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On the solution of linear evolution equations by a variational method. (English. Russian original) Zbl 0782.35030

Sov. Math., Dokl. 43, No. 3, 735-737 (1991); translation from Dokl. Akad. Nauk SSSR 318, No. 3, 545-547 (1991).
In Dokl. Akad. Nauk SSSR 151, 292-294 and 511-512 (1963; Zbl 0199.453 and Zbl 0211.445) V. M. Shalov has shown that the linear equation \(Au=f\) in a Hilbert space can be solved according to the Friedrichs method by a direct variational method if the operator \(A\) is symmetric and positive definite relative to some operator \(B\). It has been possible to construct such an operator \(B\) in connection with some boundary value problems for the string equation and for the heat equation. The essence of this note is the description of an operator \(B\) for a certain class of linear evolution problems of the form \[ L(t)u\equiv u'+ C(t)u= F(t), \qquad t\in S=[0,T]; \tag{1} \]
\[ \Gamma u\equiv \sum_ 0^ m \gamma_ i u(t_ i)=f. \tag{2} \] Here \(C(t): X_ 1\to X_ 1'\) is a family of linear operators defined on the Banach space \(X_ 1\), and the points \(t_ i\) of the interval \(S\) and the numbers \(\gamma_ i\) are fixed, \(0=t_ 0<t_ 1<\cdots< t_ m\leq T\), and \(\gamma_ 0=1\). The function \(F: S\to X_ 1\) and the element \(f\) (\(\in X_ 1\)) are also known. Condition (2) is a generalization of the initial condition and arises in connection with the problem of long-term weather forecasting.

MSC:

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35A15 Variational methods applied to PDEs
86A10 Meteorology and atmospheric physics
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