Abstract
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MATHEMATICS
M. A. KRASNOSEL’SKII, S. G. KREIN, and P. E. SOBOLEVSKII
ON DIFFERENTIAL EQUATIONS WITH UNBOUNDED OPERATORS IN HILBERT SPACE
(Presented by Academician A. N. Kolmogorov on 25 IX 1956)
We shall consider the equation
\[ \frac{dx}{dt}+A(t)x=f(t) \tag{1} \]
in a Hilbert space \(H\), where \(A(t)\), for each \(t\) \((0\leq t\leq b)\), is a self-adjoint operator, and \(f(t)\) is a function with values in \(H\).
In the more general case of a Banach space, this equation was investigated by T. Kato \((^{1})\). Under certain assumptions he showed, using the theory of semigroups, that the solution of the homogeneous equation
\[ \frac{dx}{dt}+A(t)x=0 \tag{2} \]
with initial condition \(x_0\) from the domain of definition \(D(A)\) of the operator \(A(s)\):
\[ x(s)=x_0 \tag{3} \]
can be represented in the form
\[ x(t)=U(t,s)x_0, \tag{4} \]
where \(U(t,s)\) is an operator, bounded and jointly continuous in \(t\) and \(s\). The solution of equation (1) is then constructed by Kato in the form
\[ x(t)=U(t,0)x_0+Qf(t), \tag{5} \]
where
\[ Qf(t)=\int_0^t U(t,s)f(s)\,ds \tag{6} \]
under stringent restrictions on the function \(f(t)\).
In the authors’ paper \((^{2})\), the operator \(Q\) was subjected to further study. In particular, it was shown that (5) gives a solution of equation (1) for any continuously differentiable \(f(t)\). Judging from the review in Math. Rev., 17, No. 1 (1956), Kato also obtained an analogous result. With the aid of the operator \(Q\), in \((^{2})\) an equation of type (1) with nonlinearities was also considered.
In the particular case considered below, the operators \(U\) and \(Q\) can be studied in greater detail. We note that by other methods equation (1) with operators \(A(t)\) acting in a Hilbert space was studied in the works of M. Vishik and O. Ladyzhenskaya \((^{3,4})\).
- We assume that the following conditions are satisfied.
a) The operators \(A(t)\) are self-adjoint and uniformly bounded below:
\[ (A(t)x,x)\geq (x,x). \]
b) The operators \(A^{-\alpha}(t)\), for \(0\leq \alpha \leq 1\), are strongly differentiable with respect to \(t\), and the operators
\[ C_\alpha(t)=A^\alpha(t)\frac{d}{dt}A^{-\alpha}(t) \tag{7} \]
are bounded uniformly with respect to \(\alpha\) and \(t\).
c) The operator \(C_1(t)\) is strongly continuous with respect to \(t\) and bounded.
Conditions b) and c) are satisfied in an obvious way if \(A(t)\) does not depend on \(t\). Condition b) is easily verified if the operators \(A(t)\) commute with one another for different \(t\).
It is of interest to ask under what conditions b) follows from a) and c).
A bounded operator \(F\) acting in \(H\) will be called absolutely bounded with respect to an orthonormal basis \(\{e_i\}\) if the infinite matrix with entries \(|(Fe_i,e_j)|\) defines a bounded operator in Hilbert space. We shall denote the norm of this operator by \(N(F,e_i)\). We note that operators having finite absolute norm (see \((5)\)) are absolutely bounded with respect to any basis.
Theorem 1. Suppose that the operator \(A^{-1}(t)\) is strongly differentiable with respect to \(t\) and is completely continuous for each \(t\). Suppose that the operators \(C_1(t)\) are absolutely bounded with respect to the basis \(\{e_i(t)\}\) formed from eigenvectors of the operators \(A(t)\), and that the function \(N[C_1(t),e_i(t)]\) is uniformly bounded on \([0,b]\).
Then b) follows from conditions a) and c).
The proof of the theorem is based on a generalization of the formula for differentiating, with respect to a parameter, functions of self-adjoint operators, obtained in \((6)\).
- Formula (4) makes it possible to construct solutions of problem (2)—(3) also in the case when \(x_0\in \overline{D(A)}\).
Theorem 2. The function \(x(t)\) defined by formula (4) satisfies, for \(t>s\), the homogeneous equation (2) for any \(x_0\in H\).
A more general assertion for the case of a constant operator \(A\) was established in \((7)\) (see also \((8)\)).
Theorem 2 is based on the fact that \(U(t,s)x_0\in D(A)\) for \(t>s\) and for any \(x_0\in H\). In the proof of the subsequent theorems a stronger fact is used, which, as it seems to us, is of independent interest.
Theorem 3. For \(t>s\) and \(0\leq \alpha<2\), the operators \(A^\alpha(t)U(t,s)\) are bounded, and the estimate
\[ \bigl\|A^\alpha(t)U(t,s)\bigr\|\leq M(t-s)^{-\alpha}. \tag{8} \]
holds.
Estimate (8) is also valid for \(\alpha=2\), if \(C(t)\) satisfies the Lipschitz condition \(\mathrm{Lip}\,\beta\):
\[ \|C(t)-C(s)\|\leq L|t-s|^\beta . \tag{9} \]
If the operator \(A\) is constant, then (8) holds for all \(\alpha>0\).
It follows from Theorem 3, in particular, that when condition (9) is satisfied, the solution of the homogeneous equation (2) is twice differentiable with respect to \(t\) for \(t>s\).
- Theorem 4. Suppose the function \(f(t)\) satisfies the condition \(\mathrm{Lip}\,\varepsilon\), \(\varepsilon\leq 1\). Then formula (5) determines, for \(t>0\), a solution of equation (1) for any initial condition \(x_0\in H\).
If \(x_0\in D(A)\), then this solution has the property
\[ \left\|A^\alpha(t)\frac{dx}{dt}\right\|\leq M|t|^{-\alpha} \]
for \(\alpha<\varepsilon\).
In this theorem, for the case of equations in Hilbert space, the corresponding result from the paper \((2)\), where equations in Banach spaces were considered, is strengthened.
- Let \(C\) denote the space of functions \(f(t)\), continuous on \([0,b]\), with values in \(H\), with norm \(\|f\|_{C}=\max \|f(t)\|\), and let \(C_0^1\) denote the space of continuously differentiable functions, equal to zero for \(t=0\), with norm \(\|f\|_{C_0^1}=\max \|f'(t)\|\). The operator \(Q\), defined by formula (6), acts and is bounded in these spaces.
Theorem 5. If \(f(t)\in C\), then
\[ \|Qf(t+\Delta t)-Qf(t)\|\leq K_1\Delta t\,|\ln \Delta t|\,\|f\|_{C}. \]
If \(f(t)\in C_0^1\), then
\[ \left\|\frac{d}{dt}Qf(t+\Delta t)-\frac{d}{dt}Qf(t)\right\| \leq K_2\Delta t\,|\ln \Delta t|\,\|f\|_{C_0^1}. \]
From Theorems 4 and 5 there follows the theorem:
Theorem 6. If the operator \(A^{-1}(t)\) is completely continuous, then the operator \(Q\) is completely continuous in the spaces \(C\) and \(C_0^1\).
- We turn to the consideration of the nonlinear equation
\[ \frac{dx}{dt}+A(t)x=f(t,x). \tag{10} \]
We shall assume that the operator \(A(t)\) satisfies conditions a), b), c), and that for each \(t\) the operator \(A^{-1}(t)\) is completely continuous. Concerning \(f(t,x)\), we shall assume that it is a continuous bounded operator with values in \(H\), defined on \([0,b]\times T\), where \(T\) is the ball of radius \(r\) with center at \(x_0\).
In parallel with equation (10), consider the integral equation
\[ x(t)=U(t,0)x_0+Qf[t,x(t)]. \tag{11} \]
If the solution \(x(t)\) of the integral equation (11) is such that \(f[t,x(t)]\) satisfies the condition \(\operatorname{Lip}\alpha\;(\alpha\leq 1)\), then, by Theorem 4, the function \(x(t)\) is a solution of the differential equation (10), satisfying the initial condition \(x(0)=x_0\).
Theorem 7. Equation (11) has a continuous solution on some interval \((0,h]\) \((h\leq b)\). If the condition
\[ \|f(t+\Delta t,x+\Delta x)-f(t,x)\|\leq K\bigl(|\Delta t|^\alpha+\|\Delta x\|^\alpha\bigr)\quad(\alpha\leq 1) \]
is fulfilled, then every continuous solution of equation (11) is, for \(t>0\), a solution of equation (10).
- The requirement of continuity of the operator \(f(t,x)\) is, for a Hilbert space, very stringent.
If, for example, one considers the operator \(f(x)=x^2\), then on \(L_2\) it is not everywhere defined and is not continuous. However, this operator is bounded and continuous in the space of continuous functions, which is a part of \(L_2\). Therefore it is important to study the case of operators \(f(t,x)\) that are bounded and continuous on some linear set \(M\) of the space \(H\), which is a Banach space with respect to another norm \(\|x\|_M\). In this case the inequality
\[ \|x\|\leq K\|x\|_M\quad (x\in M) \tag{12} \]
usually holds.
Suppose that the operator \(f(t,x)\) acts from \(M\) into \(H\) and is continuous, while the operator \(U(t,0)x_0+Qy\) acts from \(H\) into \(M\). Then in equation (10) one may make the substitution
\[ x(t)=U(t,0)x_0+Qy(t) \tag{13} \]
and pass to the integral equation
\[ y(t)=f[t,U(t,0)x_0+Qy(t)] \tag{14} \]
with a continuous operator in \(H\) on the right-hand side.
Theorem 8. Let the operators \(A(t)\) satisfy the conditions of the preceding paragraph and, in addition, let the operators \(A^{-\varepsilon}(t)\), for some \(\varepsilon\), \(0<\varepsilon<1\), act from \(H\) into \(M\), with
\[ \|A^{-\varepsilon}(t)x\|_M \leq q\|x\|_H \quad (x\in H), \]
where \(q\) does not depend on \(t\). Let the operator \(f(t,x)\), with values in \(H\), be defined and continuous in a neighborhood of the point \(\{0,x_0\}\) of the space \([0,b]\times M\). Finally, let \(x_0\in D(A^\varepsilon)\).
Then on some interval \([0,h]\), \(h\leq b\), there exists a solution, continuous in \(H\), of the integral equation (14). This solution, by formula (13), determines a solution \(x(t)\) of equation (10) that is continuous in \(M\) and differentiable in \(H\), provided the function \(f(t,x)\) satisfies the condition
\[ \|f(t+\Delta t,x+\Delta x)-f(t,x)\|_H \leq K\left(|\Delta t|^\alpha+\|\Delta x\|_M^\alpha\right). \]
7. In conclusion we give an example. Consider the mixed problem with zero boundary conditions for the heat equation with nonlinearity:
\[ \frac{\partial u}{\partial t}=\Delta u+f(t,s,u) \]
in a bounded domain of \(n\)-dimensional space. For simplicity we shall assume that the function \(f(t,s,u)\) is continuously differentiable with respect to all \(n+2\) variables. From Theorem 8 there follows a local theorem on the existence of bounded solutions of this mixed problem for spaces of dimension \(n=2,3\). If the dimension \(n>3\), then the existence of solutions can be proved under restrictions on the growth of the function \(f(t,s,u)\):
\[ |f(t,s,u)|\leq a+b|u|^m, \]
where \(m<1+4/(n-4)\).
In the investigation of the mixed problem just presented, the results of V. A. Il’in \({}^{9}\) and M. Riesz on fractional powers of the Laplace operator and the general theorems on the splitting of a linear operator \({}^{10}\) are used essentially.
Received
2 IX 1956
REFERENCES
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