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DIFFERENTIAL EQUATIONS
APRIL 1967, VOLUME III, № 4
A LINEAR PARABOLIC PROBLEM IN AN INFINITE CYLINDER: STABILIZATION OF SOLUTIONS
We consider a general linear parabolic problem in an infinite cylinder $Q = G \times R^+$, where $G$ is a finite $n$-dimensional domain in space bounded by an $(n-1)$-dimensional surface $\Gamma$. We assume $\Gamma$ is infinitely smooth, locally rectifiable, and oriented. The time domain $R^+$ is the semi-infinite line $t > 0$. The problem is formulated as follows:
$$\begin{aligned} \frac{\partial u}{\partial t} &= A(x, \frac{\partial}{\partial x}) u + f(x, t), \quad (x, t) \in Q \\ B(x, t) u &= \phi(x, t), \quad (x, t) \in S \\ u|_{t=0} &= u_0(x), \quad x \in G \end{aligned}$$
Here, $u$ and $f$ are vector columns of height $m$, and $S = \Gamma \times R^+$ represents the lateral surface of the cylinder. The operator $A$ is a square matrix differential operator of the form:
$$A(x, \frac{\partial}{\partial x}) = \sum_{|k| \le 2p} a_k(x) D^k$$
where $k = (k_1, k_2, \dots, k_n)$ is a multi-index, $|k| = \sum k_i$, and $D^k = \frac{\partial^{|k|}}{\partial x_1^{k_1} \dots \partial x_n^{k_n}}$. The boundary conditions are defined by the operator $B(x, t)$, which typically involves derivatives with respect to the spatial coordinates.
The primary objective of this study is to investigate the behavior of the solution $u(x, t)$ as $t \to \infty$. Specifically, we analyze the conditions under which the solution stabilizes, meaning it approaches a steady-state or follows a predictable asymptotic trajectory as time increases indefinitely. This involves examining the spectral properties of the operator $A$ and the regularity of the source terms $f(x, t)$ and boundary data $\phi(x, t)$.
The system $\frac{\partial u}{\partial t} = \dots$ satisfies the condition of parabolicity in the sense of I. G. Petrovskii (see \cite{1}) in the domain $Q = \Omega \times (0, T)$, where $T > 0$. Specifically, the principal part of the matrix $D$ satisfies the condition $\det A(x, t, \xi) \neq 0$ for $|\xi| \neq 0$. Here, $2m$ denotes the order of the system (1), $p$ represents the parabolic weight (which is an integer), and $B(x, t)$ is a rectangular matrix satisfying the "regular solvability" condition (see \cite{2,3,4,5,6}). Specifically, this refers to a one-dimensional boundary value problem involving matrices, where the principal parts of the matrices $A(x, t)$ and $B(x, t, p)$ are given by:
$$ A(x, t) \frac{\partial^2 z}{\partial x^2} + B(x, t, p) \frac{\partial z}{\partial x} + \dots $$
written in the local coordinate system, is uniquely solvable. The functions $a(x, t)$ and $b(x, t)$ are infinitely smooth in $Q$. Under these assumptions, M. S. Agranovich and M. I. Vishik (see \cite{6}) studied the parabolic problem in a finite cylinder with operators $A(x, t)$ and $B(x, t)$ of general form:
$$\begin{aligned} A(x, t)u &= f(x, t) \text{ in } Q_{0T} \\ B(x, t)u &\to g(x', t) \text{ on } Q'_{0T} \end{aligned}$$
where $Q_{0T} = G \times (0 < t < T)$ and $Q'_{0T} = \Gamma \times (0 < t < T)$. They also examined the parabolic problem in an infinite cylinder for the case where the coefficients $a(x, t) = a(x)$ and $b(x, t) = b(x)$ depend only on $x$.
In the aforementioned work, the problem (7)–(9) is investigated by reducing it to a stationary (semi-bounded) problem.
LINEAR PARABOLIC PROBLEM IN A CYLINDER INFINITE IN TIME
Along with equations $(1)$–$(3)$, we will consider the problem:
$$\begin{aligned} A(x, D)u = A(x, D + p)u = f(x), & \quad (x, t) \in Q_{\pm} \\ B(x, D)u = B(x, D + p)u = g(x), & \quad t \to +0 \end{aligned}$$
This is obtained from $(1)$–$(3)$ by substitution. In particular, $A(x)u \sim A(x, D + p)u = f(x)$.
For $(x', t) \in Q_{+}$, as $t \to +0$ and $p \to +0$ with $p < -1$ and $x \in G$, we will primarily employ the notation established in works \cite{5, 6, 7, 8}. Let $l$ be an arbitrary fixed number satisfying the condition that $l$ is an integer such that $l > \max(2m, m_j + 1)$. We define $\mu = l - 2m$ and $\mu_j = l - m_j - \frac{1}{2}$ for $x \in G$.
The space is defined as $\mathcal{H}^l(Q_{T_1 T_2}, Q'_{T_1 T_2}) = H^l(Q_{T_1 T_2}) \times \prod_{j=1}^m H^{\mu_j}(G)$. Here, $H = W_2^l$ denotes the Sobolev space (see \cite{8}) and $H_\mu$ denotes L. N. Slobodetskii spaces (see \cite{5} and \cite{7}). The norms of the vectors $(f, g) \in (G; \Gamma)$ are defined as the sum of the norms of their components:
$$\| (f, g) \|_{(G; \Gamma)} = \| f \|_{W^s_p(G)} + \sum \| g_j \|_{W^{s-1/p}_p(\Gamma)}$$
Let $\mathcal{A}$, $\mathcal{A}_j$ ($j=1, 2, 3$), and $\mathcal{A}_4$ be the operators corresponding to problems (1)–(3), (7)–(9), (12)–(14), and (15)–(17), respectively, acting on the appropriate spaces. Let $\mathcal{L}$ be the operator corresponding to problem (4)–(6), acting from the subspace of the space $H$ consisting of functions consistent with zero at $t=0$, into the subspace of the space of functions $(f, g)$ consistent with zero at $t=0$. Finally, let $\Pi$ be the operator:
$$\Pi(x): w \to (A(x)w, B(x)w|_{x'}) \in G^{(p)}$$
corresponding to problem (10), (11). Estimates related to functions belonging to $H(e^{-pt})$ or $H$ will be written with constants that are independent of the functions themselves and the parameter $p$. We present here the formulation of the main results from the cited work by M. S. Agranovich and M. I. Vishik:
Theorem I. For a finite $T$, the operator $\Pi(x, t)$ possesses an inverse $\Pi^{-1}(x, t)$ acting from the subspace of elements $g$ in $H$ that are compatible with zero at $t=0$ to the subspace of elements $u$ in $H$ that are also compatible with zero at $t=0$. Moreover, the following estimate holds:
$$C(\gamma) \|\Pi(x, t)u\| \le \|u\| \le C'(\gamma) \|\Pi(x, t)u\|$$
Theorem II. For each operator $\Pi(x)$, one can specify a number $\delta = \delta(\Pi(x))$, depending only on $\Pi(x)$, such that:
a) For any $p$ satisfying the inequality $\text{Re } p > \delta$, the operator $\Pi(x)$ possesses an inverse $\Pi^{-1}(x)$ acting from $H_s(G)$ to $H_{s+2b}(G)$, and the following estimate holds:
$$C_1 \|u\|_{H_{s+2b}(G)} \le \|\Pi(x)u\|_{H_s(G)} \le C_2 \|u\|_{H_{s+2b}(G)}$$
b) For any $p$ satisfying the inequality $\text{Re } p > \delta$, the operator $\Pi(x)$ possesses an inverse acting on $H(e^{-pt})(Q_+; \Omega_+)$, and the following estimate holds:
$$\|u\|_{H(e^{-pt})(Q_+; \Omega_+)} \le C \|\Pi(x)u\|_{H(e^{-pt})(Q_+; \Omega_+)}$$
We aim to extend these results to the general case of operators $\Pi(x, t)$ and to investigate the stabilization property of solutions as $t \to +\infty$.
EXISTENCE AND UNIQUENESS THEOREM. APRIORI ESTIMATES
In this section, we assume that the derivatives of the functions $b(x, t)$ satisfy the following condition: for any $\epsilon > 0$, one can specify a $T(\epsilon)$, depending only on the fixed $\epsilon$, such that for $t > T(\epsilon)$ and $x \in G$:
$$|a_{ij}(x, t) - a_{ij}(x)| < \epsilon \tag{24}$$
The Lemmas I–V presented in this section will be proven using a method entirely analogous to the proofs of the corresponding propositions in work \cite{6}.
Lemma I
The operator $L(x, t)$ (corresponding to problem (1)–(3)) acts boundedly from $H^{l+2, \frac{l+2}{2}}(Q_\infty)$ to $H^{l, \frac{l}{2}}(Q_\infty)$ for any $\rho$ satisfying the specified conditions. More precisely, the following estimate holds:
$$ \| e^{-\rho t} L(x, t) u \|_{H^{l, \frac{l}{2}}(Q_\infty)} \leq C \| e^{-\rho t} u \|_{H^{l+2, \frac{l+2}{2}}(Q_\infty)} \tag{26} $$
We now consider the following problem in the domain $\Omega_\Lambda(x, \tau + T)$:
$$\begin{aligned}
u &= A(x, \tau + T) w = f(x, \tau), \quad (x, \tau) \in \Omega \\
B(x, \tau + T) u &= \phi(x, \tau), \quad (x, \tau) \in \Omega^+
\end{aligned}$$
This corresponds to the operator $\Pi(T): u \to (A(x, x + T)u|_{Q(T)}; B(x)u|_{Q(T)})$. For any $T > 0$, an estimate analogous to (26) holds:
$$\|u - \Pi(x, x + T)u\| \le C \|u\| \tag{26'}$$
Lemma II
For any $\epsilon > 0$, there exists a $T(\epsilon)$ depending only on $\epsilon$ such that for $T > T(\epsilon)$, the operator $\Pi(x, x+T) - \Pi(x): u \to ((A(x, x+T) - A(x))u; (B(x, x+T) - B(x))u)$ satisfies:
$$ \|\Pi(x, x+T) - \Pi(x)\| \le \epsilon \|e^{-\beta T} u\| \tag{26''} $$
Lemma III
There exists a $T_0$ such that for $T \ge T_0$, the operator $A(x, \tau + T)$ satisfies the following a priori estimate:
$$ \|u\|_{W_2^{2b, \tau}(\Omega)} \le 2C_1 \|e^{-p\tau} A(x, \tau + T) u\|_{L_2(Q)} \tag{34} $$
Lemma IV
There exists a $T_0$ such that for $T \ge T_0$, the operator $A(x, \tau + T)$ possesses a bounded inverse $A^{-1}(x, \tau + T)$.
Proof. According to Theorem II, b) of M. S. Agranovich and M. I. Vishik, the operator $A(x)$ for $\text{Re } p > \delta$ possesses an inverse acting as a bounded operator. We write $A(x, \tau + T) = A(x) + (A(x, x + T) - A(x))$. Multiplying on the right by $A^{-1}(x)$, we obtain $A(x, \tau + T) A^{-1}(x) = I + S(x, \tau + T)$, where $S(x, \tau + T) = (A(x, x + T) - A(x)) A^{-1}(x)$. By virtue of (26') with $\epsilon = \frac{1}{4}$, the operator $I + S(x, \tau + T)$ possesses a bounded inverse.
Lemma V
For $\text{Re } p > \sigma$, where $\sigma = \sigma(\Lambda(x))$, the following a priori estimate holds for the operator $\Lambda(x, T + \tau)$:
$$\| (I - P) u \|_{H} \leq C \| \Lambda(x, T + \tau) u \|_{H}$$
SPATIAL STABILIZATION OF SOLUTIONS
According to the theory developed by L. N. Slobodetskii (see \cite{7}), one can construct a continuation into the domain $\Omega_{T, \infty}$. Denoting this continuation by $\tilde{u}$, it satisfies:
$$\| \tilde{u} \|_{H} \leq C e^{\rho(t-T)} \| f \|_{H}, \quad T \in \Omega_\infty$$
It is clear that $e^{P(t-T)}$ holds, and by applying the substitution $x = t - T$, we obtain $(x + T)(u - u_{app}) e^{P(t-T)}$. Then, according to $(34)$, for $T > T_0$ and $\text{Re } p > \delta$, we have:
$$ \| (x, x + T) (u - u_{app}) e^{-P(t-T)} \| < \text{const } R \| e^{-P(t-T)} \Pi(x, t) u_{app} \| $$
Theorem. For any $p$ satisfying the condition $\text{Re } p > \delta$, the operator $L(p)$ corresponding to problem (1)–(3) possesses an inverse $L^{-1}(p)$ acting in $H_s$, and the following estimate holds:
$$\|u(x, t)\| \le C \|e^{-\mu t} f(x, t)\| \tag{38}$$
Lemma. Let $\alpha > 0$ be a constant. Then the function $z = \int_{0}^{x} \frac{f(t)}{(1+t)^\alpha} dt$ possesses the property that for any constant $\epsilon > 0$, the limit exists in the $L_p$ metric.
Lemma. Let condition (45) be satisfied for some constant. Then the solution to the problem (15)–(17) and the solution $v$ to the problem (7)–(9) possess the property that for any constant $\epsilon > 0$, the limit $\lim_{t \to \infty} u$ exists.
Theorem. Suppose that for some constant $(P)$, the condition holds. Then, for any $p$ satisfying the condition $\text{Re } p > \delta$, the solution $u$ of the problem (14) stabilizes in the metric to the solution $w$ of the corresponding stationary problem (10), (11).
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