MATHEMATICS

S. Ya. YAKUBOV

ON THE SOLVABILITY OF THE CAUCHY PROBLEM FOR EVOLUTION EQUATIONS

(Presented by Academician I. G. Petrovskii, 24 I 1964)

The paper studies the equation

\[ \frac{d^{n}u(t)}{dt^{n}}+A_1(t)\frac{d^{n-1}u(t)}{dt^{n-1}}+\ldots+A_n(t)u(t)=f(t) \tag{1} \]

with initial conditions

\[ u^{(i)}(0)=u_i \qquad (i=0,\ldots,n-1), \tag{2} \]

where \(u(t)\) is the unknown function with values in a Banach space \(E\); \(A_i(t)\) are operators acting in \(E\).

The investigation of problem (1)—(2) is carried out by methods of the theory of semigroups. By a solution of problem (1)—(2) we shall mean a function \(u(t)\), \((n-1)\) times continuously differentiable on \([0,T]\), \(n\) times continuously differentiable on \((0,T]\), satisfying equation (1) for every \(t\in(0,T]\) and the initial conditions (2), and possessing, in addition, the property that the functions
\(A_i(t)d^{\,n-i}u(t)/dt^{\,n-i}\) \((i=1,\ldots,n)\) and
\(A_1(t)d^i u(t)/dt^i\) \((i=1,\ldots,n-2)\) are continuous on \((0,T]\).

1. Let us first consider the case of a first-order equation

\[ \frac{du(t)}{dt}=Au(t)+f(t) \tag{3} \]

with the initial condition

\[ u(0)=u_0. \tag{4} \]

Problem (3)—(4), when \(A\) generates a semigroup of class \(C_0\), was studied by R. Phillips \((^2)\). The case when \(A\) is a strongly positive operator, i.e. generates a semigroup of class \(H(\Phi_1,\Phi_2)\), was considered by M. Z. Solomyak \((^3)\) and K. Yosida \((^4)\). Problem (3)—(4) with a variable operator \(A(t)\) has been studied in detail in the works of T. Kato \((^5)\), M. A. Krasnosel’skii, S. G. Krein, and P. E. Sobolevskii \((^6)\), P. E. Sobolevskii \((^7)\), and others. The case in which \(A(t)\), for every \(t\in[0,T]\), is a generating operator of class \(C_0\) or \((C_0)_u\) has been studied in detail \((^1)\).

Lemma. Let \(A\) be the generating operator of some semigroup \(T(t)\) of class \((0,A)\) \([(1,A)]\)*. If \(f(t)\) is continuously differentiable (continuous) on \([0,\infty)\), then

\[ g(t)=\int_0^t T(t-\tau)f(\tau)\,d\tau \]

is continuous on \([0,\infty)\). If, however, \(f(t)\) is twice continuously differentiable (continuously differentiable), then \(g(t)\) is continuously differentiable and, for \(t>0\),

\[ g'(t)=T(t)f(0)+\int_0^t T(t-\tau)f'(\tau)\,d\tau =f(t)+A\int_0^t T(t-\tau)f(\tau)\,d\tau . \]

* In \((^1)\), a generating operator is called an infinitesimal generating operator.

Let us note that for semigroups of class \(C_0\) this lemma was proved by R. Phillips \((^2)\).

Theorem 1. Suppose that the following conditions are satisfied: 1) the closed linear operator \(A\) is the infinitesimal generator of some semigroup \(T(t)\) of class \((0,A)\) \([(1,A)]\); 2) \(f(t)\) is a twice continuously differentiable (continuously differentiable) vector-function; 3) \(u_0\in D(A)\).

Then the solution of problem (3)—(4) exists and is given by the formula

\[ u(t)=T(t)u_0+\int_0^t T(t-\tau)f(\tau)\,d\tau, \tag{5} \]

and the equality

\[ \frac{du(t)}{dt}=AT(t)u_0+T(t)f(0)+\int_0^t T(t-\tau)f'(\tau)\,d\tau \]

holds.

If, moreover, \(u_0\in D(A^2)\) and \(f(0)\in D(A)\), then equation (3) is satisfied also for \(t=0\).

Remark. If \(u_0\in E\), then formula (5) will be called a generalized solution of equation (3). The expediency of this definition follows from Theorem 1. We replace the initial condition (4) by the condition

\[ \lim_{\lambda\to\infty}\lambda\int_0^T e^{-\lambda t}u(t)\,dt=u_0 \tag{6} \]

(the continuity of \(u(t)\) at zero in the Abel sense). It is easy to see that the generalized solution of problem (3)—(6) is continuous on \((0,T]\).

1. In the work of B. S. Mityagin \((^8)\) the equation

\[ \frac{d^n u(t)}{dt^n}+A_1\frac{d^{\,n-1}u(t)}{dt^{\,n-1}}+\cdots+A_nu(t)=f(t) \tag{7} \]

with operators \(A_i\) independent of \(t\) \((i=1,\ldots,n)\) was investigated. In the case \(n=2\), problem (1)—(2) was investigated by P. E. Sobolevskii \((^{9,10})\) and A. Balakrishnan \((^{11})\).

In the present article the method of reducing a higher-order equation to a system is used (this method differs from the method developed in \((^{8-10})\)). For brevity we shall illustrate the method for \(n=2\):

\[ \frac{d^2u(t)}{dt^2}+A(t)\frac{du(t)}{dt}+B(t)u(t)=f(t), \tag{8} \]

\[ u(0)=u_0,\qquad u'(0)=u_1. \]

With the aid of the substitution \(v(t)=du(t)/dt\) and \(w(t)=A(0)u(t)\), problem (8) is reduced to the Cauchy problem for a system of first-order evolutionary equations

\[ \frac{dw(t)}{dt}-A(0)v(t)=0, \]

\[ \frac{dv(t)}{dt}+A(t)v(t)+B(t)A^{-1}(0)w(t)=f(t), \tag{9} \]

\[ w(0)=A(0)u_0, \]

\[ v(0)=u_1. \]

In the topological product \(E\times E\), problem (9) can be written in the form

\[ \frac{dU(t)}{dt}+\mathfrak{A}(t)U(t)=F(t), \]

\[ U(0)=U_0, \]

where

\[ U(t)=\binom{w(t)}{v(t)},\qquad F(t)=\binom{0}{f(t)},\qquad U_0=\binom{A(0)u_0}{u_1}, \]

\[ \mathfrak{A}(t)= \begin{pmatrix} 0 & -A(0)\\ B(t)A^{-1}(0) & A(t) \end{pmatrix} = \begin{pmatrix} 0 & -A(0)\\ 0 & A(t) \end{pmatrix} + \begin{pmatrix} 0 & 0\\ B(t)A^{-1}(0) & 0 \end{pmatrix} = \mathfrak{A}_1(t)+\mathfrak{A}_2(t). \]

In the topological product \(E \times E\) the resolvent of the operator \(\mathfrak A_1(t)\) is expressed in terms of the resolvent of the operator \(A(t)\):

\[ R(\lambda,-\mathfrak A_1(t))= \left( \begin{array}{cc} \dfrac{1}{\lambda} A(0)A^{-1}(t)\left[\dfrac{1}{\lambda}-R(\lambda,-A(t))\right] \\[6pt] 0\quad R(\lambda,-A(t)) \end{array} \right). \tag{10} \]

Formula (6) makes it possible to express any function of the operator \(\mathfrak A_1(t)\) in terms of the same function of the operator \(A(t)\); namely, the formula

\[ f(\mathfrak A_1(t))= \left( \begin{array}{cc} \beta & A(0)A^{-1}(t)\,[\beta-f(A(t))] \\ 0 & f(A(t)) \end{array} \right), \]

is proved, where \(\beta\) is a certain number depending on the function \(f(z)\).

Applying perturbation theory for various classes of semigroups and Theorem 1, we prove Theorem 2.

Theorem 2. Suppose the following conditions are satisfied: 1) the closed linear operator \(-A_1\) is the infinitesimal generator of a certain semigroup of class \((0,A)\) \([(1,A)]\); 2) the operators \(A_iA_1^{-1}\) \((i=1,\ldots,n)\) are bounded; 3) \(f(t)\) is a twice continuously differentiable (continuously differentiable) vector function; 4) \(u_i\in D(A)\) \((i=0,\ldots,n)\).

Then problem (1)—(2) has a solution, and moreover a unique one.

In the analogous theorem of B. S. Mityagin ((8), Theorem 4), under the assumption that \(-A_1\) is the infinitesimal generator of a semigroup of class \(C_0\), more is required, namely, the possibility of the decomposition

\[ \sum_{j=1}^{n} A_j\lambda^{\,n-j} = A_1(\lambda-F_1)(\lambda-F_2)\cdots(\lambda-F_{n-1}), \]

where \(F_i\) \((i=1,\ldots,n-1)\) are bounded operators, and their spectra are pairwise disjoint.

Theorem 3. Suppose the following conditions are satisfied: 1) the operator \(A_1(t)\) \((t\in[0,T])\) acts in \(E\), has a domain of definition everywhere dense and independent of \(t\), and the estimate \(\|[A_1(t)+\lambda I]^{-1}\|\le \dfrac{1}{\lambda+1}\) holds \((\lambda>-1)\); 2) the operator-functions \(A_i(t)A_1^{-1}(0)\) \((i=1,\ldots,n)\) are once strongly continuously differentiable on \([0,T]\); 3) the vector function \(f(t)\) is continuously differentiable on \([0,T]\); 4) \(u_i\in D\) \((i=0,\ldots,n-1)\).

Then problem (1)—(2) has a solution, and moreover a unique one.

Applying the results of (7), one can formulate an analogous theorem when the operator \(A_1(t)\) is strongly positive. For \(n=2\), Theorem 2 was proved by P. E. Sobolevskii (10) under twice continuous differentiability of the operator function \(A_1(t)A_1^{-1}(0)\).

1. We turn to consideration of the nonlinear equation

\[ \frac{d^n u(t)}{dt^n} + A_1(t)\frac{d^{\,n-1}u(t)}{dt^{\,n-1}} +\cdots+ A_n(t)u(t) = f(t,u(t)). \tag{11} \]

For \(n=2\), in the work (10) a local existence theorem was proved for the solution of problem (11)—(2). The following theorem is the \(n\)-dimensional analogue of this theorem. Denote by \(S_0\) a certain ball of the space \(E\) with center at \(A(0)u_0\).

Theorem 4. Suppose conditions 1), 2), and 4) of Theorem 2 are satisfied. Suppose the operator \(f(t,A^{-1}(0)u)\) on \([0,T]\times S_0\) has continuous partial derivatives with respect to the aggregate of variables \(f'_t(t,A^{-1}(0)u)\), \(f'_u(t,A^{-1}(0)u)\) (the derivative is understood in the sense of Fréchet), satisfying in \(u\) a Lipschitz condition \((f'_u(t,A^{-1}(0)u)\) with respect to the norm of the space of linear operators over \(E)\).

Then there exists, and moreover is unique, a solution of problem (11)—(2) on some interval \([0,t_0]\subset[0,T]\).

1. In some problems of mathematical physics one encounters the partial differential equation \(^{12}\)

\[ \frac{\partial^2 u}{\partial t^2} - \frac{\partial}{\partial t}\Delta u - \Delta u = f(x,y,z,t) \qquad \left( \Delta \equiv \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right). \tag{12} \]

In \(^{12}\), formal solutions are constructed for the Cauchy problem (for \(t=0\), \(u=u_0(x,y,z)\), \(\partial u/\partial t=u_1(x,y,z)\)) and for one mixed problem (for \(t=0\), \(u=u_0(x,y,z)\), \(\partial u/\partial t=u_1(x,y,z)\); for \(z=0\), \(\partial u/\partial z=0\)). Applying the results of article \(^{13}\) and of the present article, one can obtain existence theorems for equations more general than \(^{12}\).

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
21 I 1964

CITED LITERATURE

\(^{1}\) E. Hille, R. Phillips, Functional Analysis and Semi-Groups, IL, 1962.
\(^{2}\) R. S. Phillips, Trans. Am. Math. Soc., 74, No. 2 (1953).
\(^{3}\) M. Z. Solomyak, DAN, 122, No. 5 (1958).
\(^{4}\) K. Josida, Proc. Japan Acad., 34, No. 6 (1958).
\(^{5}\) T. Kato, J. Math. Soc. Japan, 5, No. 2 (1953).
\(^{6}\) M. A. Krasnosel’skii, S. G. Krein, P. E. Sobolevskii, DAN, 111, No. 1 (1956).
\(^{7}\) P. E. Sobolevskii, Tr. Mosc. Math. Soc., 10, 297 (1961).
\(^{8}\) B. S. Mityagin, Izv. AN AzerbSSR, Phys.-Math. Ser., No. 1 (1961).
\(^{9}\) P. E. Sobolevskii, DAN, 146, No. 4 (1962).
\(^{10}\) P. E. Sobolevskii, Uch. Zap. Azerb. State Univ., Phys.-Math. Ser., No. 3 (1962).
\(^{11}\) A. V. Balakrishnan, Pacif. J. Math., 10, No. 2 (1960).
\(^{12}\) E. M. Dobryshman, A. F. Dyubyuk, DAN, 111, No. 1 (1956).
\(^{13}\) S. Agmon, L. Nirenberg, Comm. on Pure and Appl. Math., 16, No. 2 (1963).