Vol 55, No 1 (2019)

We study a differential inclusion unsolved for the derivative of the unknown function and prove a theorem on the solvability of an abstract inclusion generated by a multivalued orderly covering mapping. We use this result to obtain sufficient solvability conditions and estimate the solutions of the Cauchy problem for the implicit differential inclusion.

Inverse problems of spectral analysis are studied for second-order differential pencils on arbitrary compact graphs. The uniqueness of recovering operators from their spectra is proved, and a constructive procedure for the solution of this class of inverse problems is provided.

We study the well-posedness of a linear inverse problem for a multidimensional mixed-type equation including the classical equations of elliptic, hyperbolic, and parabolic types as special cases. For this problem, using the “ε-regularization,” a priori estimate, and successive approximationmethods, we prove the existence and uniqueness theorems for the solution in some function class.

We consider a problem with some boundary and initial conditions for an equation arising in the theory of ion-sound waves in plasma. We prove that if the spatial (one-dimensional) variable ranges on an interval, then this problem has a unique nonextendable classical solution which in general exists only locally in time. If the spatial variable varies on the half-line, then, for the problem in question, we obtain an upper bound for the lifespan of its weak solution and find initial conditions for which there exist no solutions even locally in time (instantaneous blow-up of the weak solution). A similar result is obtained for the classical solution.

We study the extremals of problems of the classical calculus of variations. We obtain a new necessary condition for a strong minimum under the assumption that the Weierstrass and Legendre conditions degenerate, i.e., are satisfied as equalities, on the extremal under study; this condition is called a generalized Weierstrass condition. We also give a modified condition that is a necessary condition for a weak minimum.

We consider the feedback control problem for a nonlinear distributed equation with memory. An algorithm for solving this problem based on constructions of the theory of positional differential games is proposed under the assumption that the equation is subjected to an unknown dynamic disturbance.

We find a class of practically important differential games where several most important parameters any change in which significantly changes the functional form of the original differential game can nevertheless be varied broadly without affecting the structure (or principle) of the players’ optimal behavior. We use a new method for solving differential games based on the decomposition of the original game into finitely many local static games.

We consider the Cauchy problem in three-dimensional space for a first-order almost linear differential equation with discontinuous coefficients of the derivatives. Two special cases related to the behavior of characteristics are singled out and studied.

We consider the inhomogeneous Tricomi-Neumann problem for a parabolic-hyperbolic equation with noncharacteristic type change line and degenerate hyperbolic part. The auxiliary function method is used to obtain an a priori estimate for the solution. The existence of a classical solution is proved for the case in which the right-hand side of the equation and the boundary functions are smooth. The unique generalized L₂ solvability is established for the case of nonsmooth conditions.