UDC 517.9

MATHEMATICS

A. A. ARSEN'EV

ON THE BEHAVIOR OF A GENERALIZED SOLUTION OF A MIXED PROBLEM FOR THE WAVE EQUATION IN A DOMAIN CLOSE TO CLOSED

(Presented by Academician A. N. Tikhonov on 10 VII 1968)

Our purpose is to study the behavior of the solution of the mixed problem for the wave equation in the exterior of a domain \(\Omega\) of “trap” type (see Fig. 1), if the support of the initial data belongs to a set \(S\) inside the “trap.” The main result of our work is formulated in Theorem 2.

I. Let \(R_N\) be \(N\)-dimensional Euclidean space \((N \ge 3)\), \(\Omega\) an open bounded set in \(R_N\) (not necessarily simply connected), and \(B\) an operator given on the boundary of the set \(\overline{\Omega}\). By the symbol \(\Omega_1\) we denote the largest open connected set contained in \(R_N / \overline{\Omega}\) and containing infinitely distant points, and \(\Omega_2 = R_N \setminus (\Omega \cup \Omega_1)\). We assume that the domain \(\Omega\) and the operator \(B\) are such that:

1) the boundary of the set \(\overline{\Omega}\) belongs to the class \(C_{[1,\alpha]}\), \(\alpha > 0\)*;

2) \(\Omega_2\) is either empty, or \(\operatorname{mes}\Omega_2 > 0\) and \(\rho(\Omega_1,\Omega_2) > 0\);

3) the operator \(B\) has the form \((Bu)(x) = u(x)\,|\,x \in \operatorname{bd}\overline{\Omega}\), or
\[ (Bu)(x) = [\partial u/\partial n + \sigma(x)u](x)\,|\,x \in \operatorname{bd}\overline{\Omega}, \]
where the function \(\sigma(x) \ge 0\) is continuous.

Fig. 1

II. Consider the Cauchy problem for the heat equation.
\[ \partial u/\partial t = \Delta u,\quad x \in R_N/\overline{\Omega},\quad t > 0,\quad u(x,t) \in L^\infty;\quad (Bu)u(x,t)=0, \]
\[ u(x,+0)=u_0(x). \tag{1} \]

Let \(G(x,y,t)\) be the Green’s function of problem (1). We extend the function \(G(x,y,t)\) by putting \(G(x,y,t)=0\), if either \(x \in \Omega\), or \(y \in \Omega\), and put
\[ G_0(x,y,t)=(4\pi t)^{-N/2}\exp\bigl(-(x-y)^2/4t\bigr);\quad g(x,y,t)=G_0(x,y,t)-G(x,y,t). \]

The operator
\[ (G(t)f)(x)=\int G(x,y,t)f(y)\,dy \]
is a bounded operator in any \(L^p\), \(1 \le p \le \infty\). On functions from \(L^p\) we define the operator
\[ Hu=(\text{strong})\lim_{t\to +0} t^{-1}(E-G(t))u. \tag{2} \]

The operator \(H\), considered in \(L^2\), is self-adjoint, semibounded, and is an extension of the Laplace operator originally given on finite functions in \(R_N\setminus\Omega\) satisfying the boundary conditions \(B\).

III. The function \(u(x,k)\) is called a solution of the scattering problem for the operator \(H\) if \(u(x,k)\in L^\infty\) satisfies the equation
\[ Hu=\lambda u,\qquad \lambda=k^2,\qquad k\in R_N \tag{3} \]

* For the definition of this class see \((^1)\).

and is representable in the form

\[ u(x,k)=\exp(ikx)+\varphi(x,k), \tag{4} \]

where the function \(\varphi(x,k)\) satisfies the radiation conditions

\[ \varphi(x,k)=O\left(|x|^{(1-N)/2}\right);\qquad (\partial/\partial |x|-i\sqrt{\lambda})\varphi(x,k)=o\left(|x|^{(1-N)/2}\right), \quad |x|\to\infty . \tag{5} \]

Let

\[ T^{+}(\lambda)=\left[\lim_{\varepsilon\to +0}\left(e^{-(\lambda+i\varepsilon)t}-G_0(t)\right)^{-1}\right]g, \quad t>0,\quad \lambda>0 . \]

Theorem 1. 1. In order that the function \(u(x,k)=\exp(ikx)+\varphi(x,k)\) be a solution of the scattering problem, it is necessary and sufficient that the function \(\varphi(x,k)\) satisfy the equation

\[ \varphi=T^{+}(\lambda)(\exp(ikx)+\varphi). \tag{6} \]

1. Either the operator

\[ (E-T^{+}(\lambda))^{-1}\in [L^p\to L^p,\; 2N/(N-1)<p<\infty] \]

exists and equation (6) has a unique solution, or in \(L^p,\; 2N/(N-1)<p<\infty,\) there exists a nontrivial solution of the equation

\[ T^{+}(\lambda)\psi=\psi, \tag{7} \]

and the function \(\psi(x,\lambda)\) is a solution of equation (7) if and only if it satisfies the equation

\[ H\psi=\lambda\psi . \]

1. Those numbers \(\lambda\) for which there exists a nontrivial solution of equation (7) are eigenvalues of the discrete spectrum of the operator \(H\) and form at most a countable set \(\{\lambda_i\}\), while the functions \(\psi(x,\lambda_i)\) satisfying (7) belong to \(L^\infty\) and are nonzero only when \(x\in\Omega_2\). (It follows from this that if \(\Omega_2\) is empty, then \(\{\lambda_i\}\) is empty as well.)

With the aid of the functions \(u(x,k)\) and \(\psi(x,\lambda_i)\) one constructs the spectral resolution of the operator \(H\). The spectral function \(E(\lambda,H)\) of the operator \(H\) is an integral operator with kernel

\[ \theta(\lambda,x,y)=(2\pi)^{-N}\int_{|k|^2<\lambda}u^*(x,k)u(y,k)\,dk +\sum_{\lambda_i<\lambda}\psi(x,\lambda_i)\psi(y,\lambda_i). \]

IV. Definition. The function \(v(x,t)\) is called a generalized solution of the mixed problem for the wave equation if it is twice differentiable with respect to \(t\), for \(t>0\), strongly in the sense of the metric of \(L^2\), satisfies the equation

\[ \partial^2 v/\partial t^2=Hv \tag{8} \]

and the initial conditions

\[ \lim_{t\to +0}\|v(\,\cdot\,,t)-\beta\|_2=0,\qquad \lim_{t\to +0}\|v_t(\,\cdot\,,t)-\alpha\|_2=0. \tag{9} \]

Lemma 1. If

\[ \langle \alpha,\alpha\rangle+\langle \alpha,H\alpha\rangle-\langle \beta,\beta\rangle+\langle H\beta,H\beta\rangle=C_0<\infty, \]

then the function

\[ v(\,\cdot\,,t)=\int_0^\infty \cos\sqrt{\lambda}\,t\,d_\lambda E(\lambda,H)\beta +\int_0^\infty \frac{\sin\sqrt{\lambda}\,t}{\sqrt{\lambda}}\,d_\lambda E(\lambda,H)\alpha \]

is a generalized solution of problem (8)—(9).

V. Let \(\Omega^{(\tau)}\) and \(B^{(\tau)}\), \(0\le \tau\le 1\), be a family of domains and boundary operators depending on the parameter \(\tau\) and for each \(\tau\) satisfying

conditions 1)—3). Let \(G(\tau\mid x,y,t)\) be the corresponding Green functions of problem (1), and let \(H(\tau)\) be the operators defined by formula (2). Suppose that the domain \(\Omega\) and the operator \(B\) are such that, for some \(t=t_0>0\), the following additional conditions are satisfied:

4) There exist constants \(C_1, C_2\), and \(C_3\), independent of \(\tau\), such that for all \(\tau\in[0,1]\) the estimates

\[ |g(\tau\mid x,y,t_0)|\leq C_1\exp\left(-\frac{1}{8t_0}(x-y)^2\right),\qquad x,y\in R_N; \]

\[ \int |g(\tau\mid x,y,t_0)|\,dy\leq C_2\exp\left(-\frac{1}{8t_0}|x|^2\right),\qquad x\in R_N; \]

\[ \int |\nabla_x g(\tau\mid x,y,t_0)|\,dy\leq C_3\exp\left(-\frac{1}{8t_0}|x|^2\right),\qquad |x|\geq 2R_0, \]

hold, where \(R_0\) is the radius of a ball containing any of the sets \(\Omega^{(\tau)}\), \(0\leq\tau\leq 1\).

5)

\[ \lim_{|\tau_1-\tau_2|\to 0}\int_\Omega |g(\tau_1\mid x,y,t_0)-g(\tau_2\mid x,y,t_0)|\,dx\,dy=0, \qquad \tau_1,\tau_2\in[0,1]. \]

We give an example of a family satisfying conditions 1)—5).

\[ \Omega^{(\tau)}=\{x=(r\sin\theta\sin\varphi,\ r\sin\theta\cos\varphi,\ r\cos\theta); \]

\[ R_1<r<R_2,\quad 0\leq\varphi\leq 2\pi,\quad (1-\tau)\pi/2<\theta\leq\pi\}. \]

\[ (B^{(\tau)}u)(x)=u(x)\mid x\in \operatorname{gr}\overline{\Omega}^{(\tau)}. \]

For simplicity we shall assume, in addition, that for any \(\tau\in[0,1)\) the domain \(\Omega^{(\tau)}\) is simply connected, while for \(\tau=1\) it is not simply connected.

Let \(\{\lambda_i\}\) be the eigenvalues of the discrete spectrum of the operator \(H(1)\). If conditions 1)—5) are satisfied, then the following is valid.

Theorem 2. Let \(v(\tau\mid x,t)\) be the solution of problem (8)—(9),

\[ S\subset \Omega^{(1)}_2;\quad \inf_{0\leq\tau\leq 1}\rho(S,\Omega^{(\tau)})>0;\quad \alpha(x)\in W^1_2;\quad \beta(x)\in W^2_2; \]

the support of the functions \(\alpha(x)\) and \(\beta(x)\) is contained in the set \(S\), \(\varepsilon\) is an arbitrary positive number, and

\[ \lambda(\varepsilon)>8\varepsilon^{-2}\bigl[\|\alpha\|_{W^1_2}+\|\beta\|_{W^2_2}\bigr]. \]

Then there exist a \(\delta<1\) and a function \(\sigma(\tau)\to 0\) as \(\tau\to 1\), such that for all \(\tau\in[\delta,1)\), \(t>0\), the inequality

\[ \left\|v(\tau\mid,t)- \sum_{\lambda_j<\lambda(\varepsilon)} \left[ \int_{\lambda_j-\sigma(\tau)<\lambda<\lambda_j+\sigma(\tau)} \cos\sqrt{\lambda}\,t\,d_\lambda E(\lambda,H(\tau))\beta +\right.\right. \]

\[ \left.\left. +\int_{\lambda_j-\sigma(\tau)<\lambda<\lambda_j+\sigma(\tau)} \frac{\sin\sqrt{\lambda}\,t}{\sqrt{\lambda}}\,d_\lambda E(\lambda,H(\tau))\alpha \right]\right\|_2<\varepsilon \]

is satisfied.

The proof of Theorem 2 is based on Theorem 1 and Lemma 2.

Lemma 2. The operator \(T^{+}(\tau\mid\lambda)\) is continuous in the uniform operator topology of the space

\[ [L^p\to L^p,\ 2N/(N-1)<p<\infty] \]

with respect to the aggregate of variables \(\tau\in[0,1]\), \(\lambda\in[0,\infty)\).

It follows from Theorem 2 that if the domain \(\Omega^{(\tau)}\) is sufficiently close to a closed one (for this it is sufficient to require that the integral

\[ \int |g(1\mid x,y,t_0)-g(\tau\mid x,y,t_0)|\,dx\,dy \]

be sufficiently small), then in the continuous spectrum of the operator \(H(\tau)\) only intervals in neighborhoods of the discrete eigenvalues of the closed domain are essential.

Lemma 3. If \(\lambda_j\) is a simple eigenvalue of the discrete spectrum of the operator \(H(1)\), then the point \(\mu=1\) is a pole of first order for the function

\[ (\mu E-T^{+}(1\mid\lambda_j))^{-1}. \]

* We note that A. A. Samarskii had already proved earlier (2) that a perturbation of the spectrum of the Helmholtz operator depends on the change in the capacity of the boundary.

It is easy to see that the function \((\mu E - T^{+}(1|\lambda_j) - \Delta T)^{-1}\), for
\(\Delta T = T^{+}(\tau|\lambda) - T^{+}(1|\lambda_j)\) sufficiently small in a neighborhood of the point \(\mu = 1\), has a pole of first order. Expanding the function \((\mu E - T^{+}(1|\lambda_j) - \Delta T)^{-1}\) in powers of \(\Delta T\) in a neighborhood of this pole, it is easy to obtain the asymptotics of the function \(u(\tau|x,k)\) as \(\tau \to 0\) and \(|k^{2} - \lambda_j| \to 0\).

The author expresses deep gratitude to A. N. Tikhonov and A. A. Samarskii for discussing the results.

Moscow State University
named after M. V. Lomonosov

Received
5 VI 1968

References

¹ O. A. Ladyzhenskaya, N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type, “Nauka,” 1964. ² A. A. Samarskii, DAN, 63, 631 (1948).