Abstract Generated abstract
This paper studies generalized solutions of first order differential equations in a Banach space generated by strongly positive operators, extending earlier Hilbert space results. It establishes coercivity for equations with operators sharing a common domain, then treats nonautonomous equations with variable strongly positive operators and relatively small perturbations, proving existence, uniqueness, and a priori estimates in Bochner spaces. For the associated evolution operator, the paper derives smoothing and point singularity estimates in interpolation spaces under minimal regularity assumptions on the operator function, including conditions applicable to elliptic operators in \(L_p\) spaces with normal boundary conditions. It also gives an integral representation for inhomogeneous solutions and corresponding smoothness estimates for the integral term.
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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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