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For the first time, the theory of strongly continuous cosine operator functions (COF) has been applied to study the correct solvability of boundary value problems for second-order linear differential equations in a Banach space (elliptic case). The correct solvability of the Cauchy problem (hyperbolic case) is usually formulated in COF terms. The conditions on the order of COF growth are specified under which the Dirichlet boundary value problem is correct on a finite interval. An integral representation of the solution and its sharp estimate are given.

The oblique-derivative spectral problem for the Laplacian in a disk D is studied. The asymptotic properties of its eigenvalues are established, and the system of root functions of this problem is proved to be a basis with brackets in L2 (D).

The study of well-solvable operator equations in a Banach space, which was initiated by the authors in [4, 5], is continued. Namely, it is proved by means of Maslov’s operator method that a polynomial equation with abstract Newton polynomials is well solvable in the sense of Hadamard. The obtained results are applied to prove that a large class of problems for differential equations with variable coefficient having a singularity (such equations are called generalized Euler equations in the paper) are well solvable.

As is known, beginning in the fundamental papers by E. Hille, R. Phillips, and other authors, an impor� tant place in the theory of equations of parabolic type is occupied by oneparameter semigroups of linear transformations U(t), T ≥ 0, said to be canonical and determined by the relation U(α ⊕ β) U(α)U(β), where α and β are real or complex numbers (8, p. 275 of the Russian translation). In the system of numbers under consideration, there is a subset determining semigroups corresponding to various addition opera� tions.

The spatiotemporal structures induced by a weak localized stimulus in excitable contractile fibers are studied by using a two-component reaction–diffusion model with global coupling. The character of the induced structures is analyzed, and the regimes of excitation spreading over the fiber are determined depending on the global coupling strength and the contraction type (associated with the mechanical fiber fixation conditions). It is shown that the global coupling can lead to long-lasting transient dynamics and new oscillatory attractor modes.

A scheme is proposed for analyzing bifurcations of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic foundation in the case of bimodal degeneration. For a homogeneous beam, a similar problem was solved by Darinskii and Sapronov in [1, 2]. Their analysis scheme is based on the condition that the eigenfunctions e 1 and e 2 of the second differential (at zero) of the energy functional are constant. In the case of an inhomogeneous beam, however, Darinskii and Sapronov’s analysis scheme has to be modified, since this condition is violated; moreover, it cannot be directly generalized, for example, in the form of an existence condition for a continuous family of eigenfunctions. We propose a scheme in which the constancy condition for eigenfunctions is replaced by the existence of a pair of smooth vector fields and whose linear span is invariant under the second differential at zero. Such a pair is sufficient for constructing the principal part of the key function and, as a consequence, for analyzing bifurcations of equilibrium configurations of the beam. The construction of the required pair of vector fields relies heavily on the formula from [3] for an orthogonal projector (onto the linear span of and ). In the case of a weakly inhomogeneous beam, the symmetry of the key function is reduced and the dimension of its strain basis is increased by one (a quadratic term is added to the principal part of the key function). In this situation, the analysis of bifurcation diagrams and bifurcating critical points is somewhat complicated [4] but reveals new bifurcation effects.