Abstract
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MATHEMATICS
I. I. SHMULEV
BOUNDED SOLUTIONS OF BOUNDARY-VALUE PROBLEMS WITHOUT INITIAL CONDITIONS FOR PARABOLIC EQUATIONS AND INVERSE BOUNDARY-VALUE PROBLEMS
(Presented by Academician S. L. Sobolev on 24 VII 1961)
Bounded classical solutions of the Dirichlet and Neumann problems without initial conditions for the equation
\[ u_t=\Delta u+qu+F(x,t) \qquad (q=\mathrm{const}) \]
were studied in \((^1)\). In the present note, for the study of these problems both for linear and for nonlinear parabolic equations, a simple method is proposed, borrowed from the theory of elliptic equations and originating in the works \((^{2-4})\).
Let \(D\) be a bounded \(m\)-dimensional domain of the space of variables \((x_1,\ldots,x_m)=x\), and let \(\Gamma\) be the boundary of \(D\). Denote by \(Q=D\times(-\infty,+\infty)\) the straight cylinder of \(m+1\) dimensions, and by \(S\) the lateral surface of \(Q\). The part of \(Q\) enclosed between the planes \(t=t_1\) and \(t=t_2\), where \(t_1\) and \(t_2\) are arbitrary numbers from \((-\infty,+\infty)\) and \(t_2>t_1\), will be denoted by \(Q_{[t_1,t_2]}\), and by \(S_{[t_1,t_2]}\) the lateral surface of \(Q_{[t_1,t_2]}\). Finally, denote by \(L\) the elliptic operator
\[ L=\sum_{i,j=1}^{m} a_{ij}(x,t)\frac{\partial^2}{\partial x_i\partial x_j} +\sum_{i=1}^{m} b_i(x,t)\frac{\partial}{\partial x_i} +c(x,t), \]
where
\[ \sum_{i,j=1}^{m} a_{ij}(x,t)\xi_i\xi_j \geq \alpha \sum_{i=1}^{m}\xi_i^2,\quad a_{ij}(x,t)=a_{ji}(x,t) \quad\text{and}\quad -c(x,t)\geq c_0, \]
for \((x,t)\in\overline Q\); \(\alpha\) and \(c_0\) are positive numbers.
Below, by a solution of one or another problem we shall mean a classical solution of the problem under consideration.
Consider in \(Q\) the Dirichlet problem without initial conditions:
\[ u_t=Lu+f(x,t), \tag{1} \]
\[ u\big|_S=\varphi(x,t). \tag{2} \]
Theorem 1. Let the coefficients of \(L\) and the bounded function \(f(x,t)\) be such that, in any \(Q_{[t_1,t_2]}\), the boundary-value problem
\[ u_t=Lu+f(x,t),\qquad u\big|_{S_{[t_1,t_2]}}=\Phi(x,t),\qquad u(x,t_1)=\Psi(x) \]
has a solution, whatever the continuous functions \(\Phi(x,t)\), \(\Psi(x)\) may be \((\Psi|_{\Gamma}=\Phi)\). Then, if \(\varphi(x,t)\) is continuous and bounded, problem (1), (2) has in \(Q\) a unique bounded solution.
We indicate a brief scheme of the proof. Let \(v(x,t)\) be the solution in \(Q^{-}=D\times(-\infty,0]\) of the boundary-value problem
\[ v_t=Lv+\lambda v+e^{\lambda t}f(x,t), \tag{3} \]
\[ v\big|_{S^-}=e^{\lambda t}\varphi(x,t), \tag{4} \]
admitting in \(\overline{Q^-}\) the estimate
\[ |v(x,t)|=O\left(e^{\lambda t}\right). \tag{5} \]
The number \(\lambda>0\) is chosen according to the condition \(c_0-\lambda>0\). We shall prove the existence of \(v(x,t)\).
Construct in \(Q^-\) a sequence \(\{v_n(x,t)\}\) as follows. Let \(\{Q_n^-\}\) be a sequence of nested cylinders:
\[
Q_n^-=D\times [t_n,0],\qquad t_n\to -\infty.
\]
Define \(v_n(x,t)\) in \(Q_n^-\) as the solution of the boundary-value problem
\[ v_{nt}=Lv_n+\lambda v_n+e^{\lambda t}f(x,t), \tag{6} \]
\[ v_n\big|_{S_{[t_n,0]}}=e^{\lambda t}\varphi_n(x,t), \tag{7} \]
\[ v_n(x,t_n)=0, \tag{8} \]
where \(\varphi_n(x,t)\) is defined by the equalities
\[ \varphi_n(x,t)= \begin{cases} \varphi(x,t), & t_n+\delta\leq t\leq 0,\\[4pt] \varphi(x,t)\cos\left(\dfrac{t-t_n-\delta}{\delta}\right)\dfrac{\pi}{2}, & t_n\leq t\leq t_n+\delta, \end{cases} \]
in which the number \(\delta>0\) is sufficiently small. In the domain \(\overline{Q^-}-\overline{Q_n^-}\) set \(v_n(x,t)=0\).
By the hypothesis of the theorem, problem (6)—(8) has a solution in \(Q_n^-\), for which, as is not difficult to show, estimate (5) is valid. With the aid of this estimate one then establishes the uniform convergence in \(\overline{Q^-}\) of the sequence \(\{v_n(x,t)\}\) to some function \(v(x,t)\), which, as is shown, will be a solution of equation (3) in every bounded part of \(Q^-\).
It is immediately clear that the function \(u^-=ve^{-\lambda t}\) will be a bounded solution in \(Q^-\) of the boundary-value problem:
\[ u_t^-=Lu+f(x,t),\qquad u^-\big|_{S^-}=\varphi(x,t), \]
where \(S^-\) is the lateral surface of \(Q^-\).
Define \(u^+(x,t)\) as the solution in \(Q^+=D\times[0,+\infty)\) of the boundary-value problem:
\[ u_t^+=Lu^+ + f(x,t), \tag{9} \]
\[ u^+\big|_{S^+}=\varphi(x,t), \tag{10} \]
\[ u^+(x,0)=u^-(x,0), \tag{11} \]
where \(S^+\) is the lateral surface of \(Q^+\). The function
\[ u(x,t)= \begin{cases} u^+(x,t), & 0\leq t<+\infty,\\ u^-(x,t), & -\infty<t\leq 0, \end{cases} \]
is a bounded solution of problem (1), (2).
We shall prove the uniqueness of the bounded solution of problem (1), (2). Let \(u_1\) and \(u_2\) be two bounded solutions of problem (1), (2), and let \(u=u_2-u_1\). Denote by \(u^-\) the part of \(u\) defined in \(Q^-\). The function \(v=e^{\lambda t}u^-\), where \(0<\lambda<c_0\), is a solution in \(Q^-\) of the boundary-value problem
\[ v_t=Lv+\lambda v,\qquad v\big|_{S^-}=0. \]
Whatever \(T\in(-\infty,0)\) may be, in \(Q^-_{[T,0]}\) the estimate \(|v(x,t)|=O(e^{\lambda T})\) holds, following from the maximum principle for parabolic equations. The indicated estimate makes it possible to conclude that in \(Q^-\) the function \(v=0\), i.e. \(u^-=0\). But then, on the basis of the uniqueness theorem for the problem
\[ u_t=Lu, \tag{12} \]
(9)—(11), the part \(u\) defined in \(Q^{+}\) is also equal to zero: \(u^{+}=0\). Thus in \(\overline Q\), \(u_2 \equiv u_1\). Let us note an important special case of Theorem 1.
Theorem 2. If the conditions of Theorem 1 are satisfied, and the coefficients \(L\) and the functions \(f(x,t)\) and \(\varphi(x,t)\) are periodic in \(t\) with period \(T\), then in \(Q\) there exists a unique solution of problem (1), (2) that is periodic in \(t\) with period \(T\).
Let us now consider in \(Q\) the Neumann problem without initial conditions:
\[ \left(\frac{\partial u}{\partial \gamma}-a(x,t)u\right)\bigg|_{S} =\varphi(x,t) \qquad (a(x,t)\geqslant a_0=\text{const}>0), \tag{13} \]
where \(\gamma\) is the direction of the conormal to \(S\).
Theorem 3. Let \(S\) belong to the class \(A^{(2)}\), and let the coefficients \(L\) and the continuous and bounded function \(a(x,t)\) be such that in any \(Q_{[t_1,t_2]}\) the boundary-value problem
\[ u_t=Lu,\qquad (\partial u/\partial\gamma-a(x,t)u)\big|_{S_{[t_1,t_2]}}=\Phi(x,t), \qquad u(x,t_1)=\Psi(x) \]
has a solution, whatever the continuous functions \(\Phi(x,t)\) and \(\Psi(x)\) may be. Then, if \(\varphi(x,t)\) is continuous and bounded, problem (12), (13) has in \(Q\) a unique bounded solution.
The proof of this theorem repeats the main points of the proof of Theorem 1; however, in addition to the maximum principle for equation (12), Theorem 1 of paper \((^5)\) is also used.
Theorem 4. If the conditions of Theorem 3 are satisfied, and the coefficients \(L\) and the functions \(a(x,t)\) and \(\varphi(x,t)\) are periodic in \(t\) with period \(T\), then in \(Q\) there exists a unique solution of boundary-value problem (12), (13) that is periodic in \(t\) with period \(T\).
The method of proof of Theorems 1 and 3, with some modification, may be used in the study of bounded solutions of nonlinear boundary-value problems.
Let us begin with the first boundary-value problem:
\[ u_t=\sum_{i,j=1}^{m} a_{ij}(x,t,u)\frac{\partial^2 u}{\partial x_i\partial x_j} +\sum_{i=1}^{m} b_i(x,t,u)\frac{\partial u}{\partial x_i} +c(x,t,u)u+f(x,t), \tag{14} \]
\[ u\big|_{S}=0 \qquad (s\in A^{(2,\lambda)}), \tag{15} \]
where
\[ \sum_{i,j=1}^{m} a_{ij}(x,t,u)\xi_i\xi_j \geqslant \alpha\sum_{i=1}^{m}\xi_i^2 \]
for \((x,t)\in \overline Q\) and \(u\in(-\infty,+\infty)\); \(a_{ij}=a_{ji}\) and \(-c(x,t,u)\geqslant c_0\) for the indicated values of \(x,t\), and \(u\); \(\alpha\) and \(c_0\) are positive numbers.
Theorem 5. In the cylinder \(Q\) there exists at least one bounded solution of boundary-value problem (14), (15), if the following conditions are satisfied:
-
For \((x,t)\in\overline Q\), \(|u|\leqslant C_0\) \((C_0=\mathrm{const}>0)\), both the functions \(a_{ij}(x,t,u)\), \(b_i(x,t,u)\), \(c(x,t,u)\) and their derivatives with respect to \(x,u\) up to the fourth order inclusive are continuous, bounded, and satisfy a Hölder condition in \(x,u\), with
\[ \max_{(x,t,u)}|\partial a_{ij}/\partial u| \leqslant \alpha e\sqrt{3}/12\, m C_0\,^*. \] -
The function \(f(x,t)\) and its derivatives with respect to \(x\) up to the fourth order inclusive are continuous and bounded.
The proof of this theorem uses a priori estimates of solutions and their derivatives for the first boundary-value problem for equation (14), obtained in papers \((^6,^7)\).
Theorem 6. In the strip \(Q=\{0<x<l,\ -\infty<t<+\infty\}\) there exists at least one bounded solution of the boundary-value problem
* This condition can apparently be replaced by the condition that \(|\partial a_{ij}/\partial u|\) is simply bounded (see \((^8)\)).
\[ u_t = a(x,t,u)_{xx} + b(x,t,u)u_x + c(x,t,u)u + f(x,t), \tag{16} \]
\[ u_x(0,t)=\varphi_1(u(0,t),t), \tag{17} \]
\[ u_x(l,t)=\varphi_2(u(l,t),t), \tag{18} \]
if the following conditions are satisfied:
-
For \((x,t)\in \overline Q,\ |u|\le C_0\) \((C_0=\mathrm{const})\), the functions \(a(x,t,u)\), \(b(x,t,u)\), \(c(x,t,u)\), \(f(x,t)\) and their derivatives with respect to \(x,u\) up to and including the fourth order are continuous and bounded, and the derivatives of second order with respect to \(x,u\) have bounded first-order derivatives with respect to \(t\).
-
For \((x,t)\in \overline Q,\ u\in(-\infty,+\infty)\), the inequalities \(a(x,t,u)\ge \alpha\), \(-c(x,t,u)\ge c_0\) hold, where \(\alpha,c_0\) are positive numbers.
-
For \(t\in(-\infty,+\infty),\ |u|\le C_0\), the functions \(\varphi_1(u,t)\), \(\varphi_2(u,t)\) are continuous and bounded together with their derivatives with respect to \(u\) up to and including the second order and with their first-order derivatives with respect to \(t\), and \(\varphi_1(0,t)=\varphi_2(0,t)=0\).
-
For \(t\in(-\infty,+\infty),\ u\in(-\infty,+\infty)\), the inequalities \(\varphi_{1u}(u,t)\ge \beta_1\), \(-\varphi_{2u}(u,t)\ge \beta_2\) hold, where \(\beta_1,\beta_2\) are positive numbers.
The proof of the theorem is based on the results of paper (9).
We indicate the connection between the questions considered and inverse boundary-value problems.
Theorem 7. If the hypotheses of Theorem 1, referred to \(Q^-\), are satisfied, then the inverse boundary-value problem
\[ u_t=Lu+f(x,t), \tag{19} \]
\[ u|_{S^-}=\varphi(x,t)\qquad (t\in(-\infty,0]), \tag{20} \]
\[ u(x,0)=\psi(x)\qquad (\psi|_{\Gamma}=\varphi) \tag{21} \]
has a bounded solution in \(Q^-\) if and only if the continuous function \(\psi(x)\equiv u^-(x,0)\), where \(u^-(x,t)\) is the solution of problem (19), (20) bounded in \(Q^-\).
Theorem 8. If the hypotheses of Theorem 3, referred to \(Q^-\), are satisfied, then the inverse boundary-value problem
\[ u_t=Lu, \tag{22} \]
\[ (\partial u/\partial \gamma-a(x,t)u)|_{S^-}=\varphi(x,t)\qquad (t\in(-\infty,0]), \tag{23} \]
\[ u(x,0)=\psi(x) \tag{24} \]
has a bounded solution in \(Q^-\) if and only if the continuous function \(\psi(x)\equiv u^-(x,0)\), where \(u^-(x,t)\) is the solution of problem (22), (23) bounded in \(Q^-\).
Since the function \(u^-(x,t)\) appearing in each of these theorems is determined uniquely, as was established earlier, it follows from these theorems that the inverse Dirichlet and Neumann boundary-value problems are ill-posed in the class of bounded functions.
I take this opportunity to express my gratitude to S. G. Krein, who drew my attention to certain questions touched upon in the present note.
Voronezh Forestry Engineering Institute
Received
10 VII 1961
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