Vol 72, No 1 (2026): Differential and Functional Differential Equations

In this paper, we consider the third boundary-value problem for a third-order nonhomogeneous equation with multiple characteristics in three-dimensional space. We prove the uniqueness of the solution using the energy integral method, and we prove the existence of the solution using the separation of variables method. We present the solution explicitly using the constructed Green's function. We find the conditions on the given functions that ensure the regularity of the solution. Proving the uniform convergence, we establish that the ``small denominator'' is nonzero.

In this work, we study the problem on blocking fire by constructing a wall $\zeta$ whose shape is spiral-like. This is supposed to be the best strategy in the case where a single firefighter is constructing the wall with a finite construction speed $\sigma$: the barriers that satisfy this restriction on the construction speed are called admissible. We prove a sharp version of Bressan's Fire Conjecture [5] in this case, i.e. when admissible barriers are spiral-like curves: namely, there exists a spiral-like barrier confining the fire in a bounded domain of $R^2$ if and only if the speed of construction of the barrier $\sigma$ is strictly larger than a critical speed $\bar \sigma = 2,614\ldots$ The existence of confining spiral barriers for $\sigma > \bar \sigma$ is already known [7,15], while we concentrate on the negative side, i.e. if $\sigma \leq \bar \sigma,$ then no admissible spiral blocks the fire. The proof of these results relies on: 

1. the precise definition of a spiral barrier and its representation;
2. the analysis of saturated spiral barriers as a Retarded Differential Equation (RDE) in the spirit of [15];
3. the equivalent reformulation of the conjecture as a minimum problem of a functional for a prescribed functional;
4. the construction of the optimal closing spiral;
5. the analysis of a differentiable path of admissible spirals along which the functional is differentiable, and in particular increasing when moving from the optimal spiral to any other one (homotopy argument). 

Due to the complexity of the solution, the evaluation of the quantities needed to prove that the functional is increasing is performed numerically.

We give an analytical and topological description of the manifolds of eigenfunctions and potentials generated by a family of one-dimensional stationary Schrödinger operators and the manifold of self-adjoint nonseparated boundary conditions. In particular, we prove Arnold's conjecture on the codimension of the manifold of those potentials that correspond to double eigenvalues under the chosen self-adjoint nonseparated boundary conditions.

In this paper, we discuss two approaches to the study of the long-time behaviour and infinite-time behaviour of solutions for integrable hamiltonian systems under small stochastic perturbations. Then we compare these results with the results for deterministic perturbations of integrable systems.

In this paper, we consider Burau representations of braid groups and Gassner representations of pure braid groups, and their realization as monodromy representations of Jordan-Pochhammer systems with suitable parameters. Jordan-Pochhammer systems are Fuchs-type systems, which demonstrates the solvability of the Riemann-Hilbert problem for Burau and Gassner representations. We consider an extension of this statement to generalized braid groups and their representations.

We construct the first terms of the formal asymptotic expansion in a small parameter of the solution to the Cauchy problem with ``narrow cap'' initial conditions for a singularly perturbed system of wave equations in the critical case. The asymptotic expansion is constructed as a sum of traveling-wave terms and boundary-value functions. The traveling-wave terms are described by generalized Korteweg--de Vries equations. Under certain assumptions, the remainder term is estimated by the residual.

The constant $3/2$, which determines sufficient conditions for the stability of linear nonautonomous differential equations with delay, was first obtained by A. D. Myshkis in the middle of the twentieth century, and these results initiated the research of equations with aftereffect. The goal of such a research is stability conditions explicitly expressed in terms of the equation parameters. In the last decade of the twentieth century, the first discrete analogues of the results of such investigations were established: effective sufficient conditions for the stability of difference equations with aftereffect. Recently, stability conditions for differential equations with aftereffect have been obtained, which strengthen the well-known theorems, expressed through estimating the values of functionals of the parameters of the equation by the constant $3/2$. In this paper, we obtain discrete analogs of these results, i.e., sufficient conditions for the stability of a linear nonautonomous difference equation with delay expressed in terms of the equation parameters (coefficients and delays) and significantly strengthening the well-known conditions of this kind.

In this expository paper, we collect many recent advances in analytic function spaces of several complex variables related with the trace problem in tubular domains over symmetric cones and bounded strongly pseudoconvex domains with smooth boundary. We consider various function spaces of analytic functions of several variables in various domains in $\mathbb C^n$ and provide either complete descriptions of traces or estimates of traces of various analytic function spaces in various domains obtained in recent years, by various authors. The problem to find sharp estimates of traces of Hardy analytic function spaces in the unit polydisk first was posed by W. Rudin in 1969. Since then many papers appeared. We collect in this expository paper not only already many known results on traces of various analytic function spaces in product domains but also discuss various new interesting results related with this problem. Related to the trace problem various results were provided previously by G. Henkin, E. Amar, H. Alexander and various other authors. Finnaly, note that our trace theorems are closely related with the Bergman type projections acting between function spaces with different dimensions. This expository paper contains mainly new results concerning traces in tubular and bounded strongly pseudoconvex domains, proofs of these theorems are based in particular also on various properties of Bergman type projection. In this expository paper we will also shortly discuss some new results obtained by the first author on Bergman type projections in these complicated domains in $\mathbb C^n.$ This is the second part of our notes related to the trace problem. In the first part, we provided a large list of recent sharp results on traces in the polydisk and polyball; it was published in[43]. In the last section of this expository paper, we collect many interesting remarks and put various open interesting questions for interested readers and also add various interesting short comments related with this problem of traces in various simple and complicated domains in $\mathbb C^n$ in various analytic function spaces of several variables. Some new results concerning traces in harmonic function spaces of several variables will be also added.