Kushnir V. Absolute stability of solutions of linear parabolic differential-difference equations

The thesis is devoted to establishment the necessary and sufficient conditions of absolute stability to solutions to linear parabolic differential-difference equations with fixed coefficient and delays. The mixed problems on the segment with homogeneous boundary conditions for linear parabolic differential-difference equations of the delayed and neutral types and for linear elliptic differential-difference equation of the delayed type are investigated. The mixed problems for a system of linear parabolic differential-difference equations with one delay and for linear elliptic differential-difference equation with even order derivatives delayed type are considered. For all these problems for linear elliptic differential-difference equations there are founded the necessary and sufficient conditions of absolute exponential stability to solutions in different pairs of norms. The necessary and sufficient conditions for absolute asymptotic stability of solutions and absolute boundedness of solutions for the delayed mode equation are also established. For the system of linear parabolic equations with one delay the sufficient conditions for absolute exponential stability of solutions are established.