MATHEMATICS

F. O. PORPER

ON THE STABILIZATION OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A PARABOLIC EQUATION WITH VARIABLE COEFFICIENTS

(Presented by Academician I. G. Petrovsky, 15 VI 1963)

In the present article it is shown how, from Nash’s estimates (²) for the fundamental solution itself of a parabolic equation and for its modulus of continuity, one can establish the fact of stabilization of a bounded solution of the Cauchy problem for which the initial function \(u(x,0)=\varphi(x)\) has a “uniform mean” over the whole space \(x\).

Consider the equation

\[ \frac{\partial u}{\partial t} = \sum_{i,j=1}^{n} \frac{\partial}{\partial x_i} \left( a_{ij}(x,t)\frac{\partial u}{\partial x_j} \right), \quad x=(x_1,\ldots,x_n)\in E^{(n)} \quad (0<t<\infty), \tag{1} \]

where \(\|a_{ij}(x,t)\|\) is a symmetric matrix for which

\[ c_1|\xi|^2 \le \sum_{i,j=1}^{n} a_{ij}(x,t)\xi_i\xi_j \le c_2|\xi|^2, \quad 0<c_1\le c_2. \tag{2} \]

Let \(a_{ij}(x,t)\) and \(\partial a_{ij}(x,t)/\partial x_i\) be continuous in \(t\), with the continuity in \(t\) of the coefficients \(a_{ij}(x,t)\) uniform with respect to \(x\in E^{(n)}\). Suppose, furthermore, that \(a_{ij}(x,t)\) and \(\partial a_{ij}(x,t)/\partial x_i\) are bounded and satisfy the Hölder condition in \(x\) with exponent \(\alpha_1\), \(0<\alpha_1\le 1\). Then it follows from the results of (¹,²) that every bounded solution \(u(x,t)\) of equation (1), having second continuous derivatives with respect to \(x\) and a first continuous derivative with respect to \(t\) for \(t>0\), is representable in the form

\[ u(x,t)=\int_{E^{(n)}} Z(x,t,\xi,0)\varphi(\xi)\,d\xi, \tag{3} \]

where \(\varphi(x)=u(x,0)\), and for the fundamental solution \(Z(x,t,\xi,\tau)\) the estimates

\[ Z(x,t,\xi,\tau) \le k_1(t-\tau)^{-n/2} \exp\left[ -k_2 \frac{|x-\xi|}{(t-\tau)^{1/2}} \ln\left( k_3\frac{|x-\xi|}{(t-\tau)^{1/2}} \right) \right]; \tag{4} \]

\[ |Z(x,t,\xi_1,\tau)-Z(x,t,\xi_2,\tau)| \le k_4(t-\tau)^{-n/2} \left( \frac{|\xi_1-\xi_2|}{(t-\tau)^{1/2}} \right)^{\alpha}; \tag{5} \]

hold; the constants \(\alpha,k_1,k_2,k_3,k_4\) depend only on \(n,c_1,c_2\) from (2).

Theorem. Let \(|\varphi(x)|\le K\),

\[ \frac{1}{(2N)^n} \int_{|\gamma_i-x_i|\le N}\varphi(\gamma)\,d\gamma \underset{N\to\infty}{\longrightarrow}0 \]

uniformly with respect to \(x\in E^{(n)}\).

Then

\[ u(x,t)\underset{t\to\infty}{\longrightarrow}0 \]

uniformly with respect to \(x\in E^{(n)}\).

Proof. In (3) make the substitution \(\xi-x=t^{1/2}\beta\). We obtain

\[ u(x,t)=t^{n/2}\int_{E^{(n)}} Z(x,t,x+t^{1/2}\beta,0)\, \varphi(x+t^{1/2}\beta)\,d\beta. \]

It follows from (4) that

\[ t^{n/2}Z(x,t,x+t^{1/2}\beta,0) \le k_1\exp[-k_2|\beta|\ln(k_3|\beta|)] \tag{6} \]

uniformly in \(x\) and \(t\).

In view of the estimate \(|\varphi(x)| \leq K\) and (6), for the given \(\varepsilon\) one can choose such a \(B\) that

\[ u_1(x,t)=t^{n/2}\int_{|\beta_i|\geq B} Z(x,t,x+t^{1/2}\beta,0)\varphi(x+t^{1/2}\beta)\,d\beta \leq \frac{\varepsilon}{2} \tag{7} \]

for all \(x\) and \(t>0\).

Let us show that by taking \(t \geq T\), where \(T\) depends on \(n,c_1,c_2,K,\varepsilon\), one can obtain

\[ |u_2(x,t)|=t^{n/2}\left|\int_{|\beta_i|\leq B} Z(x,t,x+t^{1/2}\beta,0)\varphi(x+t^{1/2}\beta)\,d\beta\right| \leq \frac{\varepsilon}{2}. \]

Divide the domain \(|\beta_i|\leq B\) into \((2s)^n\) cubes by dividing \(B\) into \(s\) parts, where \(s\), as will be seen from what follows, depends only on \(n,c_1,c_2,K,\varepsilon\), and denote the \(\nu\)-th cube by \(K_\nu\). Then

\[ u_2(x,t)=\sum_{\nu=1}^{(2s)^n} t^{n/2}\int_{K_\nu} Z(x,t,x+t^{1/2}\beta,0)\varphi(x+t^{1/2}\beta)\,d\beta. \]

Decompose \(\varphi(x+t^{1/2}\beta)\) into its positive and negative parts: \(\varphi=\varphi^+-\varphi^-\). Then

\[ \begin{aligned} u_2(x,t) &=\sum_{\nu=1}^{(2s)^n} t^{n/2}\left[\int_{K_\nu} Z(x,t,x+t^{1/2}\beta,0)\varphi^+(x+t^{1/2}\beta)\,d\beta \right.\\ &\qquad\left. -\int_{K_\nu} Z(x,t,x+t^{1/2}\beta,0)\varphi^-(x+t^{1/2}\beta)\,d\beta\right] \\ &=\sum_{\nu=1}^{(2s)^n} t^{n/2}\left[ Z(x,t,x+t^{1/2}\beta_{1,\nu,x,t},0)\int_{K_\nu}\varphi^+(x+t^{1/2}\beta)\,d\beta \right.\\ &\qquad\left. - Z(x,t,x+t^{1/2}\beta_{2,\nu,x,t},0)\int_{K_\nu}\varphi^-(x+t^{1/2}\beta)\,d\beta\right] \\ &=\sum_{\nu=1}^{(2s)^n} t^{n/2}\left[\{Z(x,t,x+t^{1/2}\beta_{1,\nu,x,t},0)-Z(x,t,x+t^{1/2}\beta_{2,\nu,x,t},0)\}\right.\\ &\qquad\times \int_{K_\nu}\varphi^+(x+t^{1/2}\beta)\,d\beta + Z(x,t,x+t^{1/2}\beta_{2,\nu,x,t},0)\int_{K_\nu}\varphi(x+t^{1/2}\beta)\,d\beta\Big]. \end{aligned} \]

Since \(\beta_{1,\nu,x,t}\) and \(\beta_{2,\nu,x,t}\in K_\nu\), we have

\[ |\beta_{1,\nu,x,t}-\beta_{2,\nu,x,t}|<\frac{B}{s}\sqrt{n}. \]

Let \(\sigma_{\nu i}\) be the \(i\)-th coordinate of the center of the cube \(K_\nu\). Then \(K_\nu\) is written as

\[ |\beta_i-\sigma_{\nu i}| \leq \frac{B}{2s}. \]

For a given \(\delta\), which depends on \(n,c_1,c_2,K\) and \(\varepsilon\) and which we shall define later, one can find such a \(T\), depending on \(n,c_1,c_2,K,\varepsilon\) and on the rate at which the mean of \(\varphi(x)\) tends to zero, that

\[ \frac{1}{(Bt^{1/2}/s)^n} \int_{|\gamma_i-x_i-\sigma_{\nu i}t^{1/2}|\leq Bt^{1/2}/2s} \varphi(\gamma)\,d\gamma \leq \delta \quad \text{for } t\geq T. \tag{8} \]

Make in the last integral the change of variables \(\gamma_i=x_i+t^{1/2}\beta_i\). We obtain

\[ \int_{|\beta_i-\sigma_{\nu i}|\leq B/2s} \varphi(x+t^{1/2}\beta)\,d\beta < \delta\left(\frac{B}{s}\right)^n \quad \text{for } t\geq T. \]

This means that

\[ \int_{K_\nu}\varphi\left(x+t^{1/2}\beta\right)\,d\beta<\delta V_{K_\nu} \]

for \(t\geq T\), where \(V_{K_\nu}\) is the volume of the cube \(K_\nu\).

Let us estimate \(u_2(x,t)\):

\[ \left|u_2(x,t)\right|\leq \sum_{\nu=1}^{(2s)^n}\left[k_4\left|\beta_{1,\nu,x,t}-\beta_{2,\nu,x,t}\right|^\alpha K V_{K_\nu} +M\delta V_{K_\nu}\right]\leq \]

\[ \leq \left(k_4\frac{B^\alpha}{s^\alpha}n^{\alpha/2}K+M\delta\right) \sum_{\nu=1}^{(2s)^n}V_{K_\nu} = \]

\[ =\left(k_5K\frac{B^\alpha}{s^\alpha}+M\delta\right)(2B)^n, \quad \text{where } M=\max_\beta\{k_1\exp[-k_2|\beta|\ln(k_3|\beta|)]\}, \]

and therefore \(M\) depends only on \(n,c_1\), and \(c_2\).

We now choose \(s\) and \(\delta\) so that

\[ k_5K\frac{B^\alpha}{s^\alpha}(2B)^n\leq\frac{\varepsilon}{4}, \qquad M\delta(2B)^n\leq\frac{\varepsilon}{4}. \tag{9} \]

Thus, for \(t>T\), where \(T\) depends on \(n,c_1,c_2,K,\varepsilon\) and on the rate of convergence of the mean of \(\varphi(x)\) to zero, we obtain

\[ |u(x,t)|\leq |u_1(x,t)|+|u_2(x,t)|< \frac{\varepsilon}{2}+\frac{\varepsilon}{4}+\frac{\varepsilon}{4} =\varepsilon \]

simultaneously for all \(x\in E^{(n)}\), as was required to prove.

Remark. It can be shown that, in order to satisfy conditions (7) and (9), it is sufficient to take

\[ B=k_6\ln\frac{Kk_7}{\varepsilon},\quad s=\frac{E}{\varepsilon^{1/\alpha'}},\quad \delta=F\varepsilon^{1+\alpha''}, \]

where \(0<\alpha'<\alpha,\ \alpha''>0\); the constants \(k_6\) and \(k_7\) depend on \(n,c_1,c_2\), the constant \(E\) depends on \(n,c_1,c_2,K,\alpha'\), and the constant \(F\) depends on \(n,c_1,c_2,K,\alpha''\).

Therefore inequality (8) can be written as follows:

\[ \frac{1}{(2D)^n} \int_{|\gamma_i-x_i-a_i|\leq D}\varphi(\gamma)\,d\gamma \leq \varepsilon F^{1+\alpha''}, \tag{10} \]

where

\[ D\leq |a_i|\leq \frac{2E}{\varepsilon^{1/\alpha'}}D. \]

It is easy to show that (10) will hold for \(D\geq D_0(\varepsilon,X)\), for all \(|x_i|\leq X\), already when the initial function \(\varphi(x)\) has only angular means (see (3)).

Thus the following assertion holds:

If \(|\varphi(x)|\leq K\),

\[ \frac{1}{A}\int_{\operatorname{Rea}}\varphi(\xi)\,d\xi \longrightarrow 0 \quad \left(\text{notation as in (3)}\right), \quad \text{as } a_1\to\infty,\ldots,a_n\to\infty, \]

then

\[ u(x,t)\longrightarrow 0 \quad \text{as } t\to\infty \]

uniformly in \(x,\ |x_i|\leq X\).

The author expresses deep gratitude to E. M. Landis for supervising the work.

Note added in proof. Estimate (7) is easily obtained without (4) from the estimate of the moment of the fundamental solution \((^2)\), which in our case has the form

\[ \int |x-\xi|\,|Z(x,t,\xi,0)|\,d\xi\leq Lt^{1/2}. \]

Moscow State University
named after M. V. Lomonosov

Received
21 V 1963

References

1. S. D. Eidelman, Matem. sbornik, 53 (95), 1 (1961).
2. J. Nash, Collection of translations. Mathematics, 4, No. 1, 1960.
3. S. D. Eidelman, F. O. Porper, Izv. vyssh. uchebn. zaved., Matem., No. 4 (1960).