Abstract
Full Text
V. P. MIKHAILOV
ON THE ASYMPTOTIC BEHAVIOR AS \(t\to\infty\) OF SOLUTIONS OF CERTAIN NONSTATIONARY BOUNDARY-VALUE PROBLEMS
(Presented by Academician I. G. Petrovskii, 4 XII 1964)
Consider in the \((n+1)\)-dimensional space \((t,x)=(t,x_1,\ldots,x_n)\) an \(n\)-dimensional cylindrical surface \(\Gamma\) with generator parallel to the axis \(Ot\) and with directrix an \((n-1)\)-dimensional, closed, \(N\)-times continuously differentiable surface \(S\), \(\Gamma=S\times(-\infty<t<\infty)\).
Let \(V_0\) be the domain of the plane \(t=0\) bounded by the surface \(S\) and containing infinity, and
\[ \Omega=V_0\times[0\le t<\infty), \]
\[ V_\tau=\Omega\cap(t=\tau),\quad \Omega_\tau=\Omega\cap(0<t<\tau),\quad 0\le \tau<\infty. \]
Denote by \(u(x,t)\) a solution in \(\Omega\) of the equation
\[ \partial^2 u/\partial t^2+(-\Delta)^m u=f(x)e^{i\omega t}, \tag{1} \]
satisfying the initial conditions
\[ u|_{t=0}=a(x); \tag{2} \]
\[ \partial u/\partial t|_{t=0}=b(x) \tag{3} \]
and the boundary conditions
\[ B_i(\partial/\partial x)u|_{\Gamma}=0,\qquad i=1,\ldots,m, \tag{4} \]
where the functions \(f(x)\), \(a(x)\), and \(b(x)\) are assumed to be finite and \(N\)-times continuously differentiable functions of their arguments; \(m\) is some integer, \(m\ge 1\); \(\omega\) is some real number; the operators \(B_i(\partial/\partial x)\), \(i=1,\ldots,m\), prescribed on the boundary \(\Gamma\), are assumed to be linearly independent and to have orders \(\mu_i<2m\), \(i=1,\ldots,m\). In addition, we shall assume symmetric the operator \(\Delta^m\), defined on functions in \(V_0\) that are \(2m\) times continuously differentiable and satisfy the boundary conditions (4). The function \(u(x,t)\), \((x,t)\in\Omega\), will be called a generalized solution of problem (1), (2), (3), (4) if \(u(x,t)\in W^{1,m}_{2,t,x}(\Omega_T)\) for any \(T>0\) and if, for the function \(u(x,t)\), the integral identity
\[ \int_{\Omega_T}u\left(\frac{\partial^2\bar v_T}{\partial t^2}+(-\Delta)^m\bar v_T\right)\,dx\,dt = \int_{\Omega_T} f(x)e^{i\omega t}\bar v_T(x,t)\,dx\,dt+ \]
\[ +\int_{V_0} b(x)\bar v_T(x,0)\,dx - \int_{V_0} a(x)\frac{\partial \bar v_T(x,0)}{\partial t}\,dx \tag{5} \]
holds for any function \(v_T(x,t)\in W^{2,2m}_{2,t,x}(\Omega_T)\) that vanishes for \(t=T\) together with its derivative with respect to \(t\) and satisfies on \(\Gamma\) the conditions (4).
Lemma 1. There exists a unique generalized solution of the problem (1), (2), (3), (4).
We shall denote equation (1) for \(f(x) \equiv 0\) by \((1_0)\), and condition (2) for \(a(x) \equiv 0\) by \((2_0)\).
The solution \(u(x,t)\) of the problem (1), (2), (3), (4) can be represented in the form
\[ u(x,t)=\frac{\partial u_a(x,t,0)}{\partial t} +\int_0^t u_{f e^{i\omega \tau}}(x,t,\tau)\,d\tau, \tag{6} \]
where \(u_b(x,t,\tau)\) is the solution in the half-space \(t>\tau\) of equation \((1_0)\), satisfying for \(t=\tau\) the conditions \((2_0)\) and (3), and for \(t\ge \tau\) the condition (4). This representation makes it possible to prove the following lemma.
Lemma 2. If the functions \(a(x)\), \(b(x)\), \(f(x)\) are sufficiently smooth, then the generalized solution of the problem (1), (2), (3), (4) is a sufficiently smooth function satisfying, in the classical sense, the conditions (1), (2), (3), and (4). Moreover, for the solution of the problem \((1_0)\), (2), (3), (4) the inequality
\[ \int_{V_t}\left(\left|\frac{\partial^s u}{\partial t^s}\right|^2 +\sum_{|\alpha|=ms}\left|D_x^\alpha u\right|^2\right)\,dx\le C_s \tag{7} \]
holds for \(s=1,\ldots,S\), where \(S\) is a sufficiently large number; \(C_s\) are certain constants independent of \(t\).
Lemma 3. The solution \(u(x,t)\) of the problem (1), (2), (3), (4) is bounded in \(\Omega\), together with all its derivatives of sufficiently high order.
This lemma follows immediately from Lemma 2 and Lemma 4.
Lemma 4. Let \(w(x)\) be a function of the \(n\) variables \(x=(x_1,\ldots,x_n)\), defined in the whole space \(R_n\) and belonging to \(W_2^{(n)}(R_n)\). Then
\[ \operatorname*{vrai\,max}_{x\in R_n}|w(x)|\le C\|w\|_{W_2^{(n)}(R_n)} \]
with a constant \(C\) independent of the function \(w(x)\).
The principal role in the subsequent considerations will be played by the following lemma.
Lemma 5. Let
\[ A=\sup_{x\in S}|x|. \]
Then for \(|x|\ge 2A\) the inequality
\[ |u(x,t)|+|\partial u/\partial t|+|\nabla_x u|+\cdots+|\nabla_x^{2m-1}u| \le Ct^{2\rho(m+2)/m(m+1)}/|x|^{2\rho/(m+1)}, \tag{8} \]
holds, where \(\nabla_x\) is the gradient with respect to \(x_1,\ldots,x_n\); \(\rho\) is any integer not exceeding \(N\) (\(N\) is the assumed smoothness of the right-hand sides in (1), (2), (3)).
We shall prove Lemma 5 for \(u(x,t)\) which is a solution of the problem \((1_0)\), \((2_0)\), (3), (4). The general result follows from this with the aid of (6).
Let \(2m>n\). Denote by \(U(x-\xi,t-\tau)\) the fundamental solution of equation (1) in the whole space. Multiplying equation \((1_0)\) by \(U(x-\xi,t-\tau)\) and integrating the resulting equality over the domain \(\Omega_t\), we obtain
\[ u(x,t)=\int_{V_0} b(\xi)U(x-\xi,t)\,d\xi+ \]
\[ +\sum_{i=1}^m\int_0^t d\tau\int_S B_i\left(\frac{\partial}{\partial \xi}\right)U(x-\xi,t-\tau)g_i(\xi,\tau)\,d\xi, \tag{9} \]
where \(g_i(x,t)=A_i(\partial/\partial x)u(x,t)\); \(A_i(\partial/\partial x)\) are certain differential operators on \(\Gamma\), whose orders do not exceed \(2m-1\). In any case, by Lemmas 4 and 5, \(g_i(x,t)\), \(i=1,\ldots,m\), are sufficiently smooth functions on \(\Gamma\), bounded together with their derivatives. From (9) one can obtain the estimate (8), if one uses the fact that, for any \(s=1,\ldots,n\), the following representation of the function \(U(x-\xi,t-\tau)\) holds:
\[ U(x-\xi,t-\tau)=\frac{1}{(2\pi)^n}\left[ \int_{|\alpha'|\geqslant 1} d\alpha'\int_{-\infty}^{+\infty} e^{i(x-\xi,\alpha)} \frac{\sin |\alpha|^m(t-\tau)}{|\alpha|^m}\,d\alpha_s + \int_{|\alpha'|\leqslant 1} d\alpha'\int_{l_2} e^{i(x-\xi,\alpha)} \frac{\sin |\alpha|^m(t-\tau)}{|\alpha|^m}\,d\alpha_s \right], \]
where \(\alpha'=(\alpha_1,\ldots,\alpha_{s-1},\alpha_{s+1},\ldots,\alpha_n)\); \(l_2\) is a contour in the complex \(\alpha_s\)-plane, consisting of the intervals of the real axis \((-\infty,-2)\), \((+2,+\infty)\) and the semicircle \(|\alpha_s|=2\), \(\operatorname{Im}\alpha_s\geqslant 0\) for \(x_s-\xi_s\geqslant 0\), and the semicircle \(|\alpha_s|=2\), \(\operatorname{Im}\alpha_s\leqslant 0\) for \(x_s-\xi_s\leqslant 0\).
If \(2m\leqslant n\), then one should first obtain, by the method described above, an estimate analogous to (8) for sufficiently high derivatives of \(u(x,t)\), and then, by standard devices, also the inequality (8).
Lemma 6. In the domain \(V_0\) there exists a unique solution of the elliptic equation
\[ (-\Delta)^m v-\omega^2 v=f(x), \tag{10} \]
satisfying the conditions (4) on \(S\) and the radiation conditions \((^1)\) at infinity. For the Green function of this problem one has the inequality
\[ |G(x,\xi)|\leqslant C/|x-\xi|^{(n-1)/2} \]
with a constant \(C\) depending only on the boundary \(S\) of the domain \(V_0\).
The purpose of this note is to prove the following theorem.
Theorem. If \(u(x,t)\) is a solution of the problem (1), (2), (3), (4), then there exists the limit
\[ \lim_{T\to\infty}\frac{1}{T}\int_0^T u(x,t)e^{-i\omega t}\,dt=v(x), \]
where \(v(x)\) is the solution of equation (10) discussed in Lemma 6.
This theorem is a weakened limiting-amplitude principle \((^2)\). For \(n=1\) for the second boundary-value problem this theorem was proved in \((^3)\), and for the first boundary-value problem \((m=1,\ n=2)\) in \((^4)\) (even for some domains with an infinite boundary)\(*\).
The function
\[ w(x,t)=\int_0^t u(x,\tau)e^{-i\omega\tau}\,d\tau \]
for any \(t>0\) is a solution of the equation
\[ \begin{aligned} (-\Delta)^m w-\omega^2 w &=t f(x)+b(x)-i\omega a(x)-u_t(x,t)e^{-i\omega t}-u(x,t)i\omega e^{-i\omega t} \\ &\equiv t f(x)+h(x,t), \end{aligned} \]
where \(h(x,t)\) is a function bounded in \(\Omega\) and satisfying the inequality
\[ \text{* Proof-correction note. Recently the author became acquainted with the note \((^5)\), in which similar questions are also considered.} \]
(8), since \(a(x)\) and \(b(x)\) are finite functions. According to Lemma 6,
\[ \frac{1}{t}w(x,t) = \frac{1}{t}\int_{0}^{t} u(x,\tau)e^{-i\omega\tau}\,d\tau = \int_{V_0} G(x,\xi)\left[f(\xi)+\frac{h(\xi,t)}{t}\right]\,d\xi = \]
\[ = v(x)+\frac{1}{t}\int_{V_0} G(x,\xi)h(\xi,t)\,d\xi, \tag{11} \]
where \(v(x)\) is the desired solution of equation (10). To complete the proof of the theorem it remains to show that the last term in (11) tends to zero as \(t\to\infty\). This is done with the aid of Lemmas 3, 5, and 6. Moreover, if one assumes that the number \(N\) is sufficiently large, then one can also give an estimate of this remainder term as \(t\to\infty\).
Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
26 XI 1964
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