Azerbaijan National Academy of Science

- Determination of well posedness of the Cauchy problem and boundary value problem for operator coefficient equations.  Determination  evolution operator for higher order variable operator equation.

- Study of spectral problems with a spectral parameter in the boudary condition for differential operator.

- Well-posedness of the Cauchy problem was determined for equations where a pary of coefficients are smooth, another part is nonsmooth;

- Theorems on behaviour of solutions of nonlinear elliptic and parabolic equations in the vicinity special points of boundaries and in unbounded domains with noncompact boundary, were proved. When in bounded and in unbounded domains the weight function satiesfies the Mackenhoupt conditions, the degenerating coefficients satisfy certain growth conditions, the guality character of the solutions of Dirichlet problem stated for higher order nonlinear parabolic equations was studied and unigueness classes for them were found. 

- Quality properties of a solutions of certain class of elliptic and parabolic type equations were studied, sufficient conditions for almost periodicity and asymptotically of solutions of operator coefficient equations and partial hyperbolic equations were determined; Global solvability conditions of one-sided problemss for quasilinear hyperbolic equations were determined;

- Radiation principles in higher order elliptic equations in cylindric domains were proved; resonans effect in justification of the limit amplitude principle was revealed, the Cauchy problem, the solutions of mixed problems were studied for Sobolev equations and for a system of well-posed by Petrovsky equations in bounded domains and in unbounded cylindric domains, and the asympthotics of solutions at layest values of time was reserached.

The following results were obtained on the second problem.

- The well-posedness of the Cauchy problem and the initial boundary value problem was studiedd for higher order variable operator coefficient evolution equations;

- Fredholm property of solvability of a boundary value problem was studied for a second order differential-operator equation with an unbounded operator in the boundary condition. 

The following results were obtained on the third problem.

- The trace of a boundary value problem was calculated for operator coefficient Sturm-Liouville equation and an unbounded operator differential operators with a spectral parameter in boundary conditions;

- Solvability, Fredholm property and spectral pecularities of boundary value problems were studied for a second order differential-operator equation with a spectral parameter in the boundary condition.