Abstract
Full Text
Mathematics
Yu. N. Drozhzhinov
ON THE STABILIZATION OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A PARABOLIC EQUATION
(Presented by Academician I. G. Petrovskii, 2 VIII 1961)
Consider the parabolic equation:
\[ \frac{\partial u}{\partial t} = \sum_{k,l=1}^{n} a_{kl}(t)\frac{\partial^2 u}{\partial x_k \partial x_l} + \sum_{k=1}^{n} b_k(t)\frac{\partial u}{\partial x_k} + g(t)u; \tag{1} \]
\[ a_{ij}(t)=a_{ji}(t), \qquad \sum_{k,l=1}^{n} a_{kl}(t)\alpha_k\alpha_l \ge \gamma(t)\sum_{k=1}^{n}\alpha_k^2, \qquad \gamma(t)>0. \tag{1'} \]
The functions \(a_{kl}(t)\), \(b_k(t)\), \(g(t)\) for all \(k,l=1,2,\ldots,n\) are assumed to be integrable in every finite interval of the argument \(t\).
Introduce the notation:
\[ A_{kl}(t)=\int_{0}^{t} a_{kl}(\tau)\,d\tau,\qquad B_k(t)=\int_{0}^{t} b_k(\tau)\,d\tau,\qquad G(t)=\int_{0}^{t} g(\tau)\,d\tau. \]
The symmetric matrix \(A=\|A_{ij}(t)\|\) is uniformly positive definite for \(t\ge t^*>0\), since the parabolicity condition \((1')\), integrated from \(0\) to \(t\), gives
\[ \sum_{k,l=1}^{n} A_{kl}(t)\alpha_k\alpha_l \ge \int_{0}^{t}\gamma(\tau)\,d\tau \sum_{k=1}^{n}\alpha_k^2. \]
The eigenvalues of \(A\) satisfy
\[ \lambda_i(t)\ge \int_{0}^{t}\gamma(\tau)\,d\tau \ge \delta>0 \quad \text{for } \quad t\ge t^*>0. \]
The fundamental solution of equation (1), according to \((^1)\), can be written in the form
\[ W(\bar{x}-\bar{\xi},t) = \frac{e^{G(t)}}{(2\pi)^n} \int \exp\left[i(\bar{y}',\bar{\alpha})-\bar{\alpha}'A\bar{\alpha}\right]\,d\alpha, \tag{2} \]
where the integral is extended over the whole space, \(d\alpha=d\alpha_1\ldots d\alpha_n\),
\[ \bar{\alpha}= \begin{pmatrix} \alpha_1\\ \vdots\\ \alpha_n \end{pmatrix}, \qquad \bar{\alpha}'=(\alpha_1,\ldots,\alpha_n), \qquad \sum_{k=1}^{n}\alpha_k^2=(\bar{\alpha}',\bar{\alpha}), \]
\[ \bar{y}= \begin{pmatrix} B_1(t)+x_1-\xi_1\\ \cdots\\ B_n(t)+x_n-\xi_n \end{pmatrix} \equiv \bar{B}+\bar{x}-\bar{\xi}. \]
With the aid of (2), the solution of the Cauchy problem for equation (1) with bounded initial condition
\[ u\big|_{t=0}=\varphi(x_1,\ldots,x_n)\equiv \varphi(\bar{x}) \tag{3} \]
can be written as follows:
\[ u(t,\bar{x}) = \tag{4} \]
\[ = \frac{e^{G(t)}}{2^n\sqrt{\pi^n\det A}} \int \varphi(\bar{\xi}) \exp\left[ -\frac14 \sum_{i,j=1}^{n} \bar{A}_{ij}(t)(B_i+x_i-\xi_i)(B_j+x_j-\xi_j) \right], \]
where \(\bar{A}_{ij}(t)\) are the elements of the matrix \(A^{-1}\). For what follows, it is more convenient to present (4) in the form
\[ u(t,\bar{x})= \frac{e^{G(t)}}{\sqrt{\pi^n}} \int \varphi(\bar{B}+\bar{x}-2PA\bar{\xi})\exp[-(\bar{\xi}',\bar{\xi})]\,d\bar{\xi}, \tag{*} \]
where
\[ \Lambda^1 \equiv \begin{pmatrix} \lambda_1(t) & 0\\ & \ddots\\ 0 & \lambda_n(t) \end{pmatrix} = P'AP \]
is the Jordan normal form of the symmetric positive-definite matrix \(A\); \(P=\|p_{ij}(t)\|\) is an orthogonal matrix. Denote by \(\lambda_{\max}(t)\) and \(\lambda_{\min}(t)\), respectively, the maximum and minimum eigenvalues of the matrix \(A\); then the estimate holds:
\[ \exp\left[-\frac{1}{4\lambda_{\min}(t)}\right] \sum_{i=1}^{n}(B_i+x_i-\xi_i)^2 \leq \]
\[ \leq \exp\left[ -\frac14 \sum_{i,j=1}^{n}\bar{A}_{ij}(t)(B_i+x_i-\xi_i)(B_j+x_j-\xi_j) \right]\leq \]
\[ \leq \exp\left[ -\frac{1}{4\lambda_{\max}(t)} \sum_{i=1}^{n}(B_i+x_i-\xi_i)^2 \right]. \tag{5} \]
Let \(\bar{e}\) be the vector \((\varepsilon_1,\ldots,\varepsilon_n)\), where each \(\varepsilon_i\), for all \(i=1,2,\ldots,n\), can take only two different values \(\pm1\); \(R_{\bar e}\) is the angle of the space \(\{\varepsilon_1x_1\geq 0,\ldots,\varepsilon_nx_n\geq 0\}\); \(\bar a\) is an arbitrary vector \((a_1,\ldots,a_n)\), with
\[ [a]=\prod_{i=1}^{n} a_i; \]
\(R_{\bar e a}\) is the parallelepiped \(\{0\leq \varepsilon_1x_1\leq a_1,\ldots,0\leq \varepsilon_nx_n\leq a_n\}\).
Following (2), we shall say that \(\varphi(\bar{x})\) has angular limiting means \(l\), if
\[ E(\varphi)\equiv \lim_{\substack{a_1\to+\infty\\ \cdots\\ a_n\to+\infty}} \frac1{[a]} \int_{R_{\bar e a}}\varphi(\bar{\xi})\,d\bar{\xi} = l \tag{6} \]
for all vectors \(\bar e\). We say that \(\varphi(\bar{x})\) has a ball limiting mean equal to \(l\), if
\[ M(\varphi)\equiv \lim_{r\to+\infty} \frac1{c_n r^n} \int_{0}^{r}\int_{\Omega} \varphi(r'\bar{\omega})(r')^{n-1}\,d\Omega\,dr' = l, \tag{7} \]
where \(c_n\) is the volume of the \(n\)-dimensional unit ball; \(\Omega\) is the \(n\)-dimensional unit sphere; \(\bar{\omega}\) is a variable unit vector. We say that \(\varphi(\bar{x})\) has a spherical limiting mean equal to \(l\), if
\[ N(\varphi)\equiv \lim_{r\to+\infty} \frac1{s_n r^{\,n-1}} \int_{\Omega}\varphi(r\bar{\omega})r^{n-1}\,d\Omega = l, \tag{8} \]
where \(s_n\) is the area of the \(n\)-dimensional unit sphere \(\Omega\).
It can be proved that the existence, for a bounded function, of angular limiting means equal to \(l\) implies the existence for it of a spherical limiting mean equal to \(l\). But, as the simple example
\[ \varphi(x,y)= \begin{cases} 1, & \text{for } x>0,\ y>0 \text{ and } x<0,\ y<0,\\ -1, & \text{for } x<0,\ y>0 \text{ and } x>0,\ y<0, \end{cases} \]
shows, the converse is not true. It is easily proved that if a bounded function has a spherical limiting mean equal to \(l\), then it also has a ball limiting mean equal to \(l\).
Let \(u(t,\vec x)\) be the solution of the Cauchy problem for equation (1) with initial condition (3). Suppose that \(M(\varphi)=l\). The question arises: what conditions must be imposed on the coefficients of equation (1) so that the solution \(u(t,\vec x)\) stabilizes as \(t\to+\infty\)?
Theorem 1. If:
1) the trace of the matrix \(A\), \(\operatorname{Sp} A\to+\infty\) as \(t\to+\infty\);
2) there exists \(K\) such that \((\operatorname{Sp} A)^n/\det A\leq K\);
3) \[ \sum_{i=1}^{n} B_i^2(t)/\operatorname{Sp} A\to 0 \quad \text{as } t\to+\infty; \]
4) \[ \lim_{t\to+\infty} G(t)=c; \]
5) \(\varphi(\vec x)\) is bounded, \(M(\varphi)=l\), and \(\varphi(\vec x)-l\) preserves its sign,
then the solution \(u(t,\vec x)\) of the Cauchy problem (1), (3) stabilizes to \(le^c\), i.e.
\[ \lim_{t\to+\infty} u(t,\vec x)=le^c \]
uniformly in \(x\) in any bounded domain.
Theorem 2. If:
1) there exist a positive function \(\varkappa(t)\) and a constant \(K\) such that
\[ \varkappa(t)\sum_{i=1}^{n}\alpha_i^2 \geq \sum_{k,l=1}^{n} a_{kl}(t)\alpha_k\alpha_l; \qquad \int_{0}^{t}\varkappa(\tau)\,d\tau-\int_{0}^{t}\gamma(\tau)\,d\tau\leq K; \]
2) \[ \int_{0}^{t}\gamma(\tau)\,d\tau\to+\infty \quad \text{as } t\to+\infty; \]
3) \[ \sum_{i=1}^{n} B_i^2(t)\bigg/ \int_{0}^{t}\gamma(\tau)\,d\tau \to 0 \quad \text{as } t\to+\infty; \]
4) \[ \lim_{t\to+\infty}G(t)=c; \]
5) \(\varphi(\vec x)\) is bounded, \(M(\varphi)=l\), then the solution of the Cauchy problem (1), (3) stabilizes to \(le^c\) uniformly in \(x\) in any bounded domain.
We give a brief proof of Theorem 2.
Let \(\varphi(\vec x)-l=\psi(\vec x)\). Clearly, \(M(\psi)=0\). Using (*), we have:
\[ u(t,\vec x)-le^{G(t)} = \frac{e^{G(t)}}{\sqrt{\pi^n}} \int_{\Xi} \psi\!\left(\vec B+\vec x-2P\Lambda\vec \xi\right) \exp\left[-(\vec \xi,\vec \xi)\right]d\vec \xi = I_1+I_2, \tag{9} \]
where \(I_1\) is the integral over the domain \((\delta>|\vec \xi|)\cup(|\vec \xi|>R)\), and \(I_2\) over the remaining part of space \(\delta<|\vec \xi|<R\). Since \(\psi(\vec x)\) is bounded, \(|\psi(\vec x)|\leq M\), for every \(\varepsilon>0\) there exist \(\delta^*\) and \(R^*\) such that \(|I_1|<\varepsilon/3\) for \(\delta<\delta^*\) and \(R>R^*\). Passing in \(I_2\) to spherical coordinates, separating out the ball mean and integrating by parts, we obtain:
\[ I_2= \frac{e^{G(t)}}{\sqrt{\pi^n}} \left\{ e^{-r^2} \int_{0}^{r} \int_{\Omega} \psi\!\left(\vec B+\vec x-2P\Lambda r'\vec\omega\right) (r')^{n-1}\,d\Omega\,dr' \bigg|_{r=\delta}^{r=R} \right. \]
\[ \left. + \int_{\delta}^{R} 2re^{-r^2} \int_{0}^{r} \int_{\Omega} \psi\!\left(\vec B+\vec x-2P\Lambda r'\vec\omega\right) (r')^{n-1}\,d\Omega\,dr'\,dr \right\} = \frac{e^{G(t)}}{\sqrt{\pi^n}}\{I_2' + I_2''\}. \tag{10} \]
For sufficiently small \(\delta^*\) and \(1/R^*\), \(|I_2'|<\varepsilon/3\). Fixing \(\delta<\delta^*\) and \(R>R^*\), we make the change of variables \(2r'\sqrt{\lambda_{\min}(t)}=\xi\) in the inner integral of the expression \(I_2''\):
\[ I_2''=\int_\delta^R 2r^{n+1}e^{-r^2} \left\{ \frac{1}{\left(2r\sqrt{\lambda_{\min}(t)}\right)^n} \int_0^{2r\sqrt{\lambda_{\min}(t)}}\int_\Omega \psi\bigl(\bar B+\bar x-\xi P\tilde\Lambda\bar\omega\bigr)\xi^{n-1}\,d\Omega\,d\xi \right\}dr, \]
where
\[ \tilde\Lambda=\frac{1}{\sqrt{\lambda_{\min}(t)}}\Lambda. \]
The expression in braces is the integral of the function \(\psi(\bar x)\) over an \(n\)-dimensional ellipsoid with semiaxes \(\{2r\sqrt{\lambda_1(t)},\ldots,2r\sqrt{\lambda_n(t)}\}\) and center at the point \((\bar B(t)+\bar x)\), rotated in some way in the space by the orthogonal matrix \(P\). Using the conditions of the theorem and the fact that \(M(\psi)=0\), one can show that the expression in braces tends to zero as \(t\to+\infty\). Consequently, \(|I_2|<2\varepsilon/3\) for sufficiently large \(t\). The proof of the theorem now follows easily from this.
In particular, the conditions of Theorem 2 are satisfied for the equation
\[ \frac{\partial u}{\partial t}=a(t)\Delta u+\sum_{k=1}^n b_k(t)\frac{\partial u}{\partial x_k}, \qquad \text{where } a(t)>0, \]
\[ \int_0^t a(\tau)\,d\tau\to\infty,\qquad \int_0^t b_k(\tau)\,d\tau \bigg/ \left[\int_0^t a(\tau)\,d\tau\right]^{1/2}\to 0 \quad \text{as } t\to\infty \text{ for } k=1,2,\ldots,n. \]
Moreover, the following stabilization theorems hold:
Theorem 3. If:
1) \[ \lim_{t\to+\infty}G(t)=c; \]
2) \[ \lim_{|\bar x|\to+\infty}\varphi(\bar x)=l; \]
3) there exists at least one \(k\), \(k=1,2,\ldots,n\), such that
\[ \lim_{t\to+\infty}\left|\frac{B_k(t)}{\sqrt{\operatorname{Sp} A}}\right|=+\infty, \]
then the solution of the Cauchy problem (1), (3) stabilizes to \(le^c\) as \(t\to+\infty\), uniformly in \(x\) in any bounded domain.
Theorem 4. If:
1) \[ \lim_{t\to+\infty}G(t)=c; \]
2) \(\gamma(t)\) in the parabolicity condition \((1')\) is such that
\[ \int_0^t \gamma(\tau)\,d\tau\to+\infty \quad \text{as } t\to+\infty; \]
3) there exist constants \(K\) and \(l\) such that, for bounded \(\varphi(\bar x)\),
\[ \left| \int_0^{x_1}\cdots\int_0^{x_n}\bigl[\varphi(\bar\xi)-l\bigr]\,d\xi \right|\le K \qquad \text{for all } \bar x, \]
then the solution of the Cauchy problem (1), (3) stabilizes to \(le^c\) as \(t\to+\infty\), uniformly in \(x\) in any bounded domain.
Let us note that, in the case of one variable \((n=1)\), condition 3) of Theorem 4 will be satisfied for every uniformly almost periodic function whose Fourier exponents do not have zero as an accumulation point (see \((^3)\), p. 89). In particular, this condition will be satisfied for a periodic function.
Received
30 VI 1961
CITED LITERATURE
- I. G. Petrovsky, Bull. Moscow State Univ., A, no. 7 (1938).
- S. D. Eidelman, F. O. Porper, Izv. Vyssh. Uchebn. Zaved., Mathematics, no. 4, 210 (1960).
- B. M. Levitan, Almost Periodic Functions, Moscow, 1953.