T. I. Zelenyak

On the Behavior as \(t\to\infty\) of Solutions of a Problem of S. L. Sobolev

(Presented by Academician S. L. Sobolev on 21 III 1961)

Consider the equation

\[ \frac{\partial^2}{\partial t^2}\left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) + \frac{\partial^2 u}{\partial y^2} =0 \tag{1} \]

and the conditions

\[ u\big|_{t=0}=u_0(x,y);\qquad \frac{\partial u}{\partial t}\bigg|_{t=0}=u_1(x,y);\qquad u(x,y,t)\big|_{\Gamma}=0, \tag{2} \]

where \(\Gamma\) is the boundary of a closed bounded domain \(\Omega\) in the \(xy\)-plane. In what follows it is always assumed that the equation of the contour \(\Gamma\) in polar coordinates has the form \(x=x(\varphi)\), \(y=y(\varphi)\), where \(x(\varphi)\), \(y(\varphi)\) are three times continuously differentiable, periodic functions with period \(2\pi\), satisfying the condition:

\[ x'y''-x''y' \geq q>0, \tag{3} \]

where \(q\) is a certain constant.

Lemma 1. There exists a function \(f_1(\varphi,\alpha)\), satisfying the equation
\(x(f_1)-\operatorname{ctg}\alpha\, y(f_1)=x(\varphi)-\operatorname{ctg}\alpha\, y(\varphi)\), defined for \(-\infty<\varphi<+\infty\), \(-\infty<\alpha<+\infty\), twice continuously differentiable, such that
\(\partial f_1(\varphi,\alpha)/\partial\varphi \leq -q_1<0\),
\(\partial f_1(\varphi,\alpha)/\partial\alpha \geq q_1>0\), where \(q_1\) is a certain constant.

Set \(\xi_n=f_n(\varphi,\alpha)=f_1(\xi_{n-1},(-1)^{n+1}\alpha)\), \(\xi_0=\varphi\).

Lemma 2. The function \(f_n(\varphi,\alpha)\) is a twice continuously differentiable function of \(\varphi\) and \(\alpha\), and moreover
\(|\partial f_k(\varphi,\alpha)/\partial\alpha|\geq q_1\),
\(|\partial f_k(\varphi,\alpha)/\partial\varphi|\geq q_1\),
while the function \(\bar f_{2N}(\varphi,\alpha)=f_{2N}(\varphi,\alpha)+4N\alpha-\varphi\) is periodic in \(\alpha\) and \(\varphi\).

Lemma 3. There exist functions \(\alpha_{N,k}(\varphi)\) satisfying the conditions:

a) \(f_{2N}(\varphi,\alpha_{N,k}(\varphi))=\varphi+2k\pi\);

b) \(\alpha_{N,k}(\varphi)\) are periodic functions with period \(2\pi\), twice continuously differentiable;

c) if
\(x(\psi)\pm \operatorname{ctg}\alpha_{N,k}(\psi)y(\psi) = x(\varphi)\pm \operatorname{ctg}\alpha_{N,k}(\psi)y(\varphi)\),
then \(\alpha_{N,k}(\varphi)=\alpha_{N,k}(\psi)\);

d) if \(k=2Nl+i\), then \(\alpha_{N,k}(\varphi)=\alpha_{N,i}(\varphi)-2l\pi\).

The functions \(f_k(\varphi,\alpha)\), for \(k=1,2,\ldots,2N\), are the polar angles of the vertices of a certain broken line inscribed in the contour \(\Gamma\). Let its vertices be \(T_k(\varphi,\alpha)\): \(T_0(\varphi)\) coincides with the point corresponding to the polar angle \(\varphi\). Denote by \(L_{k,k+1}(\varphi,\alpha)\) the straight line joining the points \(T_k(\varphi,\alpha)\), \(T_{k+1}(\varphi,\alpha)\). In \((^2)\) the existence is shown, for each \(N\), of an angle \(\alpha_N(\varphi)\) such that \(T_{2N}(\varphi,\alpha_N(\varphi))=T_0(\varphi)\), i.e. the broken line arrives at the point from which it started. This result is used in \((^2)\) for constructing, in certain domains, differential solutions of the operator \(A=\Delta^{-1}\partial^2/\partial y^2\), \(\Delta^{-1}\) restoring

function \(v\), \(v\big|_{\Gamma}=0\), from the given \(\Delta v=\partial^2 v/\partial x^2+\partial^2 v/\partial y^2\), whence follows the existence of non-almost-periodic solutions of problem (1)—(2). We must study the differential properties of the functions \(\alpha_{N,k}(\varphi)\), and therefore Lemmas 1—3 differ somewhat from those given in \((^2)\).

The proof of the following lemma, due to P. A. Aleksandryan, can be found in \((^2)\).

Lemma 4. Let \(a_1,a_2\) be such that \(T_{2N}(\varphi_1,a_1)=T_0(\varphi_1)\), \(T_{2N}(\varphi_2,a_2)=T_0(\varphi_2)\), \(|a_1-a_2|<\varepsilon\), \(|\varphi_1-\varphi_2|<\varepsilon\), \(\varepsilon>0\) and sufficiently small; then \(L_{k,k+1}(\varphi_1,a_1)\) and \(L_{k,k+1}(\varphi_2,a_2)\) do not intersect inside the domain \(\Omega\).

From Lemma 4 it follows that \(L_{k,k+1}(\varphi,\alpha_{N,i}(\varphi))\) and \(L_{k,k+1}(\varphi',\alpha_{N,i}(\varphi'))\) cannot intersect inside \(\Omega\) for any \(\varphi'\), \(T_0(\varphi)\ne T_0(\varphi')\). Using Lemmas 1—4, one can prove the following lemma:

Lemma 5. There exist functions \(\mu_{N,k,i}(x,y)\), \(N=2,3,\ldots,\infty\), \(i=1,2\), \(k=1,2,\ldots,2N\), such that:

a) \(\mu_{N,k,1}(x,y)\big|_{\Gamma}=\mu_{N,k,2}(x,y)\big|_{\Gamma}=\lambda_{N,k}(\varphi)=\operatorname{ctg}\alpha_{N,k}(\varphi)\);

b) \(\mu_{N,k,i}\,\partial\mu_{N,k,i}/\partial x=(-1)^i\,\partial\mu_{N,k,i}/\partial y\);

c) \(\mu_{N,k,i}(x,y)\) are uniquely defined at every point \(x,y\in\Omega\), and twice continuously differentiable at all points of \(\overline{\Omega}\), except for the points lying on the straight line \(y=y(\varphi')\), where \(\overline{\alpha}_{N,k}(\varphi')=0\), if such \(\varphi'\) exist.

Lemma 6. Any solution of equation (1) of the form \(u=v(x,y)e^{i\tau(x,y)t}\) with \(\partial\tau/\partial x\ne0\) can be obtained for
\(v(x,y)=\dfrac{\partial\mu}{\partial y}f(\mu)\), \(\tau=\pm\sqrt{\mu^2/(1+\mu^2)}\), where \(f\) is some twice differentiable function, \(\mu^2(\partial\mu/\partial x)^2=(\partial\mu/\partial y)^2\).

It is easy to see that the function

\[ u(x,y)=\frac{\partial}{\partial y} \int_{\mu_0}^{\mu(x,y)} f(\alpha)\exp\left[i\sqrt{\frac{\alpha^2}{1+\alpha^2}}\,t\right]\,d\alpha = \]

\[ =\pm\frac{\partial}{\partial x} \int_{\mu_0}^{\mu(x,y)} \alpha f(\alpha)\exp\left[i\sqrt{\frac{\alpha^2}{1+\alpha^2}}\right]\,d\alpha . \]

Theorem 1. The function

\[ u_{N,k}(x,y,t)= \int_{\mu_{N,k,1}(x,y)}^{\mu_{N,k,2}(x,y)} f(\alpha)\exp\left[i\sqrt{\frac{\alpha^2}{1+\alpha^2}}\,t\right]\,d\alpha, \tag{4} \]

where \(f(\alpha)\) is a differentiable finite function, is a solution of problem (1)—(2). If the domain \(\Omega\) is such that not all \(\lambda_{N,k}(\varphi)\) are constant, then \(u_{N,k}\) is not identically zero for at least one \(k\) when \(N=N^*\), \(\partial\lambda_{N^*,k}/\partial\varphi\ne0\), and for \(f(\alpha)\not\equiv0\) in the interval of integration.

It is easy to verify that \(u_{N,k}(x,y,t)=f_1(x+\mu_{N,k,1}y)+f_2(x-\mu_{N,k,2}y)\) for \(t=0\), where \(f_1\) and \(f_2\) are certain functions.

The same initial functions, for constant \(\lambda_{N,k}(\varphi)\) and naturally chosen \(f_1\) and \(f_2\), generate a periodic solution of the problem. If \(\lambda_{N,k}(\varphi)\) is not constant, solutions of the form (4) arise.

Theorem 2. Let \(\lambda_{N,k}(\varphi)\) be constant for all \(N\) and \(k\). Then the point part of the spectrum of the operator \(A\) is everywhere dense in the set formed by the whole spectrum of this operator, considered as an operator in the space of S. L. Sobolev \(W_2^1(\Omega)\).

Lemma 7. The function \(\alpha_{N,k}(\varphi)\) for each \(N\) and \(k\) is a continuous function of the domain, i.e., if \(x=x_\varepsilon(\varphi)\), \(y=y_\varepsilon(\varphi)\) and \(\rho_\varepsilon(\varphi)=\sqrt{x_\varepsilon^2+y_\varepsilon^2}\to\rho=\sqrt{x_0^2+y_0^2}\) as \(\varepsilon\to0\), then \(\alpha_{N,k}(\varphi,\varepsilon)\to\alpha_{N,k}(\varphi,0)\).

Consider the equation

\[ \sum_{i=0}^{M}\frac{\partial^i}{\partial t^i}\left(a_i\frac{\partial^2 u}{\partial x^2}+b_i\frac{\partial^2 u}{\partial y^2}\right)=0, \tag{5} \]

where \(a_i, b_i\) are constant real numbers. Let \(P(\lambda)=\sum a_i\lambda^i,\ Q(\lambda)=\sum b_i\lambda^i,\ \rho_i(\alpha)\) be the solutions of the equation

\[ P(\lambda)+\alpha^2 Q(\lambda)=0. \tag{6} \]

Theorem 3. The function

\[ u(x,y,t)= \int_{\mu_{N,k,1}(x,y)}^{\mu_{N,k,2}(x,y)} f(\alpha)\exp[\rho_i(\alpha)t]\,d\alpha, \tag{7} \]

where \(f(\alpha)\) is a differentiable finite function, is a twice continuously differentiable solution of equation (5), and moreover \(u|_{\Gamma}=0\).

The theorem is obvious in the case when all \(\rho_k(\alpha)\) are constant, i.e., when for some constant \(K\), \(P(\lambda)=KQ(\lambda)\).

In the case when \(P\) and \(Q\) have no common roots and the functions \(\rho_1(\alpha),\ldots,\rho_M(\alpha)\) are distinct, the solution of equation (5) with initial data

\[ \left.\frac{\partial^i u}{\partial t^i}\right|_{t=0}=0 \quad \text{for } i\ne j;\quad i=0,1,\ldots,j-1,j,\ldots,M-1; \]

\[ \left.\frac{\partial^j u}{\partial t^j}\right|_{t=0} = \int_{\mu_{N,k,1}(x,y)}^{\mu_{N,k,2}(x,y)} f(\alpha)\,d\alpha \tag{8} \]

can be represented in the form

\[ u(x,y,t)= \int_{\mu_{N,k,1}(x,y)}^{\mu_{N,k,2}(x,y)} \sum \varphi_i(\alpha)\exp[\rho_i(\alpha)t]\,d\alpha, \tag{9} \]

where \(\sum \varphi_i(\alpha)\exp[\rho_i(\alpha)t]\) is a differentiable function of its argument \(\alpha\).

As \(t\to\infty\), the functions \(u(x,y,t)\) defined by relation (4) decrease at infinity; \(\partial u/\partial x,\ \partial u/\partial y\), for fixed \(x,y\), are almost periodic functions of \(t\); \(\partial^2u/\partial x^2,\ \partial^2u/\partial y^2,\ \partial^2u/\partial x\,\partial y\) grow no faster than \(t\).

The asymptotics of the functions defined by equation (7) depend on the properties of the functions \(\rho_i(\alpha)\).

The author expresses deep gratitude to his supervisor, Academician S. L. Sobolev, for the formulation of the problem, attention, and assistance.

Received
15 II 1961

CITED LITERATURE

1. S. L. Sobolev, Izv. AN SSSR, Ser. Matem., 18, No. 1, 3 (1954).
2. R. A. Aleksandryan, Dissertation, Moscow State University, 1949.