Abstract Generated abstract
The paper extends earlier results on parabolic equations to Banach spaces for time dependent unbounded operators whose fractional powers satisfy uniform domain and continuity conditions. It proves existence of an evolution operator for the homogeneous problem, including semigroup composition, differentiability, representation by a multiplicative integral, and estimates involving fractional powers of the operators. The results are then applied to an inhomogeneous linear equation with Hölder continuous forcing and to a nonlinear equation with operator coefficients depending on the unknown, yielding local existence and, under stronger assumptions, uniqueness by successive approximations. As an application, the paper derives existence of a classical solution for a nonlinear parabolic boundary value problem with oblique boundary condition and without growth restrictions on the nonlinear terms.
Full Text
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.
- 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.
- 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
- P. E. Sobolevskii, DAN, 123, No. 6 (1958).
- P. E. Sobolevskii, DAN, 130, No. 2 (1960).
- M. A. Krasnosel’skii, P. E. Sobolevskii, DAN, 129, No. 3 (1959).