\partial y} +\varepsilon\psi(x,\varepsilon y,u). \]
We expand the function \(\bar w_n\) found above in a series in powers of \(t=\varepsilon y\). Substituting its expression in (7) and expanding the functions \(\varphi(x,\varepsilon y,\bar w_n)\), \(\psi(x,\varepsilon y,\bar w_n)\) in powers of \(\varepsilon\), we obtain successively
\[ \partial v_0^2/\partial y^2+\varphi_{00}(x,u_0+v_0)\,\partial v_0/\partial y=0; \tag{9} \]
\[ \partial^2 v_k/\partial y^2+\partial\left[\varphi_{00}(x,u_0+v_0)(u_k+v_k)-\varphi_{00}(x,u_0)u_k\right]/\partial y - \left(\bar E_k-E_k\right)=0,\quad (k=1,2,\ldots,n+1), \tag{10} \]
where \(\bar E_k-E_k\) depends only on \(u_i,\ v_i,\ \partial u_i/\partial y,\ \partial v_i/\partial y,\ \partial^2 v_{k-1}/\partial x^2\) \((i<k)\); \(u_k,\ \varphi_{00}\) are the coefficients of the expansions of \(\bar w_n,\ \varphi\).
Define \(v_k\) \((k=0,1,2,\ldots,n+1)\) as solutions of equations (9), (10) under the following conditions:
\[ \begin{gathered} v_0\big|_{y=0}=0,\qquad v_k\big|_{y=0}=-w_k\big|_{t=0}\quad (k=1,2,\ldots,n);\qquad v_{n+1}\big|_{y=0}=0;\\ \partial v_0/\partial y\big|_{y=0}=\varepsilon\beta(x);\qquad \partial v_k/\partial y\big|_{y=0}=-\partial w_{k-1}/\partial t\big|_{t=0}\quad (k=1,2,\ldots,n);\\ \partial v_{n+1}/\partial y\big|_{y=0}=-\partial w_n/\partial t\big|_{t=0}. \end{gathered} \tag{11} \]
For \(\varphi\geq a>0\) one can compute directly that
\[
v_0=\varepsilon\beta(x)(1-e^{-cat/\varepsilon})/ca,
\]
where \(c=c(x)\geq 1\). By induction one can prove that all the functions \(v_k\) \((k=1,2,\ldots,n+1)\) are functions of boundary-layer type.
Integrating equation (10) from \(1/\varepsilon\) to \(y\) and using the fact that
\[
\int_{1/\varepsilon}^{y}(\bar E_k-E_k)\,dy
=
B_k
\]
is a function of boundary-layer type, we obtain the initial conditions for \(w_k\):
\[
\varphi_{00}(x,u_0)w_k\big|_{t=0}
=
-B_k\big|_{t=0}-\partial w_{k-1}/\partial t\big|_{t=0}
\quad (k=1,2,\ldots,n).
\]
Theorem 1. Let \(Q\) be any triangle in the domain \(G\), bounded by characteristics of equation (1) and by a segment of the axis \(Ox\). If: 1) in \(G\times\{-\infty<u<+\infty\}\), \(\varphi,\psi\) have continuous derivatives of the form \(\partial^m/\partial x^{i_1}\partial t^{i_2}\partial u^{i_3}\) \((m=1,2,\ldots,2n+2)\); 2) \(\varphi(x,t,u)\geq a>0\) for all \((x,t,u)\in G\times\{-\infty<u<+\infty\}\); 3) \(\alpha(x)\) is continuously differentiable \(2n\) \((n=2,3,\ldots)\) times (for \(n=0,1\) it has continuous derivatives up to the third order); 4) \(\beta(x)\) has continuous derivatives up to order \((n-1)\) and \((n-2)\), respectively, for \(n=2k+1,\ n=2k+2\) \((k=2,3,\ldots)\) (for \(n=0,1,2,3,4\), \(\beta(x)\) is three times continuously differentiable); 5) \(\varphi''_{u},\psi''_{t},\psi''_{u}\) are bounded in \(Q\times\{-\infty<u<+\infty\}\), then: a) in \(Q\) there exists a unique solution \(u_\varepsilon(x,t)\) of problem (1), (2), continuous and having the continuous derivatives entering equation (1); b) the first and second iterative processes converge; c) the solution \(u_\varepsilon(x,t)\) admits the asymptotic representation
\[
u_\varepsilon=w_0+\varepsilon w_1+\cdots+\varepsilon^n w_n+v_0+\varepsilon v_1+\cdots+\varepsilon^{n+1}v_{n+1}+R_n,
\]
where the \(w_i\) are obtained by means of the first iterative process; the \(v_i\) are functions of boundary-layer type near \(t=0\), constructed by solving equations (9), (10) under conditions (11); the estimate \(\|R_n\|_{L_2(Q)}=O(\varepsilon^{n+1})\) holds everywhere in \(Q\).
2. Mixed problem
Problem \(A_\varepsilon\). In a certain rectangle
\[
R:\ \{0\leq x\leq l,\ 0\leq t\leq T\}
\]
consider the following problem:
\[ L_\varepsilon u_\varepsilon\equiv \varepsilon\left(\partial^2 u_\varepsilon/\partial x^2-\partial^2 u_\varepsilon/\partial t^2\right) -\varphi(x,t,u_\varepsilon)\,\partial u_\varepsilon/\partial t +\psi(x,t,u_\varepsilon)=0; \tag{12} \]
\[ u_\varepsilon\big|_{t=0}=\alpha(x),\qquad \partial u_\varepsilon/\partial t\big|_{t=0}=\beta(x);\qquad u_\varepsilon(0,t)=u_\varepsilon(l,t)=0; \]
\[
\varepsilon>0
\]
is a small parameter.
The degenerate problem \(A_0\) \((\varepsilon=0)\) consists in solving the problem
\[ L_0w\equiv-\varphi(x,t,w)\,\partial w/\partial t+\psi(x,t,w)=0;\qquad w\big|_{t=0}=\alpha(x). \tag{13} \]
In the present case the first iterative process is unchanged, with the only difference from the Cauchy problem case that the initial conditions are defined as follows:
\[
w_0\big|_{t=0}=\alpha(x),\qquad
w_i\big|_{t=0}=0\quad (i=1,2,\ldots,n).
\]
We shall construct functions of boundary-layer type first near the sides \(x=0,\ x=l\), then near \(t=0\). In this case a parabolic boundary layer appears near \(x=0\) and \(x=l\), i.e. the boundary-layer functions are constructed by solving parabolic equations. We shall restrict ourselves to the construction near \(x=0,\ 0\leq t\leq T\), because the boundary-layer function near \(x=l,\ 0\leq t\leq T\) can be obtained by a similar method.
Introduce a new variable \(y_1=x/\sqrt{\varepsilon}\). We split the operator for constructing the boundary-layer functions \(v_0^{(0)}, v_{1/2}^{(0)}, \ldots, v_{n+1/2}^{(0)}, v_{n+1}^{(0)}\) in the same way as we did for the Cauchy problem. Hence we have
\[ \partial^2 v_0^{(0)}/\partial y_1^2 -\varphi_{00}(t,\bar u_0+v_0^{(0)})\,\partial v_0^{(0)}/\partial t +\bigl[\varphi_{00}(t,\bar u_0) \]
\[ -\varphi_{00}(t,\bar u_0+v_0^{(0)})\bigr]\partial^2\bar u_0/\partial t +\psi_{00}(t,\bar u_0+v_0^{(0)})-\psi_{00}(t,\bar u_0)=0; \tag{14} \]
\[ v_0^{(0)}\big|_{t=0}=0,\qquad v_0^{(0)}\big|_{y_1=0}=-\bar u_0(0,t)=-w_0(0,t); \tag{15} \]
\[ \partial^2 v_{k/2}^{(0)}/\partial y_1^2 -\varphi_{00}(t,\bar u_0+v_0^{(0)})\,\partial v_{k/2}^{(0)}/\partial t -\bigl[\varphi_{01}(t,\bar u_0+v_0^{(0)})\,\partial(\bar u_0+v_0^{(0)})/\partial t+ \]
\[ +\psi_{01}(t,\bar u_0+v_0^{(0)})\bigr]v_{k/2}^{(0)} -F_{k/2}+\bar E_{k/2}-E_{k/2}=0 \quad (k=1,2,\ldots,2n+2); \tag{16} \]
\[ v_{i/2}^{(0)}\big|_{t=0}=0\quad (i=1,2,\ldots,2n+2);\qquad v_{i/2}^{(0)}\big|_{y_1=0}=0 \]
\[ (i=2k+1;\ k=0,1,\ldots,n); \tag{17} \]
\[ (\varepsilon v_k^{(0)}+\varepsilon \bar u_k+v_{k-1})\big|_{y_1=0}=0 \quad (k=1,2,\ldots,n+1), \]
where
\[ F_{k/2} =\bigl[\varphi_{00}(t,\bar u_0+v_0^{(0)})-\varphi_{00}(t,\bar u_0)\bigr]\partial\bar u_{k/2}/\partial t+ \]
\[ +\bigl[\varphi_{01}(t,\bar u_0+v_0^{(0)})\partial\bar u_0/\partial t +\varphi_{01}(t,\bar u_0+v_0^{(0)})\partial v_0^{(0)}/\partial t \]
\[ -\varphi_{01}(t,\bar u_0)\partial\bar u_0/\partial t -\psi_{01}(t,\bar u_0+v_0^{(0)})+\psi_{01}(t,\bar u_0)\bigr]\bar u_{k/2}; \]
\(\varphi_{00}, \varphi_{01}, \psi_{00}, \psi_{01}, \bar u_0, \bar u_{k/2}\) are the terms of the expansions of the functions \(\varphi,\psi,\bar w_n\) in powers of \(\sqrt{\varepsilon}\); \(\bar E_{k/2}-E_{k/2}\) depends only on \(\bar u_{i/2}, \partial \bar u_{i/2}/\partial t, v_{i/2}^{(0)}, \partial v_{i/2}^{(0)}/\partial t\) \((i<k)\) and \(\partial^2 v_{k/2-1}^{(0)}/\partial t^2\), and we obtain
\[ L_\varepsilon(\bar w_n+\bar v_n^{(0)})-L_\varepsilon \bar w_n=O(\varepsilon^{n+1}). \]
From \(L_\varepsilon \bar w_n=O(\varepsilon^{n+1})\) it follows that
\[ L_\varepsilon(\bar w_n+\bar v_n^{(0)})=O(\varepsilon^{n+1}). \]
It can be proved that \(v_0^{(0)}, v_{1/2}^{(0)}, \ldots, v_{n+1/2}^{(0)}, v_{n+1}^{(0)}\) have the character of a boundary layer.
Analogously we can construct the boundary-layer function
\[ \bar v_n^{(l)}=v_0^{(l)}+\sum_{i=1}^{2n+2}(\sqrt{\varepsilon})^{\,i}v_{i/2}^{(l)} \]
near \(x=l,\ 0\leq t\leq T\). Suppose that
\[ \tilde v_n^{(0)}=\psi_1(x/\delta)\,\bar v_n^{(0)},\qquad \tilde v_n^{(l)}=\psi_2((l-x)/\delta)\,\bar v_n^{(l)}, \]
where \(\psi_1,\psi_2\) are smoothing functions. Then everywhere in \(R\) one has
\[ L_\varepsilon(\bar w_n+\tilde v_n^{(0)})=O(\varepsilon^{n+1}),\qquad L_\varepsilon(\bar w_n+\tilde v_n^{(l)})=O(\varepsilon^{n+1}). \]
Now we pass to the construction of the sequence of boundary-layer functions near \(t=0,\ 0\leq x\leq l\). They are constructed in the same way as for the Cauchy problem, and we note only that in the present case, in a neighborhood of \(t=0\), we expand not only the functions \(w_0,w_1,\ldots,w_n\), but also the boundary-layer functions \(\tilde v_n^{(0)}, \tilde v_n^{(l)}\). Suppose that the expansions have the form
\[ \tilde v_n^{(0)}=U_0^{(0)}+\varepsilon U_1^{(0)}+\cdots +\varepsilon^{n+1}U_{n+1}^{(0)}+\varepsilon^{n+2}U_{n+2}^{(0)}, \]
\[ \tilde v_n^{(l)}=U_0^{(l)}+\varepsilon U_1^{(l)}+\cdots +\varepsilon^{n+1}U_{n+1}^{(l)} +\varepsilon^{n+2}U_{n+2}^{(l)}. \]
By virtue of the conditions (17) and the conditions of the theorem (conditions 1 and 7), the terms \(U_i^{(0)}, U_i^{(l)}\) \((i=0,1,2,\ldots,n+1)\) are equal to zero, while \(U_{n+2}^{(0)}, U_{n+2}^{(l)}\) are bounded. Therefore, when splitting the operator in a neighborhood of \(t=0\), one may regard either \(\bar v_n^{(0)}\) or \(\tilde v_n^{(l)}\) as included in \(O(\varepsilon^{n+2})\). Hence we have
\[ L_\varepsilon(\bar w_n+\tilde v_n^{(0)}+\bar v_n)-L_\varepsilon(\bar w_n+\tilde v_n^{(0)})=O(\varepsilon^{n+1}), \]
where \(\bar v_n=\psi_3(t/\delta)\bar v_n\); \(\bar v_n\) is the boundary-layer function constructed near \(t=0,\ 0\leq x\leq l\); \(\psi_3\) is a smoothing function. It is already known that
\[ L_\varepsilon(\bar w_n+\tilde v_n^{(0)})=O(\varepsilon^{n+1}). \]
From this it follows that
\[ L_\varepsilon(\bar w_n+\tilde v_n^{(0)}+\bar v_n)=O(\varepsilon^{n+1}). \]
Analogously we have
\[ L_\varepsilon(\bar w_n+\tilde v_n^{(l)}+\bar v_n)=O(\varepsilon^{n+1}). \]
By virtue of the property of the smoothing function, everywhere in \(R\) one has
\[ L_\varepsilon(\bar w_n+\tilde v_n^{(0)}+\tilde v_n^{(l)}+\bar v_n)=O(\varepsilon^{n+1}). \]
Theorem 2. If: 1) in \(R\times\{-\infty<u<+\infty\}\) the functions \(\varphi,\psi\) are continuous together with their derivatives of the form \(\partial^m/\partial x^{i_1}\partial t^{i_2}\partial u^{i_3}\) \((m=1,2,\ldots,3n+4)\); 2) on \([0,l]\) \(a(x)\) is continuously differentiable \(2n+3\) times; 3) \(\beta(x)\) is continuously differentiable \(n-1,\ n-2\) times, respectively, for \(n=2k+1,\)
\(n = 2k + 6\) \((k = 2, 3, \ldots)\) (for \(n = 0, 1, 2, 3, 4\) it is continuous together with its derivatives of the second and third orders on \([0, l]\); 4) \(\alpha(0)=\alpha(l)=\beta(0)=\beta(l)=0\), \(\alpha'(0)=\alpha'(l)=0\), \(\alpha''(0)=-\psi(0,0,0)\), \(\alpha''(l)=-\psi(l,0,0)\); 5) the functions \(\varphi(x,t,u)\), \(\psi(x,t,u)\) with respect to the arguments \(t,u\) have derivatives of the first and second orders, bounded in \(R_t \times \{-\infty < u < +\infty\}\), where \(0 \leq t^* \leq T\); 6) in \(R \times \{-\infty < u < +\infty\}\) \(\varphi(x,t,u) \geq a > 0\); 7) \(\partial^{m}\psi/dx^{i_1}dt^{i_2}=0\) at the points \((0,0,0)\), \((l,0,0)\) \((m=0,1,2,\ldots,n+2;\ i_1=0,1,2,4,6,\ldots)\); 8) \(\varphi'_u w'_t(0,t) > \psi'_u\) for \(x=0\) in \([0,T]\times\{-\infty<u<+\infty\}\), where \(w\) is the solution of the degenerate problem, then: a) there exists\({}^{4}\) a unique solution of problem (1), (2), having continuous derivatives entering the equation; b) problem (14), (15) has\({}^{5}\) a unique and bounded solution \(u\); c) the first and second iteration processes converge; d) the solution \(u_\varepsilon(x,t)\) admits the asymptotic representation
\[ u_\varepsilon=\bar w_n+\tilde v_n^{(0)}+\tilde v_n^{(l)}+\tilde v_n+R_n, \quad \text{where } \bar w_n=w_0+\sum_{i=1}^{n}\varepsilon^i w_i; \]
\(w_i\) are obtained by means of the first iteration process;
\[ \tilde v_n^{(0)}=\psi_1(x/\delta)\bar v_n^{(0)} =\psi_1(x/\delta)\left(v_0^{(0)}+\sum_{i=1}^{2n+2}(\sqrt{\varepsilon})^i v_{i/2}^{(0)}\right); \]
\(v_{i/2}^{(0)}\) are functions of the parabolic boundary layer near \(x=0,\ 0\leq t\leq T\);
\[ \tilde v_n^{(l)} =\psi_2((l-x)/\delta)\left(v_0^{(l)}+\sum_{i=1}^{2n+2}(\sqrt{\varepsilon})^i v_{i/2}^{(l)}\right); \]
\(v_{i/2}^{(l)}\) are near \(x=l,\ 0\leq t\leq T\);
\[ \tilde v_n=\psi_3(t/\delta)\bar v_n =\psi_3(t/\delta)\left(v_0+\sum_{i=1}^{n+1}\varepsilon^i v_i\right); \]
\(v_i\) are functions of the boundary layer near \(t=0,\ 0\leq x\leq l\); the estimate
\[
\|R_n\|_{L_2(R)}=O(\varepsilon^{n+1})
\]
holds everywhere in \(R\).
3. The case when the unperturbed equation is of the same order as the perturbed one.
A. An elliptic equation degenerates into a parabolic one. In the rectangle \(R: (0\leq x\leq l,\ 0\leq t\leq T)\) the following problem is considered:
\[ L_\varepsilon u_\varepsilon \equiv \varepsilon \frac{\partial^2 u_\varepsilon}{\partial t^2} +\frac{\partial^2 u_\varepsilon}{\partial x^2} -\varphi(x,t,u_\varepsilon)\frac{\partial u_\varepsilon}{\partial t} +\psi(x,t,u_\varepsilon)=0; \]
\[ u_\varepsilon|_{\Gamma}=0, \]
where \(\Gamma\) is the boundary of \(R\).
B. A hyperbolic equation degenerates into a parabolic one. In the domain
\[
G:\{-\infty<x<+\infty,\ t\geq 0\}
\]
we consider the Cauchy problem
\[ L_\varepsilon u_\varepsilon \equiv -\varepsilon \frac{\partial^2 u_\varepsilon}{\partial t^2} +\frac{\partial^2 u_\varepsilon}{\partial x^2} -\varphi(x,t,u_\varepsilon)\frac{\partial u_\varepsilon}{\partial t} +\psi(x,t,u_\varepsilon)=0; \]
\[ u_\varepsilon|_{t=0}=\alpha(x),\qquad \frac{\partial u_\varepsilon}{\partial t}\bigg|_{t=0}=\beta(x). \]
The asymptotics of the solutions of these problems is constructed in the same way as for the Cauchy problem in § 1, except that in the first iteration process \(w_0,w_1,\ldots,w_n\) are obtained as solutions of parabolic equations.
I take this opportunity to express my sincere gratitude to my scientific adviser L. A. Lyusternik for his work, advice, and systematic assistance in this work.
Received
1 XI 1960
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