UDC 517.947.43

MATHEMATICS

Yu. N. DROZHZHINOV

ON THE STABILIZATION OF THE SOLUTION OF THE HEAT-CONDUCTION EQUATION

(Presented by Academician A. N. Tikhonov on 4 VII 1968)

Consider the equation

\[ \partial u(x,t)/\partial t=\Delta u(x,t),\qquad x=[x_1,x_2,\ldots,x_n)\in R^n, \tag{1} \]

with the initial condition

\[ u(x,t)\big|_{t=0}=u_0(x). \tag{2} \]

In [4] it was proved that, in the class of bounded continuous initial functions, a sufficient condition for stabilization of the solution of the Cauchy problem (1)—(2) is the existence of a spherical limiting mean of \(u_0(x)\). In [5] it was proved that this condition is also necessary. In the present paper the generalized Cauchy problem (1)—(2) is considered in the class of initial functionals \(u_0(x)\in S'_{1/2}(R^n)\), and the behavior of the solution as \(t\to +\infty\) is studied. At the beginning of the article a generalized spherical limiting mean of the functional \(u_0(x)\) is defined, and it is shown that if it exists, then the solution stabilizes to it. Next it is proved that the generalized spherical limiting mean is a harmonic function. In the class of positive initial functionals \(u_0(x)\), the existence of a generalized spherical limiting mean is not only a sufficient but also a necessary condition for stabilization of the solution. In this class the solution can stabilize only to a constant. If the requirement of positivity of the initial functional is abandoned, the existence of a spherical limiting mean is no longer a necessary condition for stabilization of the solution.

For definitions and notation used in the paper, see [1, 2]. Let \(\Omega_\rho(x)\) be the ball in \(R^n\) of radius \(\rho\) with center at the point \(x\); \(c_n\) the volume of the unit ball in \(R^n\); \(\chi_\rho(x)\) the characteristic function of the ball, equal to \((c_n\rho^n)^{-1}\) when \(x\in \Omega_\rho(0)\), and to zero outside \(\Omega_\rho(0)\); \(u_0(x)\in S'_{1/2}(R^n)\); then, if there exists

\[ \lim_{\rho\to +\infty}\chi_\rho(x)*u_0(x)=M_{u_0(x)}(x) \quad \text{in } S'_{1/2}(R^n), \tag{3} \]

we shall call this limit the generalized spherical limiting mean of the functional \(u_0(x)\).

It follows easily from the results of [1] that the solution of the Cauchy problem (1)—(2) can be written in the form

\[ u(x,t)=\left(u_0(y),\frac{1}{(2\sqrt{\pi t})^n} \exp\left[-\frac{|x-y|^2}{4t}\right]\right),\qquad t>0, \tag{4} \]

and the space \(S'_{1/2}(R^n)\) is a class of existence and uniqueness.

Theorem 1. Let \(u_0(x)\) be continuous and satisfy the estimate: for every \(\varepsilon>0\) there exists \(C_\varepsilon\) such that

\[ |u_0(x)|\le C_\varepsilon-\exp[\varepsilon |x|^2], \tag{5} \]

and suppose that the spherical limiting mean of \(u_0(x)\) exists at the point \(x_0\), i.e.

\[ \lim_{\rho\to +\infty}(\chi_\rho(x)*u_0(x))(x_0) = \lim_{\rho\to +\infty}\frac{1}{c_n\rho^n} \int_{\Omega_\rho(0)} u_0(x_0+\xi)\,d\xi = M_{u_0(x)}(x_0). \tag{6} \]

Then the solution of the Cauchy problem (1)—(2) stabilizes at the point \(x_0\) to \(M_{u_0(x)}(x_0)\),

\[ \lim_{t\to+\infty} u(x_0,t)=M_{u_0(x)}(x_0). \]

Proof. In view of the linearity of the equation and of the operation of finding the spherical limit mean, without loss of generality we may assume \(M_{u_0(x)}(x_0)=0\). Since \(u_0(x)\) is continuous, has estimate (5), then \(u_0(x)\in S'_{1/2}(R^n)\), and, according to (4),

\[ u(x_0,t)=\int \frac{u_0(y)}{(2\sqrt{\pi t})^n} \exp\left[-\frac{|x_0-y|^2}{4t}\right]\,dy = \frac{1}{\pi^{n/2}}\int u_0(x_0+2\sqrt{t}\eta)\exp[-|\eta|^2]\,d\eta . \]

Passing to spherical coordinates and extracting the spherical mean, i.e. integrating by parts, where the nonintegral terms turn into zero, we have

\[ u(x_0,t)=\frac{2}{\pi^{n/2}}\int_0^\infty r^{n+1}\exp[-r^2]\bigl(\chi_{2\sqrt{t}\,r}(x)*u_0(x)\bigr)(x_0)\,dr . \]

Then, passing to the limit as \(t\to+\infty\) under the integral sign (and this is possible, since the integrand, by virtue of (6), is bounded on the half-axis) and taking into account that \(M_{u_0(x)}(x_0)=0\), we obtain the assertion of the theorem.

Suppose now that \(u_0(x)\in S'_{1/2}(R^n)\) and that \(M_{u_0(x)}(x)\) exists in \(S'_{1/2}(R^n)\); then the spherical limit mean of the solution of the generalized Cauchy problem (1)—(2) is itself the solution of the Cauchy problem for equation (1) with initial function equal to \(M_{u_0(x)}(x)\) (the generalized spherical limit mean of the original initial condition \(u_0(x)\)).

Indeed,

\[ \begin{aligned} \lim_{\rho\to+\infty}\chi_\rho(x)*u(x,t) &= \lim_{\rho\to+\infty} \frac{1}{c_n\rho^n} \int_{\Omega_\rho(0)} \left( u_0(y), \frac{1}{(2\sqrt{\pi t})^n} \exp\left[-\frac{|x-\xi-y|^2}{4t}\right] \right)\,d\xi \\ &= \lim_{\rho\to+\infty} \left( \chi_\rho(y)*u_0(y), \frac{1}{(2\sqrt{\pi t})^n} \exp\left[-\frac{|x-y|^2}{4t}\right] \right) \\ &= \left( M_{u_0(x)}(y), \frac{1}{(2\sqrt{\pi t})^n} \exp\left[-\frac{|x-y|^2}{4t}\right] \right) = M_{u(x,t)}(x). \end{aligned} \]

Comparing with (4), we derive the required assertion. Using Theorem 1 for \(u(x,t)\) with initial function \(u(x,t_0)\) for some \(t_0\), we obtain

\[ \lim_{t\to+\infty} u(x,t)=M_{u(x,t_0)}(x). \]

Since \(t_0\) may be chosen arbitrarily, \(M_{u(x,t)}(x)\) does not depend on \(t\) and satisfies equation (1); consequently, \(M_{u(x,t)}(x)\) is a harmonic function in the generalized sense, and then it is an ordinary harmonic function (see \((^2)\)). Thus, the following is true.

Theorem 2. Let \(u_0(x)\in S'_{1/2}(R^n)\) and let the generalized spherical limit mean (3) exist; then it is a harmonic function.

Corollary. In order that \(u_0(x)\in S'_{1/2}(R^n)\) be a harmonic function in \(R^n\), it is necessary and sufficient that

\[ \lim_{\rho\to+\infty}\chi_\rho(x)*u_0(x)=u_0(x). \tag{7} \]

Let \(u_0(x)\in S'_{1/2}(R^n)\) and be positive; then it is given by a positive measure

\[ (u_0(x),\varphi(x))=\int \varphi(x)\,d\mu_{u_0}(x), \qquad \varphi(x)\in S_{1/2}(R^n), \tag{8} \]

where the measure satisfies the condition

\[ \int \exp[-a|x|^2]\,d\mu_{u_0}(x)<+\infty \tag{9} \]

for any \(a>0\), see \((^3)\).

We note that the spherical limiting mean of a positive measure, if it exists, is a constant.

Indeed, let

\[ \lim_{\rho\to+\infty}\chi_\rho(x)*u_0(x) =\lim_{\rho\to+\infty}\frac{1}{c_n\rho^n}\int_{\Omega_\rho(x)} d\mu_{u_0}(\xi) =M_{u_0(x)}(x); \]

then \(M_{u_0(x)}(x)\) is a harmonic function. It is nonnegative and, by Liouville’s theorem, constant.

In the class of positive initial functionals from \(S'_{1/2}(R^n)\), the existence of the spherical limiting mean is not only sufficient for stabilization of the solution, but also necessary.

Theorem 3. Let \(u(x,t)\) be the solution of the Cauchy problem (1)—(2); \(u_0(x)\in S'_{1/2}(R^n)\) and positive; moreover,

\[ \lim_{t\to+\infty} u(x_0,t)=M, \tag{10} \]

then \(M_{u_0(x)}(x)\) exists and is equal to \(M\).

Proof. Without loss of generality, one may assume \(M=0\). From (4), (8), and (10) we have

\[ \lim_{t\to+\infty}u(x_0,t) =\lim_{t\to+\infty}\frac{1}{\pi^{n/2}}\int \frac{1}{(2\sqrt{t})^n} \exp\left[-\frac{|x_0-y|^2}{4t}\right]\,d\mu_{u_0}(y)=0. \tag{11} \]

The following estimate is valid:

\[ \frac{1}{c_n\rho^n}\int_{\Omega_\rho(x_0)} d\mu_{u_0}(\xi) \le \frac{e}{c_n\rho^n}\int_{\Omega_\rho(x_0)} \exp\left[-\frac{|x_0-\xi|^2}{4\rho^2}\right]\,d\mu_{u_0}(\xi) \le \frac{e}{c_n}\int \frac{1}{\rho^n} \exp\left[-\frac{|x_0-\xi|^2}{4\rho^2}\right]\,d\mu_{u_0}(\xi). \]

Passing in it to the limit as \(\rho\to+\infty\) and taking (11) into account, we obtain

\[ M_{u_0(x)}(x_0)=0. \]

Remark. Let \(u_0(x)\) be continuous, satisfy estimate (5), and be bounded below. Adding a sufficiently large constant, we make the initial condition positive; therefore Theorem 3 remains valid also for such functions \(u_0(x)\); this generalizes the corresponding results of the work \((^5)\).

Thus, in the class of positive initial functionals the solution of the generalized Cauchy problem (1)—(2) can stabilize only to a constant.

All the results obtained are valid for the generalized Cauchy problem for the ultraparabolic equation considered in work \((^1)\).

In conclusion I express my deep gratitude to V. S. Vladimirov, under whose guidance this work was carried out.

Received
27 VI 1968

CITED LITERATURE

\(^1\) V. S. Vladimirov, Yu. N. Drozhzhinov, Izv. AN SSSR, ser. matem., 31, 6, 1341 (1967).
\(^2\) I. M. Gel'fand, G. E. Shilov, Spaces of Basic and Generalized Functions, Moscow, 1958.
\(^3\) I. M. Gel'fand, N. Ya. Vilenkin, Some Applications of Harmonic Analysis, Moscow, 1961.
\(^4\) Yu. N. Drozhzhinov, DAN, 142, No. 1, 17 (1962).
\(^5\) V. D. Repnikov, S. D. Eidel'man, DAN, 167, No. 2, 298 (1966).