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UDC 517.5
MATHEMATICS
P. E. SOBOLEVSKII
GENERALIZED SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS IN A BANACH SPACE
(Presented by Academician I. G. Petrovskii on 12 IV 1965)
1. In (¹) generalized solutions of differential equations in Hilbert space were studied. The coercivity inequalities established in (²) for equations in a Banach space made it possible to obtain, for such equations, analogous and, in a certain sense, stronger results. Namely, for solutions of homogeneous equations with a variable operator, under minimal restrictions on the smoothness of this operator, it is possible to obtain the same estimates (point singularities) as in the case of a constant operator (Theorems 4 and 5). Such estimates are important in applications to nonlinear equations (see, for example, (¹, ³)).
2. Let \(A\) be a strongly positive operator in a Banach space \(E\). This means that \(A\) generates an analytic semigroup \(\exp\{-tA\}\), whose norm decreases exponentially. From the operator \(A\) we construct the spaces \(E_\alpha(A)\) \((0<\alpha<1)\) with norms \(|v|_\alpha^A\). If \(A_1\) is strongly positive and \(D(A_1)=D(A)\), then \(E_\alpha(A_1)=E_\alpha(A)\) and the norms \(|v|_\alpha^{A_1}\) and \(|v|_\alpha^A\) are equivalent (see (², ⁴)). Therefore, in what follows, in the notation of spaces and norms constructed from operators with the same domain of definition, the designation of the operator is omitted.
Consider in \(E\) the problem
\[ v' + Av = f(t) \quad (0 \le t \le T), \qquad v(0)=v_0 . \tag{1} \]
By a generalized solution of problem (1) in \(B_p([0,T],E)\)* \((1<p<\infty)\) we shall mean an absolutely continuous function \(v(t)\) on \([0,T]\) which satisfies the equation and the initial condition (1) almost everywhere and has the property:
(a) the functions \(v'\), \(Av \in B_p\), and the function \(v(t)\) is continuous in \(E_{1/q}\) \((1/p+1/q=1)\).
If for every function \(f(t)\in B_p\) and for every element \(v_0\in E_{1/q}\) there exists a unique generalized solution in \(B_p\) of problem (1) and the inequality
\[ \|v'\|_{B_p}+\|Av\|_{B_p}+\max_{0\le t\le T}|v(t)|_{1/q} \le K_p(A)\bigl(\|f\|_{B_p}+|v_0|_{1/p}\bigr), \tag{2} \]
holds, then we shall say that coercivity in \(B_p\) holds for problem (1). If this fact holds in some one \(B_{p_0}\), then it holds in every \(B_p\) (²).
Theorem 1. Let coercivity in \(B_p\) hold for problem (1). Then, for the analogous problem with an operator \(A_1\) having the same domain of definition as \(A\), coercivity in \(B_p\) also holds.
\[ \text{* } B_p([0,T],E) \text{ is the Bochner space with norm } \|v\|_{B_p}=\left(\int_0^T \|v(t)\|^p\,dt\right)^{1/p} \]
(see (⁵)).
This theorem makes it possible to establish coercivity for equations with a complex operator \(A_1\), if it holds for equations with a simple operator \(A\).
- Let us consider a problem more general than (1),
\[ v' + A(t)v + F(t)v = f(t) \quad (0 \leq t \leq T), \qquad v(0)=v_0. \tag{3} \]
Suppose that \(A(t)\), for each \(t \in [0,T]\), is strongly positive, \(D[A(t)]=D[A(0)]=D\), and the operator-function \(A(t)A^{-1}(0)\) has only discontinuities of the first kind. Suppose that \(F(t)\) is closed, \(D[F(t)] \supset D\), and the operator-function \(F(t)A^{-1}(0)\) is strongly measurable.
By a generalized solution of problem (3) in \(B_p\) we shall mean an absolutely continuous function \(v(t)\) on \([0,T]\) which, for almost all \(t\), satisfies the equation and the initial condition (4), and possesses property (a) with the operator \(A=A(0)\).
Theorem 2. Suppose that for each fixed \(t \in [0,T]\), for problem (1) with the operator \(A=A(t)\), coercivity holds in \(B_p\). Suppose that for all \(t \in [0,T]\) and \(v \in D\) the inequality \(\|F(t)v\| \leq \delta(t)\|A(t)v\| + C\|v\|\) holds, with \(\delta(t)K_p[A(t)] \leq \delta < 1\), where \(K_p[A(t)]\) is the constant occurring in inequality (2). Then, for any \(f(t) \in B_p\) and \(v_0 \in E_{1/q}\), problem (3) has a unique generalized solution in \(B_p\), and inequality (2) holds with \(A=A(0)\).
This theorem generalizes Theorem 1 of (1).
- Consider the homogeneous problem
\[ v' + A(t)v = 0 \quad (\tau \leq t \leq T), \qquad v(\tau)=v_0. \]
We shall denote its solution by \(U(t,\tau)v_0\). By Theorem 2, \(U(t,\tau)\) is an operator-function, strongly continuous jointly in \(t\) and \(\tau\), for \(0 \leq \tau \leq t \leq T\), in any space \(E_\alpha\), satisfying the condition \(U(t,\tau)=U(t,s)U(s,\tau)\) \((\tau \leq s \leq t)\). The following holds (cf. (1), Theorem 2).
Theorem 3. For any \(0<\alpha \leq \beta < 1\), \(0 \leq \tau \leq t \leq T\), the inequalities hold
\[ \left|[U(t,\tau)-I]v\right|_\alpha \leq C(\alpha,\beta)|t-\tau|^{\beta-\alpha}|v|_\beta \qquad (v \in E_\beta), \]
\[ \left[\int_\tau^t |U(s,\tau)v|_\beta^{1/(\beta-\alpha)}\,ds\right]^{\beta-\alpha} \leq C(\alpha,\beta)|v|_\alpha \qquad (v \in E_\alpha). \]
If \(\alpha<\beta\), then in the left-hand sides of these inequalities the norms \(|w|_\gamma\) may be replaced by the norms \(\|A^\gamma(0)w\|\).
In proving this theorem, inequalities of the form (2), moment inequalities from (4), and the inequality
\[ |v|_\beta \leq C(\alpha-\beta,\gamma-\beta) |v|_\alpha^{(\beta-\gamma)/(\alpha-\gamma)} \|A^\gamma(0)v\|^{(\alpha-\beta)/(\alpha-\gamma)} \]
\[ (v \in E_\delta,\ \delta=\max\{\alpha,\gamma\}), \]
valid for any \(\alpha \in (0,1)\), \(\gamma \in [0,1]\), \(\alpha \ne \gamma\), and \(\beta \in (\alpha,\gamma)\), are used.
The second of the inequalities of Theorem 3 means that the operator \(U(t,\tau)\) acts not only in \(E_\alpha\), but, for almost all \(t \geq \tau\), from \(E_\alpha\) into \(E_\beta\).
This assertion can be sharpened.
Theorem 4. For any \(0<\alpha \leq \beta < 1\) and \(0 \leq \tau \leq t \leq T\), the inequality
\[ |U(t,\tau)v|_\beta \leq C(\alpha,\beta)|t-\tau|^{\alpha-\beta}|v|_\alpha \qquad (v \in E_\alpha). \tag{4} \]
holds. If \(\alpha<\beta\), the norm \(|w|_\beta\) on the left may be replaced by the norm \(\|A^\beta(0)w\|\).
Theorem 5. Suppose that for any \(t \in [0,T]\) and \(\alpha \in (0,1)\), the operator \(A(t)\) admits closure to a bounded operator from \(E_\alpha\) to \(E_{1-\alpha}^{*}\), and suppose that this closure is an operator-function having only discontinuities of the first kind. Then, for any \(0<\alpha<1\) and \(0 \leq \tau \leq t \leq T\), the inequality
\[ |U(t,\tau)v|_\alpha \leq C(\alpha)|t-\tau|^{-\alpha}\|v\| \qquad (v \in E). \tag{5} \]
holds.
If, on the left, the norm \(|w|_\alpha\) is replaced by the norm \(\|A^\alpha(0)w\|\), then (5) is valid for \(\alpha \geqslant 0\).
We note that the conditions of Theorem 5 are satisfied when \(A(t)\) is an elliptic operator in the space \(L_p(\Omega)\) with normal boundary conditions (for the definition of a normal elliptic operator see, for example, (6)).
We outline the proofs of Theorems 4 and 5. From the identity
\[ v'(t)+A(\tau)v(t)=[A(\tau)-A(t)]v(t)\qquad (\tau \leqslant t \leqslant T), \]
which is satisfied by the function \(v(t)=U(t,\tau)v\), it follows that
\[ U(t,\tau)v=\exp\{-(t-\tau)A(\tau)\}v+ \]
\[ +\int_\tau^t \exp\{-(t-s)A(\tau)\}[A(\tau)-A(s)]U(s,\tau)v\,ds =\exp\{-(t-\tau)A(\tau)\}v+ \]
\[ +\int_{(t+\tau)/2}^t \exp\{-(t-s)A(\tau)\}[A(\tau)-A(s)] U\left(s,\frac{t+\tau}{2}\right)\,ds\, U\left(\frac{t+\tau}{2},\tau\right)v+ \]
\[ +\exp\left\{-\frac{t-\tau}{2}A(\tau)\right\} \int_\tau^{(t+\tau)/2} \exp\left\{-\left(\frac{t+\tau}{2}-s\right)A(\tau)\right\} \times \]
\[ \times [A(\tau)-A(s)]U(s,\tau)v\,ds. \tag{6} \]
For simplicity, suppose that \(A(t)A^{-1}(0)\) is continuous. Then it follows from (6) that
\[ |U(t,\tau)v|_\beta \leqslant C|t-\tau|^{\alpha-\beta}|v|_\alpha +\varepsilon \left|U\left(\frac{t+\tau}{2},\tau\right)v\right|_\beta +C|t-\tau|^{\alpha-\beta}|v|_\alpha, \]
where \(\varepsilon \to 0\) as \(t-\tau \to 0\). Here, to estimate the integral
\[ \int_\tau^{(t+\tau)/2} \]
the coercivity inequality (2) was used with \(p=\beta/(1-\beta)\), and to estimate the integral
\[ \int_{(t+\tau)/2}^{t} \]
inequality (2) was used with \(p=\alpha/(1-\alpha)\). Hence (4) follows at once. If now one uses the condition of Theorem 5 to estimate
\[ \int_\tau^{(t+\tau)/2}, \]
then, for \(\alpha>0\), one can obtain the inequality
\[ |U(t,\tau)v|_\alpha \leqslant C|t-\tau|^{-\alpha}\|v\| +\varepsilon \left|U\left(\frac{t+\tau}{2},\tau\right)v\right|_\alpha+ \]
\[ +\varepsilon |t-\tau|^{-1} \int_\tau^{(t+\tau)/2}\|U(s,\tau)v\|_\alpha\,ds. \]
From this (5) follows for \(\alpha>0\). To estimate \(\|U(t,\tau)v\|\), one must use the estimates already obtained and the identity (6).
- From (5) it follows that, under the conditions of Theorem 5, the operator-function \(U(t,\tau)\) is strongly continuous jointly in the variables in \(E\). Hence it follows (cf. (1), Theorem 3).
Theorem 6. The generalized solution in \(B_p\) of problem (3) for \(F(t)\equiv 0\) has the form
\[ v(t)=U(t,0)v_0+Qf(t),\qquad \text{where}\quad Qf(t)=\int_0^t U(t,s)f(s)\,ds. \]
The smoothness properties of the operator \(Qf(t)\) are described by the analogue of Theorem 4 from (1):
Theorem 7. Let \(0<\alpha,\ \beta<1\), \(0\le t\le t+\Delta t\le T\), \(f(t)\in B_{1/\beta}\). Then
\[ \left|Qf(t+\Delta t)-Qf(t)\right|_{\alpha} \le \Delta t^{1-\beta-\alpha} C(\alpha,\beta) \left[ \int_{0}^{t+\Delta t}\|f(s)\|^{1/\beta}\,ds \right]^{\beta} \qquad (\alpha+\beta<1), \]
\[ \left[ \int_{0}^{t} \left|Qf(s)\right|_{\alpha}^{1/(\beta+\alpha-1)}\,ds \right]^{\beta+\alpha-1} \le C(\alpha,\beta) \left[ \int_{0}^{t}\|f(s)\|^{1/\beta}\,ds \right]^{\beta} \qquad (\alpha+\beta>1). \]
If \(\alpha+\beta\ne 1\), then on the left the norm \(|w|_{\alpha}\) may be replaced by the norm \(\|A^{\alpha}(0)w\|\).
Voronezh Agricultural Institute
Received
23 III 1965
CITED LITERATURE
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- P. E. Sobolevskii, DAN, 157, No. 1 (1964).
- P. E. Sobolevskii, Tr. Mosk. Matem. Obshch., 10, 297 (1961).
- P. E. Sobolevskii, UMN, 19, issue 6 (120) (1964).
- E. Hille, Functional Analysis and Semigroups, IL, 1951.
- M. Schechter, Comm. Pure and Appl. Math., 12, No. 3 (1959).