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Some analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations. Part. 1

Pytel-Kudela M., Prykarpatsky A.K.

Opuscula Mathematica

2006

Vol. 26, no. 1

131-136

The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bilinear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.

On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Pt. 1

Prykarpatska N.K., Pytel-Kudela M.

Opuscula Mathematica

2005

Vol. 25, no. 2

299-306

The geometric structure of characteristic surfaces related with partial differential equations of first and higer orders is studied making use the vector field technique on hypersurfaces. It is shown, that corresponding characteristics are defined uniquely up to some smooth tensor fields, thereby supplying additional information about the suitable set of their solutions. In particular, it may be very useful for studying asymptotic properties of solutions to our partial differential equations under some boundary conditions.