MATHEMATICS

P. E. SOBOLEVSKII

ON EQUATIONS OF PARABOLIC TYPE IN A BANACH SPACE WITH AN UNBOUNDED VARIABLE OPERATOR WHOSE FRACTIONAL POWER HAS A CONSTANT DOMAIN OF DEFINITION***

(Presented by Academician I. G. Petrovskii on 8 XII 1960)

In this paper some results obtained in \((^{1,2})\) are carried over to the case of equations in a Banach space.

1. Consider the problem

\[ \frac{dv}{dt}+A(t)v=0\quad (\tau<t\leq T,\ \tau\in[0,T]),\qquad v(\tau)=v_0, \tag{1} \]

where \(v(t)\) is the desired function, defined on \([\tau,T]\), with values in the Banach space \(E\); \(A(t)\) \((0\leq t\leq T)\) is a linear operator acting in \(E\); \(dv/dt\) is the derivative, understood as the limit in the norm of \(E\) of the corresponding difference quotient.

Suppose that for each \(t\in[0,T]\) the operator \(A(t)\) has a domain of definition \(D[A(t)]\) everywhere dense in \(E\). Suppose that for any \(\lambda\) with \(\operatorname{Re}\lambda\geq0\) the operator \(A(t)+\lambda I\) has a bounded inverse whose norm satisfies the inequality

\[ \bigl\|[A(t)+\lambda I]^{-1}\bigr\|\leq C[|\lambda|+1]^{-1}. \tag{2} \]

Then \((^{3})\) the fractional powers of the operator \(A(t)\) are defined.

Let \(\rho\) be some number in \((0,1)\), and let \(l\) be an integer such that \(\rho_1=1-l\rho\in(0,\rho]\). Suppose that the operator \(A^\rho(t)A^{-\rho}(\tau)\) is bounded for all \(t,\tau\in[0,T]\), and that the operator \(A^{-\rho_1}(t)A^{\rho_1}(\tau)\) admits closure to a bounded operator. Suppose that

\[ \|\Delta[A(t),A(\tau)]\|\leq C|t-\tau|^{1-\rho+\varepsilon}, \tag{3} \]

where \(C>0\), \(\varepsilon\in(0,\rho]\) are some numbers, and by \(\Delta[\ldots]\) is denoted each of the bounded operators \(A^\rho(t)A^{-\rho}(\tau)-I\), \(A^\rho(t)A^{-\rho}(\tau)-\overline{A^{-\rho_1}(t)A^{\rho_1}(\tau)}\)**. In the case \(\rho\geq 1/2\), by \(\Delta[\ldots]\) is denoted the second of these operators.

Theorem 1. There exists an operator-function \(U(t,\tau)\), defined for all \(0\leq\tau\leq t\leq T\), with values in the space of bounded linear operators on \(E\), possessing the following properties:

1) \(U(t,\tau)\) is uniformly continuous jointly in \(t\) and \(\tau\) for all \(t>\tau\), and for \(t=\tau\) is strongly continuous.

2) \(U(t,t)=I\), and for any \(0\leq\tau\leq s\leq t\leq T\) the identity holds

\[ U(t,\tau)=U(t,s)\,U(s,\tau). \tag{4} \]

* The results of the present work were reported at the seminar on functional analysis of Voronezh State University in January 1959.

** A bar above denotes the closure of an operator in \(E\).

3) \(U(t,\tau)\), for \(t>\tau\), is uniformly and continuously differentiable with respect to \(t\), and

\[ \frac{\partial U(t,\tau)}{\partial t}+A(t)U(t,\tau)=0. \tag{5} \]

4) Problem (1), for any \(v_0\in E\), has a unique solution

\[ v_\tau(t)=U(t,\tau)v_0. \tag{6} \]

continuous for all \(t\ge \tau\) and continuously differentiable for \(t>\tau\).

If \(v_0\in D[A(\tau)]\), then the vector-function \(v_\tau(t)\) is continuously differentiable also at \(t=\tau\) and satisfies equation (1). Here the derivative at the point \(t=\tau\) is understood as the right derivative.

5) For any \(0\le \tau\le t\le t+\Delta t\le T\) and \(\xi\in[0,T]\) the estimates

\[ \|A^\alpha(t)U(t,\tau)A^{-\beta}(\tau)\|\le C(\alpha_0)|t-\tau|^{\beta-\alpha} \quad (0\le \beta\le \alpha\le \alpha_0<1+\varepsilon); \tag{7} \]

\[ \|A^\alpha(\xi)[U(t+\Delta t,\tau)-U(t,\tau)]A^{-\beta}(\tau)\| \le C(\alpha,\beta,\gamma)\Delta t^{\gamma-\alpha}|t-\tau|^{\beta-\gamma} \]
\[ (0\le \alpha\le \rho,\ 0\le \beta\le \gamma<1+\varepsilon,\ 0<\gamma-\alpha<1). \tag{8} \]

In the last estimate one may have \(\gamma-\alpha=1\), if either \(\xi=t\), or \(\beta<\gamma\).

6) If \(\varepsilon>\rho-\rho_1\), then \(U(t,\tau)\) is uniformly and continuously differentiable with respect to \(\tau\) for \(\tau<t\), and

\[ \frac{\partial U(t,\tau)}{\partial \tau}-U(t,\tau)A(\tau)=0. \tag{9} \]

For any \(0\le \beta\le \alpha\le \alpha_0<1+\varepsilon-\rho+\rho_1\) the estimate

\[ \|A^{-\beta}(t)U(t,\tau)A^\alpha(\tau)\|\le C(\alpha_0)|t-\tau|^{\beta-\alpha} \tag{10} \]

is valid.

7) The operator \(U(t,\tau)\) can be represented in the form of a multiplicative integral.

2. Consider the problem

\[ \frac{dv}{dt}+A(t)v=f(t)\quad (0<t\le T),\qquad v(0)=v_0. \tag{11} \]

If the vector-function \(f(t)\) is continuous, then problem (11) cannot have more than one solution continuous for all \(t\ge 0\) and continuously differentiable for \(t>0\).

Theorem 2. Suppose the vector-function \(f(t)\) satisfies the condition

\[ \|f(t)-f(\tau)\|\le C|t-\tau|^\delta \quad (t,\tau\in[0,T],\ C>0,\ 0<\delta<1). \tag{12} \]

Then the formula

\[ v(t)=U(t,0)v_0+\int_0^t U(t,\tau)f(\tau)\,d\tau \tag{13} \]

for any \(v_0\in E\) defines the unique solution of problem (11), continuous for all \(t\ge 0\) and continuously differentiable for \(t>0\). If \(v_0\in D[A(0)]\), then the vector-function \(v(t)\) is continuously differentiable also at \(t=0\) and satisfies equation (11). Here the derivative at the point \(t=0\) is understood as the right derivative.

3. Consider the problem

\[ \frac{dv}{dt}+A(t,v)v=f(t,v)\quad (0<t\le t_0,\ t_0\in(0,T]),\qquad v(0)=v_0. \tag{14} \]

Theorem 3. Let \(E_1\) be some Banach space; let \(E_2\) be a subset of \(E_1\) that is also a Banach space. Suppose \(\|v\|_{E_1}\le C\|v\|_{E_2}\) for any \(v\in E_2\).

Suppose that for every such \(v\in E_2\) that \(\|v\|_{E_1}\leq R_1\), \(\|v\|_{E_2}\leq R_2\), where \(R_1\) and \(R_2\) are some positive numbers, and every \(t\in[0,T]\), a linear operator \(A(t,v)\) acting in \(E\), with everywhere dense domain of definition, is defined. Suppose that for every \(\lambda\) with \(\operatorname{Re}\lambda\geq 0\) the operator \(A(t,v)+\lambda I\) has a bounded inverse, and suppose that

\[ \bigl\|[A(t,v)+\lambda I]^{-1}\bigr\|\leq C[|\lambda|+1]^{-1}. \tag{15} \]

Suppose that, for some \(\rho\in(0,1)\), for all such \(v,w\in E_2\) that \(\|v\|_{E_1},\|w\|_{E_1}\leq R_1\), \(\|v\|_{E_2},\|w\|_{E_2}\leq R_2\), and for all \(t,\tau\in[0,T]\), the operator \(A^\rho(t,v)A^{-\rho}(\tau,w)\) is bounded, and the operator \(A^{-\rho_1}(t,v)A^{\rho_1}(\tau,w)\) admits closure to a bounded operator. Suppose that

\[ \bigl\|\Delta[A(t,v),A(\tau,w)]\bigr\|\leq C\bigl[|t-\tau|^\mu+\|v-w\|_{E_1}^{\nu}\bigr], \tag{16} \]

where \(\mu,\nu\) are some numbers respectively from \((1-\rho,1]\), \(\left(\frac{1-\rho}{\beta-\alpha},1\right]\); \(\alpha,\beta\) are some numbers respectively from \([0,\rho)\), \((1-\rho+\alpha,1)\). Suppose

\[ v_0\in E_2,\quad \|v_0\|_{E_1}<R_1,\quad \|v_0\|_{E_2}<R_2. \]

Suppose that \(v_0\in D(A_0^\beta)\), where \(A_0=A(0,v_0)\). Suppose that each element \(v\) of \(D(A_0^\alpha)\) belongs to \(E_1\), and suppose that

\[ \|v\|_{E_1}\leq C\|A_0^\alpha v\|. \tag{17} \]

Suppose that for any such \(v,w\in E_2\) that \(\|v\|_{E_2},\|w\|_{E_2}\leq R_2\), and for all \(t,\tau\in[0,T]\),

\[ \|f(t,v)-f(\tau,w)\|\leq C\bigl[|t-\tau|^\eta+\|v-w\|_{E_2}^{\xi}\bigr], \tag{18} \]

where \(\eta\) and \(\xi\) are some numbers from \((0,1]\).

Suppose that \(E_3\) and \(E_4\) are some Banach spaces. Suppose that each element \(v\) of \(E_3\) belongs to \(E_2\) and \(E_4\), and suppose that

\[ \|v\|_{E_2}\leq C\|v\|_{E_3}^{\gamma}\|v\|_{E_4}^{1-\gamma}, \tag{19} \]

where \(\gamma\) is some number from \([0,1)\).

Suppose that each element \(v\) of \(D(A_0^\varepsilon)\), where \(\varepsilon\) is some number from \([0,\rho)\), belongs to \(E_4\), and suppose that

\[ \|v\|_{E_4}\leq C\|A_0^\varepsilon v\|. \tag{20} \]

Suppose that each element \(v\) of \(D[A^\beta(t,w)]\), where \(w\) is any such element of \(E_2\) that \(\|w\|_{E_1}\leq R_1\), \(\|w\|_{E_2}\leq R_2\), and \(t\) is any number from \([0,T]\), belongs to \(E_3\), and suppose that

\[ \|v\|_{E_3}\leq C\|A^\beta(t,w)v\|. \tag{21} \]

Finally, suppose that every set bounded in \(E_3\) is compact in \(E_2\). Then, for some \(t_0\), there exists at least one solution \(v(t)\) of problem (14), continuous for all \(t\geq 0\) and continuously differentiable for \(t>0\).

If \(E_2=E_1\) and \(\gamma=\xi=1\), then such a solution will be unique and can be found by the method of successive approximations.

1. Suppose that \(\Omega\) is an open bounded domain of \(n\)-dimensional space with boundary \(S\). With the aid of Theorem 3, an existence theorem is proved

of a classical solution of the boundary-value problem

\[ \frac{\partial v}{\partial t} - \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left[ a_{ik}(t)x,v)\frac{\partial v}{\partial x_k} \right] = f\left(t,x,v;\frac{\partial v}{\partial x_1},\ldots,\frac{\partial v}{\partial x_n}\right) \quad (0<t\leqslant t_0,\ x\in\Omega), \]

\[ \sum_{i,k=1}^{n} a_{ik}(t,y,v)\frac{\partial v}{\partial y_k}\cos(N_y,y_i) +\sigma(t,y,v)v=0 \quad (0<t\leqslant t_0,\ y\in S); \tag{22} \]

\[ v(0,x)=v_0(x) \quad (x\in\overline{\Omega}) \]

without any restrictions on the growth of the nonlinearities. Here \(N_y\) is the vector of the exterior normal to the surface \(S\) at the point \(y\).

Voronezh Agricultural
Institute

Received
7 XII 1960

References

1. P. E. Sobolevskii, DAN, 123, No. 6 (1958).
2. P. E. Sobolevskii, DAN, 130, No. 2 (1960).
3. M. A. Krasnosel’skii, P. E. Sobolevskii, DAN, 129, No. 3 (1959).