arXiv subjects
Publications and source records attributed to S. G. Pyatkov.
Quasilinear parabolic boundary value problems are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness and consistency conditions on the data and some inequalities ensuring the maximum principle for a solution. The proof relies on a priori bounds and the method of continuation in a parameter.
Under consideration are mathematical models of heat and mass transfer. We study inverse problems of recovering lower-order coefficients in a second order parabolic equation. The coefficients are representable in the form of a finite segments of the series whose coefficients depending on time are to be determined. The linear case is also considered. The overdetermination conditions are the integrals over the boundary of the domain of a solution with weights. The main attention is paid to existence, uniqueness, and stability estimates for solutions to inverse problems of this type. The problem is reduced to an operator equation which is studied with the use of the fixed point theorem and a priori estimates. A solution has all generalized derivatives occurring into the equation summable to some power. The method of the proof is constructive and it can be used for developing new numerical algorithms of solving the problem.
The initial-boundary value problems for linear non-autonomous first order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change with time. We study existence, uniqueness and maximal regularity of solutions in Sobolev spaces. In contrast to the previous results we use only the continuity assumption on the operators in the main part of the equation.
We describe necessary and sufficient conditions for a $J$-dissipative operator in a Krein space to have maximal semidefinite invariant subspaces. The semigroup properties of the restrictions of an operator to these subspaces are studied. Applications are given to the case when an operator admits matrix representation with respect to the canonical decomposition of the space. The main conditions are given in the terms of the interpolation theory of Banach spaces.