MATHEMATICS

V. D. REPNIKOV

ON UNIFORM STABILIZATION OF THE SOLUTION OF THE CAUCHY PROBLEM FOR PARABOLIC EQUATIONS

(Presented by Academician I. G. Petrovskii on 28 II 1964)

Here we shall prove a theorem which gives an exhaustive solution to the question of the so-called uniform stabilization of the solution of the Cauchy problem for a Petrovskii-parabolic system whose coefficients depend only on time \(t\).

Consider the system

\[ \frac{\partial u}{\partial t}=\sum_{|k|=2b} A_k D^k u. \tag{1} \]

Here \(D^k\) denotes \((-i)^k \dfrac{\partial^k}{\partial x_1^{k_1}\ldots \partial x_n^{k_n}}\), \(k:=k_1+k_2+\cdots+k_n\); \(A_k\) is a square matrix with constant elements. S. D. Eidelman showed in \((^1)\) that for Green’s matrix of (1)

\[ G(t,x)=\frac{1}{(2\pi)^n}\int \exp\left[i(x,\sigma)-t\sum_{|k|=2b} A_k\sigma_1^{k_1}\sigma_2^{k_2}\cdots\sigma_n^{k_n}\right]\,d\sigma \tag{2} \]

the estimate

\[ \left|\frac{\partial^k G(t,x)}{\partial x_1^{k_1}\ldots \partial x_n^{k_n}}\right| < c_1 t^{-(n+k)/2b}\exp\{-c_2(|x|t^{-1/2b})^q\}, \tag{3} \]

is valid,

\[ (x,\sigma)=x_1\sigma_1+x_2\sigma_2+\cdots+x_n\sigma_n,\quad |x|=(x_1^2+x_2^2+\cdots+x_n^2)^{1/2},\quad q=\frac{2b}{2b-1}. \]

It is known that, in the uniqueness class, the solution of problem (1), constructed from a bounded measurable initial vector-function

\[ u(0,x)=u_0(x), \tag{4} \]

is given by the Poisson integral:

\[ u(t,x)=\int G(t,x-\xi)u_0(\xi)\,d\xi \quad (d\xi=d\xi_1\,d\xi_2\ldots d\xi_n). \tag{5} \]

From the fact that the identity matrix \(E\) satisfies system (1), it follows that

\[ \int G(t,x-\xi)\,d\xi=E. \]

Before formulating the main result, let us introduce some definitions.

Definition 1. We shall say that a bounded measurable vector-function \(u_0(x)\) has a uniform limiting mean equal to \(l\) \((l=(l_1,l_2,\ldots,l_n))\), if for any \(\varepsilon>0\) there is an \(N_0(\varepsilon)\) such that for every cube \(V_N^\eta\) with side equal to \(2N\) and center at the point \(\eta(\eta_1,\eta_2,\ldots,\eta_n)\)

\[ \left|\frac{1}{(2N)^n}\int_{V_N^\eta} u_0(x)\,dx-l\right|<\varepsilon \]

for all \(\eta\) and all \(N>N_0\).

Definition 2. A vector function \(u(t,x)\) is said to be uniformly stabilized to \(l\) if, for every \(\varepsilon>0\), one can indicate a \(T_0(\varepsilon)\), independent of \(x\), such that for \(t>T_0\)
\[ |u(t,x)-l|<\varepsilon . \]

Theorem. For the uniform stabilization of the solution of problem (1), (4), represented by the Poisson integral, it is necessary and sufficient that the initial vector function \(u_0(x)\) have a uniform limiting mean.

Proof. Necessity. For simplicity we shall assume that the moduli of the components of the initial vector function \(u_0(x)\) are bounded by one \(\bigl(|u_0^j|<1,\ j=1,2,\ldots,m\bigr)\) and that \(l\) is the zero vector. We shall prove by contradiction.

Let
\[ u(t,x)=\int G(t,x-\xi)u_0(\xi)\,d\xi \longrightarrow 0 \quad \text{as } t\to\infty \]
uniformly in \(x\), while the initial vector function has no uniform limiting mean. The latter means that, for some \(\varepsilon_0>0\) and any positive \(N_k\) \((k=1,2,\ldots)\), there exist \(N>N_k\) and points \(\eta^k(\eta_1^k,\eta_2^k,\ldots,\eta_n^k)\) such that, for at least one of the components of the vector function \(u_0(x)\) (for definiteness, \(u_0^1(x)\)),
\[ \left|S_1(N_k,\eta^k)\right| = \left| \frac{1}{(2N_k)^n} \int_{V_{N_k}^{\eta^k}} u_0^1(\xi)\,d\xi \right| >\varepsilon_0 . \tag{6} \]

We shall show that it follows from (6) that \(u_1(t,x)\) does not tend to zero as \(t\to\infty\). Indeed, choose the sequence of times
\[ t_k=\left(\frac{\varepsilon_0 N_k}{4b^{\,n+1}mn}\right)^{2b} \]
(\(N_k\) is some sequence tending to infinity, and \(B\) is a constant which will be chosen below) and average \(u_1(t,x)\) in the following way:
\[ \left| \frac{1}{(2N_k)^n} \int_{V_{N_k}^{\eta^k}} u_1(t_k,x)\,dx \right| = \]
\[ = \left| \frac{1}{(2N_k)^n} \int_{V_{N_k}^{\eta^k}} \int_{E_n} \sum_{j=1}^{m} G_{1j}(t,x-\xi) \bigl[u_0^j(x)+u_0^j(\xi)-u_0^j(x)\bigr] \,d\xi\,dx \right| = \]
\[ = \left| S_1(N_k,\eta^k) + t_k^{\,n/2b} \int_{V_B^0} \sum_{j=1}^{m} G_{1j}(t_k,yt_k^{1/2b})\,dy\, \frac{1}{(2N_k)^n} \int_{V_{N_k}^{\eta^k}} \bigl[u_0^j(x-t_k^{1/2b}y)-u_0^j(x)\bigr]\,dx \right. \]
\[ \left. \quad+ t_k^{\,n/2b} \int_{E_n-V_B^0} \sum_{j=1}^{m} G_{1j}(t_k,yt_k^{1/2b})\,dy\, \frac{1}{(2N_k)^n} \int_{V_{N_k}^{\eta^k}} \bigl[u_0^j(x-t_k^{1/2b}y)-u_0^j(x)\bigr]\,dx \right| = \]
\[ = |S_1+I_1+I_2| \ge |S_1|-|I_1|-|I_2|. \]

We now choose \(B\) so that
\[ t_k^{\,n/2b} \int_{E_n-V_B^0} \sum_{j=1}^{m} \left|G_{1j}(t_k,yt_k^{1/2b})\right|\,dy < \frac{\varepsilon_0}{8}. \]

This can be done by using estimate (3). Then, obviously, \(|I_2|<\varepsilon_0/4\). It is asserted that \(|I_1|<\varepsilon_0/4\).

Indeed, since \(u_0^j(x-t^{1/2b}y)\) differs from \(u_0^j(x)\) on a set whose measure is no greater than \(2^n n N_k^{\,n-k}Bt^{1/2b}\), we have

\[ \left| \frac{1}{(2N_k)^n}\int_{V_{N_k}^{\eta^k}} \bigl[u_0^j(x-t_k^{1/2b}y)-u_0^j(x)\bigr]\,dx \right| \leq \frac{nBt_k^{1/2b}}{N_k}. \]

Since

\[ t_k^{n/2b}\int_{V_B^0}\sum_{i=1}^m \left|G_{1j}(t_k,yt_k^{1/2b})\right|\,dy < mB^n, \]

the desired smallness of \(|I_1|\) is proved.

Thus it has been proved that, for some \(\varepsilon_0>0\), there exists a sequence of times \(t_k\) tending to infinity and a sequence of points
\(\eta^k(\eta_1^k,\eta_2^k,\ldots,\eta_n^k)\) such that

\[ \left| \frac{1}{(2N_k)^n}\int_{V_{N_k}^{\eta^k}} u_1(t_k,x)\,dx \right|>\frac{\varepsilon_0}{2}. \]

It follows from this that, for any arbitrarily large time \(T_0\), one can indicate such \(t_k>T_0\) and such points \(\eta_0^k\), lying inside \(V_{N_k}^{\eta^k}\), that
\[ |u(t_k,\eta_0^k)|>\varepsilon_0/2. \]
The necessity is thereby proved.

Sufficiency*. Let the initial vector-function \(u_0(x)\) have a uniform limiting mean equal to the zero vector.

We shall show that for every \(\varepsilon>0\) one can indicate a \(T_1\), independent of \(x\), such that for \(t>T_1\),
\[ |u(t,x)|<\varepsilon. \]
For this purpose write \(u(t,x)\) in the form

\[ \begin{aligned} u(t,x) &=\int G(t,y)\, \frac{\partial^n}{\partial y_1\ldots \partial y_n} \int_0^{y_1}\cdots\int_0^{y_n} u_0(x-\eta)\,d\eta\,dy \\ &=\int \frac{\partial^n G(t,y)}{\partial y_1\ldots \partial y_n} \int_0^{y_1}\cdots\int_0^{y_n} u_0(x-\eta)\,d\eta\,dy, \end{aligned} \tag{7} \]

since the remaining terms on the right-hand side of (7) are equal to zero by virtue of estimate (3).

Proceeding to estimate \(u(t,x)\), represent the latter as the sum of two integrals, then use estimate (3) and make the change of variable \(y=z\tau\) (where \(\tau\) denotes \(t^{1/2b}\)):

\[ \begin{aligned} |u(t,x)| &=\left| \int_{|y_1|\leq \alpha\tau}\cdots\int_{|y_n|\leq \alpha\tau} \frac{\partial^n G(t,y)}{\partial y_1\ldots \partial y_n} \int_0^{y_1}\cdots\int_0^{y_n} u_0(x-\eta)\,d\eta\,dy \right. \\ &\qquad\left. +\int_{|y_1|>\alpha\tau}\cdots\int_{|y_n|>\alpha\tau} \frac{\partial^n G(t,y)}{\partial y_1\ldots \partial y_n} \int_0^{y_1}\cdots\int_0^{y_n} u_0(x-\eta)\,d\eta\,dz \right| \\ &< c_1\int_{|z_1|\leq \alpha}\cdots\int_{|z_n|\leq \alpha} \exp\{-c_2|z|^q\} \left| \frac{1}{\tau^n} \int_0^{z_1\tau}\cdots\int_0^{z_n\tau} u_0(x-\eta)\,d\eta\,dz \right| \\ &\qquad +c_1\int_{|z_1|>\alpha}\cdots\int_{|z_n|>\alpha} \exp\{-c_2|z|^q\} \left| \frac{1}{\tau^n} \int_0^{z_1\tau}\cdots\int_0^{z_n\tau} u_0(x-\eta)\,d\eta\,dz \right| =I_3+I_4. \end{aligned} \]

* This part of the theorem follows from the general stabilization theorem of F. O. Porper and S. D. Eidelman \((^3)\); for completeness of exposition we give its proof.

Noting that

\[ I_3=c_1 \int_{|z_1|\leq a}\cdots \int_{|z_n|\leq a} \exp\{-c_2|z|^q\}|z_1z_2\cdots z_n|\,S(z_1\tau,\ldots \]

\[ \ldots,z_n\tau,x)\,dz \]

(where \(S(z_1\tau,\ldots,z_n\tau;x)\) is the mean of the function \(u_0(x)\) over the parallelepiped of dimensions \(\tau(z_1\times z_2\times\cdots\times z_n)\) with center at the point \(x\)), we choose \(a\) so small that \(I_3<\varepsilon/2\).

To complete the proof it remains to note that, by virtue of the assumption on the existence of a zero uniform limiting mean of \(u_0(x)\), one can indicate such a \(T_1\) that for all \(t>T_1\)

\[ S(\tau a,x)<\frac{\varepsilon}{2} \left\{ c_1 \int \exp[-c_2|z|^q]\,|z_1z_2\cdots z_n|\,dz \right\}^{-1}, \]

therefore \(I_4<\varepsilon/2\), \(|u(t,x)|<\varepsilon\). The theorem is completely proved.

Remark. The proof given remains valid for an arbitrary Petrovsky-parabolic system of the form

\[ \frac{\partial u}{\partial t} = \sum_{1\leq |k|\leq 2b} A_k(t)D^k u, \]

for whose Green matrix the estimate

\[ D^mG(t,0,x)\leq c_m a(t)^{-m-n}\exp\{-c(|x|a^{-1}(t))^q\} \qquad (a(t)\to\infty \text{ as } t\to\infty) \]

is valid.

Various sufficient conditions for the validity of the latter estimate are given in \((^2)\).

In conclusion I express my sincere gratitude to S. D. Eidelman for posing the problem solved here, and also for valuable suggestions given in the course of its solution.

Voronezh
Polytechnic Institute

Received
21 I 1964

REFERENCES

1. S. D. Eidelman, Mat. sborn., 33, No. 3, 359 (1953).
2. S. D. Eidelman, Mat. sborn., 44, No. 4, 481 (1958); F. O. Porper, S. D. Eidelman, Izv. vyssh. uchebn. zaved., Matematika, No. 4, 210 (1960).