Mathematics

V. P. MIKHAILOV

ON THE STABILIZATION OF THE SOLUTION OF THE CAUCHY PROBLEM FOR THE HEAT EQUATION

(Presented by Academician S. L. Sobolev on 12 V 1969)

The purpose of this note is to establish necessary and sufficient conditions under which the solution of the Cauchy problem for the heat equation stabilizes as (t \to \infty). Let (u(x,t)), (x=(x_1,\ldots,x_n)), be a solution of the equation

[
u_t=\Delta u,\qquad x\in R_n=(-\infty<x_i<\infty,\ i=1,\ldots,n),
\tag{1}
]

satisfying the initial condition

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

We shall assume that (f(x)\in C(R_n)\cap T^2(R_n)), i.e. (f) is continuous and satisfies the condition: for every (\varepsilon>0) there exists (C(\varepsilon)>0) such that (|f(x)|\le C(\varepsilon)e^{\varepsilon |x|^2}) for all (x\in R_n). It is known (\left({}^{1}\right)) that for (f\in T^2(R_n)\cap C(R_n)) the solution (u(x,t)) of problem (1)—(2) exists for all (t\in(0,\infty)), is unique, and is represented in the form of the Poisson integral

[
u(x,t)=\frac{1}{(2\sqrt{\pi t})^n}\int_{R_n} f(\xi)e^{-|x-\xi|^2/4t}\,d\xi \equiv \bigl(\mathfrak{B}^{(2)}_t f\bigr)(x).
\tag{3}
]

By stabilization of the solution (u(x,t)) as (t\to\infty) we shall, in different places of this note, mean I or II.

I. The existence of a limit uniform with respect to (x\in R_n),

[
\lim_{t\to\infty} u(x,t)=A(x).
\tag{4}
]

II. The existence of the limit (4), uniform on every compact set (K\subset R_n).

Together with these two types of stabilization we shall also automatically consider stabilization of the following kind:

III. The limit (4) exists for every (x\in R_n), since stabilization in the sense III follows from stabilization in the sense II.

A criterion for pointwise stabilization of the solution of problem (1)—(2) in the case of bounded (f(x)) is due to N. Wiener (\left({}^{2}\right)) (see also the works (\left({}^{3,4}\right))) and consists in the fact that a necessary and sufficient condition for stabilization, in the sense III, of the solution of problem (1), (2) is the existence, as (R\to\infty), at each point (x\in R_n), of the limit of the expression

[
\bigl(S^1_R f\bigr)(x)=\frac{n}{\omega_n R^n}\int_{|x-y|\le R} f(y)\,dy,
\tag{5}
]

(\omega_n=2\pi^{n/2}/\Gamma(n/2)), equal to (A(x)). (We note that in this case (A(x)\equiv \mathrm{const}).) Recently, in (\left({}^{5}\right)), the same result was established, but already for the case of semibounded (f(x)) (either (f(x)\ge M), or (f(x)\le M) for all (x\in R_n), for some constant (M)). In this case (A(x)) is also identically equal to a constant.

Before formulating the result of the article, it is convenient to introduce some concepts. Let

[
(F_\rho f)(x)=\frac{n}{\omega_n}\int_{|x-y|=\rho} f(y)\,dy
=\frac{n\rho^{\,n-1}}{\omega_n}\int_{|\omega|=1} f(x+\rho\omega)\,d\omega
\tag{6}
]

(i.e., on (C(R_n)) an operator (F_\rho) is defined which, obviously, maps (C(R_n)) into (C(R_n))). In terms of the operator (F_\rho), the spherical mean (5) can be written as follows:

[
(S_R^1 f)(x)=\frac{1}{R^n}\int_0^R (F_\rho f)(x)\,d\rho .
]

For the function ((F_\rho f)(x)) one can define Cesàro means with respect to (\rho) of arbitrary order (\alpha>0). With their aid we construct the operators

[
(S_R^\alpha f)(x)=\frac{1}{nB(\alpha,n)R^{n+\alpha-1}}
\int_0^R (R-\rho)^{\alpha-1}(F_\rho f)(x)\,d\rho
\tag{7}
]

for all (\alpha>0,\ R>0). Note that the domain of definition of the operators (S_R^\alpha) is all of (C(R_n)), without any restrictions on the growth of (f(x)). We shall also introduce the scale of operators

[
(B_R^\sigma f)(x)=
\frac{\sigma}{\Gamma(n/\sigma)(4R)^{n/\sigma}}
\int_0^\infty e^{-\rho^\sigma/4R}(F_\rho f)(x)\,d\rho;
\tag{8}
]

(R>0,\ \sigma>0). The operators (B_R^\sigma) are no longer defined on all functions in (C(R_n)), but only on (C(R_n)\cap T^\sigma(R_n)), where the set (T^\sigma(R_n)) consists of those functions which have the following property: for any (\varepsilon>0) there exists a constant (C(\varepsilon)>0) such that (|f(x)|\le C(\varepsilon)e^{\varepsilon |x|^\sigma}). In particular, for (\sigma=2), (B_R^\sigma) coincides with the operator (B_R) in (3).

Theorem 1. In order that (4) hold, where the limit is understood in sense I (i.e., uniformly in (x\in R_n)), it is necessary and sufficient that there exist, uniformly in (x\in R_n),

[
\lim_{R\to\infty}(S_R^1 f)(x)=A(x).
]

The following example convinces us that, for convergence in sense II (i.e., uniformly in (x) on any compact set (K\subset R_n)), Theorem 1 may fail to hold. Let (n=1), and let (f(x)=x\sin x\alpha) for some real (\alpha). Then the solution of problem (1)—(2) has the form

[
u(x,t)=e^{-\alpha^2 t}(x\sin \alpha x+2\alpha t\cos \alpha x).
]

(u(x,t)\to0) ((t\to\infty)) uniformly for (|x|\le a) for any (a>0), but not uniformly in all (x\in R_1). At the same time

[
(S_R^1 f)(x)=-\frac{\cos \alpha R\cos \alpha x}{\alpha}+\varphi(x,R),
]

where (|\varphi(x,R)|\le c|x|/R) with some constant (c>0), i.e. ((S_R^1 f)(x)) has no limit at any point except (x=\pi/2\alpha+k\pi/\alpha), (k=0,\pm1,\ldots). However, for example, the following holds.

Theorem 2. In order that the limit (4) exist in sense II, i.e., uniformly on any compact set (K) in (x), in the case when (|f(x)|\le c(|x|^s+1)) for some (s\ge0,\ c>0), it is necessary and sufficient that, uniformly in (x\in K) for any compact (K\subset R_n), there exist the limit ((S_R^\alpha f)(x)) as (R\to\infty) for (\alpha>s). Analogously, in the case under consideration, a necessary and sufficient condition is also the existence, uniformly in (x\in K) for any (K\subset R_n), of the limit ((B_R^\sigma f)(x)) as (R\to\infty) for any (\sigma>0).

The proofs of Theorems 1 and 2 are carried out in the same way and are based on a number of auxiliary assertions. Denote by (\mathfrak R^\alpha) the linear space of functions (f \in C(R_n)) (without any growth restriction) for which there exists, in the sense of II, the limit

[
\lim_{R\to\infty} (S_R^\alpha f)(x)=A(x)
\tag{9}
]

((\mathfrak R^\alpha) is the domain of definition of the operator (S_\infty^\alpha), defined in the corresponding way), and by (\mathfrak R_0^\alpha) its subspace for which, for (f\in \mathfrak R^\alpha), (A(x)\equiv 0). Obviously, (\mathfrak R_0^\alpha) is a nonempty subspace, since any finite function (f\in \mathfrak R_0^\alpha).

Lemma 1. For (\alpha>0) and convergence understood in the sense of II (uniform on every compact (K\subset R_n)), (\mathfrak R^\alpha/\mathfrak R_0^\alpha=G), where (G) is the linear space of all possible harmonic functions in (R_n). In other words, in order that (f\in \mathfrak R^\alpha), it is necessary and sufficient that there exist (uniquely) such functions (f_0\in \mathfrak R_0^\alpha) and (g\in G), i.e. (\Delta g=0), that

[
f(x)=f_0(x)+g(x).
]

From this lemma, in particular, it follows that, if the limit (9) exists in the sense of II, then the function (A(x)) is necessarily a harmonic function. If (f) is bounded or semibounded and (f\in \mathfrak R^\alpha), (\alpha>0), then (A(x)) obviously has the same property (and with the same constants), and then, by Liouville’s theorem for harmonic functions, (A(x)\equiv \mathrm{const}).

Similarly, if we denote by (\mathfrak B^\sigma), (\sigma>0), the linear space consisting of those (f(x)\in C(R_n)\cap T^\sigma(R_n)) for which there exists, in the sense of II,

[
\lim_{R\to\infty} (B_R^\sigma f)(x)=A(x).
\tag{10}
]

By (\mathfrak B_0^\sigma) we denote the subspace of (\mathfrak B^\sigma) for whose elements (A(x)) in (10) is identically zero. (It is clear that the set of all finite functions from (C(R_n)) is contained in (\mathfrak B_0^\sigma), i.e. (\mathfrak B_0^\sigma) is nonempty.)

Lemma 2. For (\sigma>0) and convergence understood in the sense of II (uniform on every compact (K\subset R_n)), (\mathfrak B^\sigma/\mathfrak B_0^\sigma=G), i.e. in order that (f\in \mathfrak B^\sigma), it is necessary and sufficient that there exist (uniquely) such functions (f_0\in \mathfrak B_0^\sigma) and (g\in G) that

[
f=f_0+g.
]

Lemmas 1 and 2 are consequences of Lemma 3.

Lemma 3. Let the function (K(\alpha)) be a function of one variable (\alpha\in(0,\infty)) such that (\alpha^{n-1}K(\alpha)\in L_1(0,\infty)) for some integer (n>0), and

1) (K(\alpha)\geq 0) for (\alpha\in(0,\infty)),

2) [
\int_0^\infty \alpha^{n-1}K(\alpha)\,d\alpha=1.
]

Then, if for all (x\in R_n) and (R>0) the integral

[
(K_R f)(x)=\frac{1}{nR^n}\int_0^\infty K!\left(\frac{\rho}{R}\right)(F_\rho f)(x)\,d\rho
]

exists and there exists

[
\lim_{R\to\infty} (K_R f)(x)=A(x),
]

then (A(x)) is a harmonic function (the function ((F_\rho f)(x)) is defined from (f(x)) by means of equality (6)).

Since the existence of the limit (4) in the sense of II (and, all the more, in the sense of I) is membership of (f(x)) in the space (\mathfrak B^{(2)}) (understood in the sense of the corresponding convergence), Theorem 1 is a consequence of Lemmas 1, 2, and 4.

Lemma 4. If convergence is understood as uniform in (R_n), then (\mathfrak{P}_0^{(2)}=\mathfrak{N}_0^1).

Theorem 2 is an analogous consequence of Lemmas 1, 2, and 5.

Lemma 5. If convergence is understood as uniform on every compact set (K \subset R_n), then (\mathfrak{P}_0^{(2)}=\mathfrak{N}_0^\alpha) for any (\alpha>s), provided only that (|f(x)|\leqslant \mathrm{const}\,(|x|^{\alpha'}+1)) for (\alpha'<\alpha).

REFERENCES

1. A. N. Tikhonov, Matem. sborn., 42, 2, 199 (1935).
2. N. Wiener, Ann. Math. (2), 33, 1 (1932).
3. V. D. Repnikov, S. D. Èidelman, DAN, 167, No. 2 (1966).
4. V. D. Repnikov, S. D. Èidelman, Matem. sborn., 73, No. 1, 155 (1967).
5. Yu. N. Drozhzhinov, Izv. AN SSSR, ser. matem., 33, No. 2 (1969).