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Initial value problem for pseudo-differential operators. (English. Russian original) Zbl 0946.35121
J. Math. Sci., New York 98, No. 6, 629-653 (2000); translation from Probl. Mat. Anal. 18, 3-42 (1998).
Under consideration is the initial-value problem \[ \frac{\partial u }{\partial t}=Hu,\quad u(0)=u_{0}, \] with \(H\) a differential operator in \(L_2({\mathbb{R}}^n)\) whose coefficients are of complicated behavior at infinity. The main results are obtained with the help of the theory of pseudodifferential operators. The article consists of five sections. The first one is devoted to describing the algebra of Weyl symbols. The second section contains some new results concerning the theory of smooth Weyl symbols. In particular, new examples of \(\sigma\)-temperate Riemannian metric are presented which are the bases for constructing the corresponding algebras. The results are applicable to the problem of inversion of pseudodifferential operators and to studying initial-value problems. Sections 3 and 4 contain conditions ensuring the fact that a given pseudodifferential operator is a generator of a continuous semigroup. Conditions are presented guaranteeing the similarity of the spectra for the operator \(H\) and its symbol. The main spaces in question are spaces of test functions and the corresponding distribution spaces. Section 5 is devoted to the initial-value problem for second-order differential operators. The basic obstacle for completely describing this problem for higher-order differential operators is the absence of simple conditions ensuring that the spectrum of a function (a Weyl symbol) belongs to a given sector in the complex plane.

35S10 Initial value problems for PDEs with pseudodifferential operators
47G30 Pseudodifferential operators
35K25 Higher-order parabolic equations
35S15 Boundary value problems for PDEs with pseudodifferential operators