Abstract Generated abstract
This paper studies the large-time behavior of the energy of solutions to a mixed initial-boundary value problem for the wave equation in an unbounded exterior-type domain in Euclidean space. Using scattering eigenfunctions for the corresponding exterior domain, the solution and its energy are represented through a generalized Fourier transform, and asymptotic estimates are derived for time-dependent forcing terms with almost periodic structure. The main results give formulas for the limiting mean energy growth, or average power, for finite and certain absolutely convergent infinite sums of harmonic sources, expressed in terms of the transformed spatial profiles on frequency spheres. A final lemma interprets this spectral quantity as the asymptotic energy increment produced by a single monochromatic compactly supported source.
Full Text
UDC 517.947
MATHEMATICS
A. A. ARSEN’EV
ON THE BEHAVIOR OF THE ENERGY OF A SOLUTION OF THE WAVE EQUATION IN AN UNBOUNDED DOMAIN FOR LARGE TIMES
(Presented by Academician A. N. Tikhonov, February 3, 1970)
1. Let \(R_N\) be \(N\)-dimensional Euclidean space, \(N \geq 3\), and let \(\Omega\) be an open domain in \(R_N\) with boundary of class \(A^{(1,\alpha)}\), \(\alpha>0\). We shall assume that \(\Omega\) contains the exterior of some ball. In the cylinder \(\Omega \times [0,\infty)\) consider the mixed problem
\[ \frac{\partial^2 v}{\partial t^2}-\Delta v=F(x,t), \quad x\in \Omega,\quad t>0,\quad v(x,t)\in \dot W_2^1(\Omega), \]
\[ v(x,+0)=v_t(x,+0)=0,\quad x\in \Omega. \tag{1} \]
Let \(v(x,t)\) be the solution of problem (1); call the energy \(E(t)\) the integral
\[ E(t)=\int_\Omega \left(|v_t(x,t)|^2+|\nabla_x v(x,t)|^2\right)\,dx. \]
The main result of this work is formulated in Theorems 1 and 2; it consists in calculating the asymptotics of the function \(E(t)\) as \(t\to\infty\) for certain almost periodic functions \(F(x,t)\).
2. Let \(u(x,k)\) be the solution of the scattering problem for the domain \(R_N\setminus\Omega\):
\[ -\Delta u=\lambda u,\quad x\in\Omega,\quad \lambda=k^2,\quad k\in R_N,\quad u(x,k)\in L^\infty(\Omega)\cap C(\overline{\Omega}), \]
\[ u(x,k)=0,\quad x\in \operatorname{fr}\Omega,\quad u(x,k)=\exp(ikx)+\varphi(x,k), \]
\[ \varphi(x,k)=O\left(|x|^{(1-N)/2}\right),\quad \left(\frac{\partial}{\partial |x|}-i\sqrt{\lambda}\right)\varphi(x,k) =o\left(|x|^{(1-N)/2}\right),\quad |x|\to\infty. \]
The existence and the properties needed by us of the functions \(u(x,k)\) have been proved, for example, in the work \((^1)\).
We also note the following very simply proved lemma:
Lemma 1. For \(x\in\Omega\) and \(\beta>N/2\), the inequality
\[ \int (1+k^2)^{-\beta}|u(x,k)|^2\,dk \leq (\pi)^{N/2}\Gamma(\beta)^{-1}\Gamma\left(\beta-\frac{N}{2}\right). \]
3. Let \(F(x,t)\in L^2(\Omega)\) for all \(t\geq 0\). Put
\[ \widehat F(k,t)=\lim_{R\to\infty}\int_{|x|\leq R} u(x,k)F(x,t)\,dx. \tag{2} \]
(The existence of the limit (2) in the metric of \(L^2\) was proved in \((^1)\).)
Lemma 2. If \(F(x,t)\) is continuous in \(t\) in the metric of \(L^2(\Omega)\), then the solution \(v(x,t)\) of problem (1) exists, and its energy \(E(t)\) can be calculated by the formula
\[ E(t)=(2\pi)^{-N} \int \left[ \left|\int_0^t \widehat F(k,\tau)\sin |k|(t-\tau)\,d\tau\right|^2 + \right. \]
\[ \left. + \left|\int_0^t \widehat F(k,\tau)\cos |k|(t-\tau)\,d\tau\right|^2 \right]\,dk. \tag{3} \]
Proof. By the symbol \(\langle f,g\rangle\) we denote the scalar product in \(L^2(\Omega)\). Let us note that it is sufficient to prove (3) for functions \(F(x,t)\) from \(C_0^\infty(\Omega)\) that are continuous in \(t\) (since they are dense in \(L^2(\Omega)\)). For such functions \(F(x,t)\) in (1) the following equality is valid:
\[ E(t)=\langle v_t,v_t\rangle+\langle \nabla_x v,\nabla_x v\rangle =\langle v_t,v_t\rangle-\langle \nabla_x^2 v,v\rangle =\langle v_t,v_t\rangle+\langle F(,t)-v_{tt},v\rangle . \tag{4} \]
But
\[ v(x,t)=(2\pi)^{-N}\int u^*(x,k) \left[\int_0^t |k|^{-1}\sin |k|(t-\tau)\,\hat F(k,\tau)\,d\tau\right]\,dk . \tag{5} \]
By virtue of Lemma (1), the derivatives with respect to \(t\) of the function \(v(x,t)\) may be computed by differentiating (5) under the integral sign; substituting them into (4) and using Parseval’s equality
\[ \langle f,g\rangle=(2\pi)^{-N}\int \hat f^{*}(k)\hat g(k)\,dk, \]
we obtain (3).
- Let \(\omega_n,\ n=1,\ldots,\) be real numbers, \(\omega_n\ne 0\),
\[ \alpha_n(t,r)=\int_0^t \exp(i\omega_n\tau)\sin r(t-\tau)\,d\tau, \]
\[ \beta_n(t,r)=\int_1^t \exp(i\omega_n\tau)\cos r(t-\tau)\,d\tau, \]
\[ \gamma_{nm}(t,r)=\alpha_n(t,r)\alpha_m^{*}(t,r)+\beta_n(t,r)\beta_m^{*}(t,r). \]
Lemma 3. 1) If \(\omega_n\ne\omega_m\) and \(\varphi(r)\in L^1(0,\infty)\), then
\[ \lim_{t\to\infty} t^{-1}\int_0^\infty \gamma_{nm}(t,r)\varphi(r)\,dr=0. \]
2) If \(|\omega_n|\) is a Lebesgue point of the function \(\varphi(r)\in L^1(0,\infty)\), then
\[ \lim_{t\to\infty} t^{-1}\int_0^\infty \gamma_{nn}(t,r)\varphi(r)\,dr =\pi\varphi(|\omega_n|). \]
Let us also note that if \(f(x)\in L^2(\Omega)\), then for almost all \(r>0\) the function
\[ \theta(f)(r)=\int_{|n|=1} |\hat f(nr)|^2 r^{N-1}\,dn \]
is defined, and
\[ \int_0^\infty \theta(f)(r)\,dr =\int |\hat f(k)|^2\,dk =(2\pi)^N\int |f(x)|^2\,dx . \]
- We now consider the behavior of the energy \(E(t)\) in several typical cases. Let \(W(t)=E(t)/t\). Put in (1)
\[ F(x,t)=\sum_{n=1}^{M} f_n(x)\exp(i\omega_n t). \tag{6} \]
Theorem 1. If: 1) each of the functions \(f_n(x)\in L^2(\Omega)\); 2) \(\forall n:\ \omega_n\ne 0\) and \(|\omega_n|\) is a Lebesgue point of the function \(\theta(f)(r)\), then
\[ \lim_{t\to\infty} W(t)=(2\pi)^{-N}\pi\sum_{n=1}^{M}\theta(f_n)(|\omega_n|). \]
Let
\[ F(x,t)=f(x)\sum_{n=1}^{\infty} a_n \exp(i\omega_n t). \tag{7} \]
Theorem 2. If \(f(x)\in L^1\cap L^2\), the series \(\sum a_n\) converges absolutely, and at least one of the following two conditions is fulfilled: 1) all numbers \(\omega_n\) satisfy the inequality
\[ 0<a<|\omega_n|<b<\infty; \]
2) there exists a constant \(C<\infty\), independent of \(x\) and \(k\), such that
\[ |u(x,k)|<C,\qquad x\in \operatorname{supp} f(x),\qquad k\in R_N, \]
then
\[ \lim_{t\to\infty} W(t)=(2\pi)^{-N}\pi\sum_{n=1}^{\infty}|a_n|^2\theta(f)(|\omega_n|). \]
Theorems 1 and 2 are proved in the same way: it suffices to substitute (6) and (7) into (3) and use Lemma 3.
The physical meaning of the quantity \(\theta(f)(\omega)\) is explained by the following
Lemma 4. If \(F(x,t)=f(x)\exp(i\omega t)\), \(f(x)\in L^2(\Omega)\), and the support of \(f(x)\) is compact and lies in \(\Omega\), then
\[ \lim_{t\to\infty}(E(t+\Delta t)-E(t))=(2\pi)^{-N}\pi\theta(f)(|\omega|)\Delta t. \]
Theorems 1 and 2 may be regarded as assertions on the existence of an average limiting power for an almost periodic source of oscillations, and in this they are analogous to the well-known limiting-amplitude principle \((^2)\).
The author is deeply grateful to the participants of the seminar of V. A. Il’in and A. A. Samarskii for discussion of the results.
Moscow State University
named after M. V. Lomonosov
Received
7 I 1970
CITED LITERATURE
\(^1\) N. A. Shenk II, Arch. Rat. Mech. and Anal., 21, No. 2, 121 (1966). \(^2\) A. N. Tikhonov, A. A. Samarskii, ZhETF, 18, No. 2, 243 (1948).