Reports of the Academy of Sciences of the USSR
1970, Volume 194, No. 3

UDC 517.946

MATHEMATICS

A. K. GUSHCHIN, V. P. MIKHAILOV

ON THE STABILIZATION OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A PARABOLIC EQUATION

(Presented by Academician A. N. Tikhonov on March 2, 1970)

Let \(u(x,t)\) be the solution of the Cauchy problem for the parabolic equation

\[ p(x)u_t(x,t)=\Delta u(x,t), \qquad x=(x_1,\ldots,x_n)\in R_n,\quad t>0, \tag{1} \]

\[ u(x,0)=\varphi(x), \tag{2} \]

where \(p(x)\ge a>0\).

Suppose that the functions \(p(x)\) and \(\varphi(x)\) satisfy the following conditions:

\[ p(x)\in B_{[n/2]-1}^{\alpha}(R_n)\quad \text{for } n\ge 4, \]
\[ p(x)\in B_0^{\alpha}(R_n)\quad \text{for } n<4, \tag{3} \]

\[ \varphi(x)\in B_0^0(R_n). \tag{4} \]

By the class \(B_k^{\alpha}(R_n)\), where \(k\) is an integer, we mean the set of all bounded functions satisfying a Hölder condition of order \(\alpha>0\) in \(R_n\), together with all continuous derivatives up to order \(k\).

The purpose of this note is to prove the following theorem.

Theorem. If the functions \(p(x)\) and \(\varphi(x)\) satisfy conditions (3), (4), and

\[ \frac{n}{\omega_n R^n}\int_{|x-y|\le R}|p(y)-b|\,dy\to 0 \quad \text{as } R\to\infty \tag{5} \]

uniformly with respect to \(x\in R_n\), for some constant \(b\)
\[ \left(\omega_n=\frac{2\pi^{n/2}}{\Gamma(n/2)}\right), \]
then a necessary and sufficient condition for the existence of the limit

\[ \lim_{t\to\infty}u(x,t)=A \tag{6} \]

at some \(x\in R_n\) is the existence of the limit

\[ \frac{n}{\omega_n R^n}\int_{|x-y|\le R}\varphi(y)\,dy\to A \quad \text{as } R\to\infty . \tag{7} \]

In order that condition (6) hold uniformly on any compact set \(D\subset R_n\) (or uniformly in \(R_n\)), it is necessary and sufficient that condition (7) hold uniformly on any compact set \(D\subset R_n\) (respectively, uniformly in \(R_n\)).

We note that from the results of the note \((^1)\) it follows that in the case under consideration \((\varphi(x)\in B_0^0(R_n))\), \(A\) is necessarily a constant.

It is easy to see that condition (5) is satisfied, in particular, if

\[ \lim_{|y|\to\infty}(p(y)-b)=0 \]

or if

\[ p(y)-b\in L_r(R_n) \tag{5'} \]

for some \(r\ge 1\).

Without loss of generality, one may assume that in (5) \(b=1\). Denote by \(v(x,t)\) the solution of the heat equation with initial function \(p(x)\varphi(x)\),

\[ v(x,t)=\frac{1}{(2\sqrt{\pi t})^n}\int_{R_n}\exp\left(-\frac{|x-y|^2}{4t}\right)p(y)\varphi(y)\,dy. \]

Lemma 1. Let \(p(x)\in B_0^0(R_n)\), and let conditions (4) and (5) be satisfied. Then

\[ \lim_{t\to\infty}\frac{1}{t}\int_0^t \bigl(u(x,t)-v(x,t)\bigr)\,dt=0 \tag{8} \]

uniformly with respect to \(x\in R_n\).

Let \(\tilde u(x,\lambda)\) and \(\tilde v(x,\lambda)\) be the Laplace transforms of the functions \(u(x,t)\) and \(v(x,t)\). Since \(u(x,t)\) and \(v(x,t)\) are bounded for \(t\geq 0\), \(x\in R_n\), it follows that \(\tilde u(x,\lambda)\) and \(\tilde v(x,\lambda)\) are analytic in \(\lambda\) and bounded in \(x\) for \(\operatorname{Re}\lambda>0\). The function \(\tilde u(x,\lambda)\) satisfies the differential equation

\[ -\Delta\tilde u(x,\lambda)+\lambda\tilde u(x,\lambda) = p(x)\varphi(x)-\lambda q(x)\tilde u(x,\lambda), \tag{9} \]

where \(p(x)=1+q(x)\). Equation (9) in the class \(B_0^0(R_n)\) is equivalent to the integral equation

\[ \tilde u(x,\lambda)=-\mathcal L(\lambda)\tilde u(x,\lambda)+\tilde v(x,\lambda), \tag{10} \]

where the operator \(\mathcal L(\lambda)\) is defined as follows:

\[ \mathcal L(\lambda)f(x,\lambda) = \frac{\lambda^{(n+2)/4}}{(2\pi)^{n/2}} \int_{R_n} \frac{K_{n/2-1}\bigl(\sqrt{\lambda}\,|x-y|\bigr)} {|x-y|^{n/2-1}} q(y)f(y,\lambda)\,dy, \]

\[ \tilde v(x,\lambda) = \frac{\lambda^{(n-2)/4}}{(2\pi)^{n/2}} \int_{R_n} \frac{K_{n/2-1}\bigl(\sqrt{\lambda}\,|x-y|\bigr)} {|x-y|^{n/2-1}} p(y)\varphi(y)\,dy. \]

The function \(K_\nu(z)\) is the Macdonald cylindrical function.

It can be proved that the operator \(\mathcal L(\lambda)\) is a bounded operator from \(B_0^0(R_n)\) into \(B_0^0(R_n)\) (\(B_0^0(R)\) is a Banach space with norm

\[ \|f\|_{B_0^0(R_n)}=\sup_{x\in R_n}|f(x)| \]

and

\[ \|\mathcal L(\lambda)\|_{B_0^0(R_n)}=o(1)\quad \text{as } \lambda\to 0, \tag{11} \]

when \(|\arg\lambda|\leq \pi-\sigma\) for any \(\sigma>0\). From the boundedness of the functions \(p(x)\) and \(\varphi(x)\) it follows that

\[ \|\tilde v(x,\lambda)\|_{B_0^0(R_n)}\leq \frac{C}{|\lambda|} \quad \text{for } |\arg\lambda|\leq \pi-\sigma. \tag{12} \]

Consequently, from (10) and (12) we have

\[ \|\tilde u(x,\lambda)\|_{B_0^0(R_n)} \leq \|\mathcal L(\lambda)\|\, \|\tilde u(x,\lambda)\|_{B_0^0(R_n)} +\frac{C}{|\lambda|}, \]

or, by virtue of (11),

\[ \|\tilde u(x,\lambda)\|_{B_0^0(R_n)} \leq \frac{C_1}{|\lambda|} \quad \text{for } |\arg\lambda|\leq \pi-\sigma,\ |\lambda|<\varepsilon, \]

where \(\varepsilon>0\) is a sufficiently small number. From (10) and (11) it then follows immediately that

\[ \|\tilde u(x,\lambda)-\tilde v(x,\lambda)\|_{B_0^0(R_n)} = o(1/\lambda) \]

for \(|\arg\lambda|\leq \pi-\sigma,\ |\lambda|<\varepsilon\).

Relation (8) then follows from the Tauberian theorem of Wiener \((^2)\).

Lemma 2. If the functions \(p(x)\) and \(\varphi(x)\) satisfy conditions (3), (4), (5), then

\[ \left\|\partial u(x,t)/\partial t\right\|_{B_0^0(R_n)} \leq C/t \tag{13} \]

for \(t \geq t_0 > 0\).

The function \(w(x,t)=\partial u(x,t)/\partial t\) is, for \(t>\delta\) \((0<\delta<t_0)\), a solution of equation (1) with the initial condition

\[ w(x,t)\big|_{t=\delta}=u_t(x,\delta)\equiv \psi(x). \tag{14} \]

From condition (3) it follows that

\[ \psi(x)\in B_{[n/2]}^0(R_n)\quad \text{for } n\geq 4;\qquad \psi(x)\in B_2^0(R_n)\quad \text{for } n<4. \tag{15} \]

The Laplace transform \(\widetilde w(x,\lambda)\) of the function \(w(x,t)\) satisfies equations (9) and (10), in which the function \(\varphi(x)\) is replaced by the function \(\psi(x)\).

Simultaneously with problem (1), (14), consider the following Cauchy problem for the hyperbolic equation:

\[ p(x)z_{tt}(x,t)-\Delta z(x,t)=0,\qquad t>\delta, \]

\[ z(x,\delta)=0,\qquad z_t(x,\delta)=\psi(x). \]

From relations (15) and the Sobolev embedding theorems \((^3)\) it follows that

\[ |z(x,t)| \leq Ct^m \]

for some \(m>0\), with a constant \(C\) independent of \(x,t\). Therefore the Laplace transform \(\widetilde z(x,\lambda)\) of the function \(z(x,t)\), satisfying the equation

\[ -\Delta \widetilde z(x,\lambda)+\lambda^2 \widetilde z(x,\lambda) = p(x)\psi(x)-\lambda^2 q(x)\widetilde z(x,\lambda), \tag{16} \]

is an analytic function of \(\lambda\) and bounded in \(x\) for \(\operatorname{Re}\lambda>0\), and \(|\widetilde z(x,\lambda)|\to 0\) as \(|\lambda|\to\infty\) uniformly in \(\arg\lambda\in[-(\pi-\sigma)/2,(\pi-\sigma)/2]\). Comparing equations (9) and (16) and using the uniqueness theorem in \(B_0^0(R_n)\) for the solution of equation (9), we obtain that the function \(\widetilde w(x,\lambda)\) is analytically continued into the domain \(|\arg\lambda|<\pi\) by the equality \(\widetilde w(x,\lambda)=\widetilde z(x,\sqrt{\lambda})\), and \(\widetilde w(x,\lambda)\to 0\) as \(|\lambda|\to\infty\) uniformly in \(\arg\lambda\in[-\pi+\sigma,\pi-\sigma]\). Since \(w(x,t)=u_1(x,t)\) for \(t\geq\delta\), from (10) we have

\[ \widetilde w(x,\lambda) =-\mathcal L(\lambda)\widetilde w(x,\lambda)-\mathcal L(\lambda)u(x,\delta)-u(x,\delta)+\lambda \widetilde v(x,\lambda). \]

From (11) and (12) it follows that

\[ \left\|\widetilde w(x,\lambda)\right\|_{B_0^0(R_n)}\leq \mathrm{const} \quad \text{for } |\lambda|<\varepsilon,\ |\arg\lambda|\leq \pi-\sigma. \]

Replacing, in the inverse Laplace transform for \(\widetilde w(x,\lambda)\), the contour of integration \(\operatorname{Re}\lambda=\beta>0\) by the contour \(|\arg\lambda|=3\pi/4\) (we assume that \(\sigma<\pi/4\)), we easily obtain the estimate (13).

Lemma 3. If, for a continuously differentiable function \(f(x,t)\), the conditions

\[ \frac{1}{t}\int_0^t f(x,\tau)\,d\tau\to 0 \quad \text{as } t\to\infty \tag{17} \]

hold uniformly in \(x\in R_n\), and

\[ \left|\partial f(x,t)/\partial t\right|\leq C/t \quad \text{for } t\geq t_0 \tag{18} \]

with some constant \(C>0\), then

\[ f(x,t)\to 0 \qquad \text{as } t\to\infty \]

uniformly in \(x\in R_n\).

This lemma is a consequence of Theorem 33 of the book \((^4)\) (p. 388). Let us take as the function \(f(x,t)\) the function \(u(x,t)-v(x,t)\). By Lemma 1, relation (17) holds for this function, and by Lemma 2 relation (18) holds, since the inequality \(\left|\partial v(x,t)/\partial t\right|\le C/t\) for \(t>0\) is obvious. Therefore it follows from Lemma 3 that

\[ u(x,t)-v(x,t)\to 0 \qquad \text{as } t\to\infty \]

uniformly in \(x\in R_n\). After this, the theorem formulated above follows from known results \((^{1,2,5,6})\) concerning the solution \(v(x,t)\) of the Cauchy problem for the heat equation, since from condition (5) it follows that, for relation (7) to hold, it is necessary and sufficient that the relation

\[ \frac{n}{\omega_n R^n}\int_{|x-y|\le R} p(y)\varphi(y)\,dy \to A \]

hold.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Moscow

Received
18 II 1970

REFERENCES

\(^1\) V. P. Mikhailov, DAN, 190, No. 1 (1970).
\(^2\) N. Wiener, The Fourier Integral and Certain of Its Applications, Moscow, 1963.
\(^3\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950.
\(^4\) G. Hardy, J. E. Littlewood, G. Polya, Inequalities, IL, 1948.
\(^5\) B. D. Repnikov, S. D. Eidelman, DAN, 167, No. 2 (1966).
\(^6\) Yu. N. Drozhzhinov, Izv. Akad. Nauk SSSR, Ser. Mat., 33, No. 2 (1969).