UDC 517.5

MATHEMATICS

P. E. SOBOLEVSKII

ON DIFFERENTIAL EQUATIONS WITH UNBOUNDED OPERATORS GENERATING NONANALYTIC SEMIGROUPS

(Presented by Academician I. G. Petrovskii, 21 III 1968)

In a Banach space \(E\) one considers the problem

\[ v'(t)+Av(t)=f(t)\quad (0<t\leq t_0),\qquad v(0)=v_0 . \tag{1} \]

Here \(A\) is a linear operator having an everywhere dense domain of definition \(D\). A function \(v(t)\), continuous on \([0,t_0]\), is called a solution of problem (1) if it satisfies (1) and the functions \(v'(t)\) and \(Av(t)\) are continuous on \((0,t_0]\).

It is assumed that there exists a continuously differentiable operator-function \(T(t)=\exp\{-tA\}\) \((t>0)\)—the resolving operator of problem (1)—satisfying the relation \(T'(t)=-AT(t)\) and commuting with \(A\). It is further assumed that there exists an integer \(n\geq 0\) such that \(T(t)v_0\to v_0\) as \(t\to +0\) for every \(v_0\in D(A^n)\). Then, if \(v(t)\) is a solution of problem (1) and the function \(A^n v(t)\) is continuous on \((0,t_0]\), then

\[ v(t)=T(t)v_0+\int_0^t T(t-s)f(s)\,ds =T(t)v_0+Q(t)f . \tag{2} \]

Finally, it is assumed that the operator-functions \(T(t)\) and \(AT(t)\) have power-type singularities at zero. Under these conditions it is shown that formula (2) determines a solution of problem (1), provided \(v_0\) and \(f(t)\) are sufficiently smooth. Moreover, if \(\|T(t)\|_{E\to E}\) is summable, then it suffices to require of \(f(t)\) that it satisfy a certain Hölder condition (Sec. 1). This fact is established by means of the method of fractional integration by parts (see \((^1)\)).

If \(T(t)\) has a singularity as \(t\to +0\), then problem (1) is not well posed. In Sec. 2, from \(T(t)\) a pair of Banach spaces \(E^+\) and \(E^-\), \(E^+\subset E\subset E^-\), is constructed, in which problem (1) becomes well posed. It follows from this that it is solvable in these spaces under the assumption only of Hölder continuity of \(f(t)\), for any polarity of \(AT(t)\).

The results of the first part of the work develop the investigations of the paper \((^2)\). In its second part the problem

\[ v'(t)+A(t)v(t)=f(t)\quad (0<t\leq t_0),\qquad v(0)=v_0 \tag{3} \]

is studied, with a variable operator \(A(t)\) having a constant everywhere dense domain of definition \(D\). The resolving operator \(U(t,\tau)\) of problem (3) can be sought by the method of “frozen coefficients” as a solution of the integral equations

\[ U(t,\tau)=T_\tau(t-\tau)+\int_\tau^t U(t,s)[A(\tau)-A(s)]T_\tau(s-\tau)\,ds, \tag{4} \]

\[ U(t,\tau)=T_t(t-\tau)+\int_\tau^t T_t(t-s)[A(t)-A(s)]U(s,\tau)\,ds. \tag{5} \]

Here \(T_\tau(t)=\exp\{-tA(\tau)\}\) is the resolving operator of problem (1) with \(A=A(\tau)\). If \(T_\tau(t)\) is strongly continuous in \(t\) as \(t\to +0\), while the norm

\(\|AT_\tau(t)\|_{E\to E}\) behaves like \(t^{-\beta}\) for \(\beta<2\), then equation (4) has a unique strongly continuous solution \(U(t,\tau)\) jointly in \(t\) and \(\tau\), under the assumption of a certain Hölder continuity of the operator-valued function \(A(t)A^{-1}(0)\). The operator-valued function \(U(t,\tau)\) will be the resolving operator for problem (3) if the adjoint operator-valued function \(A^*(t)\) has the same smoothness. This was established in \((^3)\). In \((^4)\) the result is generalized to the case where \(T_\tau(t)\) has a power singularity as \(t\to+0\). In \((^5)\), for \(\beta<7/5\), the resolving operator is constructed under the assumption of strong continuity of \(T_\tau(t)\) in \(t\) and Hölder continuity only of \(A(t)A^{-1}(0)\). Finally, in \((^6)\) this result is generalized to the case where \(T_\tau(t)\) has a singularity as \(t\to+0\).

In Sec. 3 the resolving operator is constructed for \(\beta<2\), under the assumption of Hölder continuity only of \(A(t)A^{-1}(0)\). We note that the condition \(\beta<2\) is necessary for the applicability of the method of “frozen coefficients.”

1. Suppose that, for some \(C,\delta,\alpha,\beta>0\), \(\beta\geq 1+\alpha\),

\[ \|T(t)\|_{E\to E}\leq ce^{-\delta t}t^{-\alpha},\qquad \|AT(t)\|_{E\to E}\leq ce^{-\delta t}t^{-\beta}. \tag{6} \]

Denote by \(C^\gamma=C^\gamma([0,t_0],E)\) \((0\leq\gamma\leq 1)\) the closure of the set \(K\) of all polynomials with coefficients in \(\bigcap_{n=1}^{\infty}D(A^n)\) in the norm

\[ \|f\|_{C^\gamma}=\max_{0\leq t\leq t_0}\|f(t)\|_E+ \sup_{0\leq t<t+\Delta t\leq t_0}\Delta t^{-\gamma}\|f(t+\Delta t)-f(t)\|_E . \]

The closure of \(K\) in the norm \(\|f\|_{C^0}+\|f^{(n)}\|_{C^\gamma}=\|f\|_{C^{\gamma+n}}\), for an integer \(n\geq0\), forms the Banach space \(C^{\gamma+n}\). Since \(D=D(A)\) is everywhere dense, \(C^{\gamma+n}\) does not depend on \(A\). The totality of all such functions \(f(t)\in C^{\gamma+n}\) \((n\geq1)\) for which \(f^{(i)}(0)\in D(A^{n-i})\) for \(i=0,\ldots,n-1\), forms the subspace \(C^{\gamma+n}(A)\). By definition we set \(C^{\gamma+0}(A)=C^\gamma\).

Theorem 1. Let \(n<\alpha<n+1\), \(f(t)\in C^{\gamma+n}(A)\), \(v_0\in D(A^{n+1})\), for some integer \(n\geq0\) and some \(\gamma\) in \(((\beta-n-1)/(\beta-\alpha),1)\). Then formula (2) defines a solution of problem (3), and the estimates

\[ \|v'(t)\|_E\leq ct^{n-\alpha},\qquad \|Av(t)\|_E\leq ct^{n-\alpha} \]

are valid.

We shall carry out the proof for the case \(n=0\). Since it is obvious that the function \(Q(t)f\) is a solution of equation (1), if \(f(t)\in K\), for such \(f(t)\) it suffices to establish the estimate

\[ \|AQ(t)f\|_E\leq ct^{-\alpha}\|f\|_{C^\gamma}. \tag{7} \]

Represent the function \(AQ(t)f\) in the form of a Stieltjes integral

\[ AQ(t)f=\int_0^t AT(t-\tau)f(\tau)\,d\tau =\int_0^t d_\tau[T(t-\tau)]f(\tau) =\int_0^1 d_s[\Phi(s)]\varphi(s). \]

Here \(\Phi(s)=T(t-ts)\), \(\varphi(s)=f(ts)\). Further, the last integral can be represented as the limit of sums

\[ s_n=\sum_{k=1}^{2^n}\left[\Phi(k/2^n)-\Phi((k-1)/2^n)\right]\varphi(k/2^n). \]

Since estimates (6) are satisfied, we have \(\|\Phi(s+\Delta s)-\Phi(s)\|_{E\to E}\leq ct^{\delta(1-\beta+\alpha)-\alpha}\Delta s^\delta\) for any \(\delta\) in \([0,1]\). Since \(f(t)\in C^\gamma\), we have \(\|\varphi(s+\Delta s)-\varphi(s)\|_E\leq ct^\gamma\Delta s^\gamma\). The last two estimates allow one to establish that, for
\(1-\gamma<\delta<(1-\alpha)/(\beta-\alpha)\), the estimate

\[ \|s_{n+1}-s_n\|_E\leq c(1-\alpha-\delta\beta+\delta\alpha)^{-1} t^{\delta(1-\beta+\alpha)+\gamma-\alpha}\,2^{-n(\delta+\gamma-1)} \|f\|_{C^\gamma},\qquad n=0,1,\ldots \]

is valid.

Finally, since \(\|s_0\|_E\leq ct^{-\alpha}\|f\|_{C^\gamma}\), for \(AQ(t)f=s=\lim s_n\) estimate (7) is valid. In the case of arbitrary \(n\), one must first transform formula (2) by integration by parts.

1. Denote by \(E^{+}(A)\) the closure of the set \(D(A^{n+1})\) in the norm
   \[ \|v\|_{E^{+}}=\sup_{t\geq 0}\|T(t)v\|_{E}. \]
   It is clear that \(\|v\|_{E}\leq \|v\|_{E^{+}}\). Further,
   \[ \|v\|_{E^{+}}\leq c\|A^{n+1}v\|_{E}. \]
   For the proof one may use the identity
   \[ A^{-n-1}=\int_{0}^{\infty}\cdots\int_{0}^{\infty}T(s_{1}+\cdots+s_{n+1})\,ds_{1}\cdots ds_{n+1}. \]

The norms of the spaces \(E\), \(E^{+}(A)\), and \(D(A^{n+1})\) are compatible; from the fundamentalness of a sequence in the strong norm and its convergence to zero in the weak norm there follows convergence to zero in the strong norm. Therefore the embeddings
\[ D(A^{n+1})\subset E^{+}(A)\subset E \]
hold.

Denote by \(E^{-}(A)\) the closure of the set \(E\) in the norm
\[ \|v\|_{E^{-}}=\sup_{t>0} t^{\alpha}\|T(t)v\|_{E}. \]
Analogously it is established that the embeddings
\[ E\subset E^{-}(A)\subset D(A^{-n-1}) \]
hold. Here by \(D(A^{-n-1})\) is meant the closure of the set \(E\) in the norm \(\|A^{-n-1}v\|_{E}\).

Theorem 2. The operator-function \(T(t)\) is a uniformly bounded and strongly continuous semigroup in the spaces \(E^{\pm}(A)\). The estimate
\[ \|AT(t)\|_{E^{\pm}\to E^{\pm}}\leq ct^{-\beta} \]
is valid.

This theorem makes it possible to investigate problem (1) in the spaces \(E^{\pm}(A)\), i.e. to study solutions with increased smoothness and generalized solutions of this problem. Problem (1) is well posed in the spaces \(E^{\pm}(A)\). The space of well-posedness for problem (1) can also be constructed by other methods, for example, with the aid of the norm
\[ \|v\|_{E^{p}}=\left(\int_{0}^{\infty}\|T(t)v\|_{E}^{p}\,dt\right)^{1/p},\qquad 1\leq p<+\infty . \]
Such a device was used earlier in the study of partial differential equations (see \((^{4})\)).

1. We turn to the consideration of problem (3). Let \(T_{\tau}(t)\), for each \(\tau\) from \([0,t_{0}]\), satisfy the estimates (6) with constants independent of \(\tau\). Suppose, moreover, that
   \[ \|A^{2}(\tau)T_{\tau}(t)\|_{E\to E}\leq ce^{-\delta t}t^{-\eta}. \tag{8} \]

Theorem 3. Let \(1<\beta<2\), \(0\leq \alpha\leq \beta-1\), \(2\beta-\alpha<\eta<2+\beta-\alpha\). Suppose that for some \(\varepsilon\) from \((\eta-2\beta+\alpha,\,2+\beta)\) the inequality
\[ \|[A(t)-A(\tau)]A^{-1}(0)\|_{E\to E}\leq c|t-\tau|^{\beta-1+\varepsilon} \]
is fulfilled.

Then there exists an operator-function \(U(t,\tau)\), defined for all \(0\leq \tau<t\leq t_{0}\), continuous in the totality of variables, continuously differentiable with respect to \(t\), and satisfying the relations
\[ U(t,\tau)=U(t,s)U(s,\tau)\quad(0\leq \tau<s<t\leq t_{0}),\qquad \frac{\partial}{\partial t}[U(t,\tau)]=-A(t)U(t,\tau). \tag{9} \]
The estimates
\[ \|U(t,\tau)\|_{E\to E}\leq c|t-\tau|^{-\alpha},\qquad \|A(t)U(t,\tau)A^{-1}(\tau)\|_{E\to E}\leq c|t-\tau|^{-\alpha}, \]
\[ \|A(t)U(t,\tau)\|_{E\to E}\leq c|t-\tau|^{-\beta} \tag{10} \]
are valid.

The operator-function \(U(t,\tau)A^{-1}(0)\) tends strongly and uniformly in \(t\) and \(\tau\) to the operator \(A^{-1}(0)\) as \(t-\tau\to +0\). For \(\tau<t\) it is continuously differentiable with respect to \(\tau\) and satisfies the relation
\[ \frac{\partial}{\partial \tau}[U(t,\tau)A^{-1}(0)]=U(t,\tau)A(\tau)A^{-1}(0). \tag{11} \]

Suppose \(f(t)\in C^{\gamma}\) for some \(\gamma\) from \(((\beta-1)/(\beta-\alpha),\,1)\) and \(v_{0}\in D\). Then there exists a unique continuous on \([0,t_{0}]\) and continuously diff-

differentiable on \((0,t_0]\) solution \(v(t)\) of problem (3), the formula

\[ v(t)=U(t,0)v_0+\int_0^t U(t,s)f(s)\,ds =U(t,0)v_0+Q(t,0)f, \tag{12} \]

is valid, and the estimates

\[ \|v'(t)\|_E\le ct^{-\alpha},\qquad \|A(t)v(t)\|_E\le ct^{-\alpha} \]

hold.

We give the scheme of the proof. The interval \([0,t_0]\) is divided into \(n\) equal parts by the points \(\tau_0=0,\tau_1,\ldots,\tau_n=t_0\). The operators \(U_n(t,\tau)\) are defined, for arbitrary \(\tau_{i-1}\le \tau<\tau_i\le \cdots \le t=\tau_k,\ \tau_{r-1}\le \tau_r\le \tau_r\), by the formula

\[ U_n(t,\tau)=T_{\tau_k}(\tau_k-\tau_{k-1})\ldots T_{\tau_i}(\tau_i-\tau). \tag{13} \]

It is shown that the operators \(U_n(t,\tau)\) satisfy an identity \((*)\) of Volterra type, analogous to relation (4) for the operator \(U(t,\tau)\). In this identity, sums occur instead of integrals. With the aid of (4) and \((*)\) it is established that \(U_n(t,\tau)\to U(t,\tau)\), the solution of (4). Next, the operators \(\widetilde U_n(t,\tau)\) are constructed analogously to the operators \(U_n(t,\tau)\), only now the points \(\tau\) must coincide with the points of the subdivision of \([0,t_0]\). The operators \(\widetilde W_n(t,\tau)=A(t)\widetilde U_n(t,\tau)A^{-1}(\tau)\) are introduced and it is shown that \(\widetilde W_n(t,\tau)\to W(t,\tau)\). Here \(W(t,\tau)\) denotes the solution of the equation into which equation (5) is transformed under the substitution \(W(t,\tau)=A(t)U(t,\tau)A^{-1}(\tau)\). Since \(U_n(t,\tau)-\widetilde U_n(t,\tau)\to 0\), it follows from this that \(U(t,\tau)=\lim U_n(t,\tau)\) and \(A(t)U(t,\tau)A^{-1}(\tau)\) are jointly continuous in the variables for \(t>\tau\) and satisfy the first two estimates (10). This is the first main fact. Since \(U_n(t,\tau)\) is constructed in the form of the product (13), \(U(t,\tau)\) satisfies the first of relations (9). This is the second main fact. Next one uses the identity following from (4), (5), and (9) \((0\le \tau<t<t+\Delta t\le t_0,\ v_0\in D)\)

\[ [U(t+\Delta t,\tau)-U(t,\tau)]v_0 = \int_t^{t+\Delta t} U(t+\Delta t,s)[A(t)-A(s)] \left\{T_t(s-\tau)v_0+\right. \]

\[ \left. +\int_\tau^t T_t(s-z)[A(t)-A(z)]U(z,\tau)v_0\,dz \right\}ds +[T_t(t+\Delta t-\tau)-T_t(t-\tau)]v_0+ \]

\[ +\int_\tau^t [T_t(t+\Delta t-s)-T_t(t-s)][A(t)-A(s)]U(s,\tau)v_0\,ds, \]

by means of which the derivative

\[ \frac{\partial}{\partial t}\,[U(t,\tau)v_0] \]

is computed.

For the proof of the second part of the theorem it is necessary to represent the function \(A(t)Q(t,0)f\) in the form of a Stieltjes integral (see (11))

\[ A(t)Q(t,0)f=\int_0^t A(t)U(t,s)f(s)\,ds =\int_0^t d_s[A(t)U(t,s)]A^{-1}(s)f(s) \]

and, for its estimate, to use the method of Section 1.

Voronezh State University

Received
13 III 1968

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