For a piecewise linear three-dimensional dynamical system of biochemical kinetics with three-steps righthand sides, we find conditions of existence of two stable cycles in the phase portrait. Toroidal neighborhoods of these cycles are constructed.

This work continues the article «On the first mixed problem for degenerate parabolic equations in stellar domains with Lyapunov boundary in Banach spaces» and studies the behavior of a solution to a second-order parabolic equation with Tricomi degeneracy on the lateral boundary of a cylindrical domain Q^T, where Q is a stellar domain whose boundary ∂Q is an (n − 1)-dimensional closed surface without an edge of class C^1+λ, 0 < λ < 1.

We consider two ways to choose the boundary condition: 1) due to the fact that Q is stellar, 2) some direction orthogonal to the boundary is allocated and the continuity of the solution as a function of a special variable with values in L_p in this direction is asserted. To do this, by determining the boundary value, while mapping the boundary ∂Q, it is necessary to take a shift not along the normal at each point x ∈ ∂Q, but to take a sufficiently small covering of the boundary and shift each piece of this covering «parallel» along the normal at one fixed point of this piece x₀.

We also consider the question of the unambiguous solvability of the first mixed problem for an equation when the boundary and initial functions belong to spaces of type L_p, p > 1.

The paper investigates the solvability of nonlocal boundary value problems with the generalized Samarsky–Ionkin condition for elliptic second order differential equations with a discontinuous coefficient in the higher part. The existence and uniqueness theorems for regular solutions to the studied problems are proved, i.e. solutions having all required generalized derivatives.

The issues of unique solvability of linear inverse coefficient problems for evolution integro-differential equations of Gerasimov type with a singular integral kernel in Banach spaces are investigated. The cases of bounded and sectorial operators at the unknown function in the equation are considered. In each case, correctness criteria were obtained for the linear inverse problem with a time-independent unknown coefficient, and sufficient conditions for solvability and correctness estimates were found for the linear identification problem with a time-dependent unknown coefficient. The abstract results obtained are illustrated by an example of a class of inverse problems for partial differential equations.

The existence of classical solutions was established in [(∗)] Tani A. and Tani H., Two-phase radial viscous fingering problem in a Hele-Shaw cell with surface tension, I: Classical solvavility,’ Mat. Zametki SVFU, 31, № 4, 82–105 (2024), for the two-phase radial viscous fingering problem in a Hele-Shaw cell under the surface tension (the original two-phase problem) by means of parabolic regularization with a small parameter ε (> 0) in the time-derivative terms and the non-homogeneous terms (the parabolic regularized two-phase problem), vanishing along some subsequence {ε_n}_n∈N of {ε > 0}. In this paper we prove the uniqueness of classical solutions to the original two-phase problem. This gives the improvement to the convergence result in [(∗)]: the convergence of the full sequence {ε > 0}, not the subsequence {ε_n}_n∈N, of classical solutions of the parabolic regularized two-phase problem to those of the original two-phase problem. Similar results for some one-phase problem have been already studied in Tani H., «Classical solvability of the radial viscous fingering problem in a Hele-Shaw cell with surface tension», Sib. J. Pure Appl. Math., 16, 79–92 (2016) (the existence) and in Tani A. and Tani H., «On the uniqueness of the classical solution of the radial viscous fingering problem in a Hele-Shaw cell with surface tension», J. Appl. Mech. Tech. Phys., 65, № 5 (2024) (the uniqueness).

This study presents a mathematical model describing the carbon cycle in wetland ecosystems of northern regions. The model characterizes carbon concentration in two key reservoirs: Live (living plants and biomass) and Mort (dead organic matter). The primary processes incorporated in the model include photosynthesis, autotrophic and heterotrophic respiration, biomass decay, and carbon transport via groundwater. These processes are formalized with respect to temperature and groundwater level. The inclusion of groundwater level allows us to consider of differences between aerobic and anaerobic organic matter decomposition processes. Numerical simulations were performed using model data. Under conditions of low temperatures and high groundwater levels, heterotrophic respiration is slowed, leading to the formation of anaerobic conditions that favor the accumulation of carbon in the soil. In contrast, under reduced water levels, increased oxygen availability to organic material stimulates aerobic decomposition, resulting in higher CO₂ emissions. Unlike models focused on global processes, this work emphasizes the specific climatic, hydrological, and biochemical conditions of northern wetlands, which is crucial for accurately modeling the carbon balance in cold regions.

The Dirichlet problem is considered, for which it is proved that it is solvable in the class of bounded functions, and regular solutions in the formulation under consideration are obtained.

We study the fractional wave equation with changing direction of evolution. Existence and uniqueness theorems of a generalized solution are proven.

An evolutionary model of HIV is considered. This model describes the dynamics of populations of healthy and infected cells. After introducing dimensionless variables and parameters a singularly perturbed partial integro-differential system arises. The order of the resulting system can be reduced.

A method for parametric identification of a system of differential equations for a mathematical model of incomplete reversibility of creep deformation has been developed. Using methods of nonlinear regression analysis, estimates of the random parameters of the system are found based on difference equations. Relationships connecting parameter estimates and coefficients of difference equations were obtained, and iterative procedures for refining parameters were developed. The method was tested on a large amount of experimental data.

Issues of the unique global solvability of the Cauchy problem for a class of quasilinear equations in Banach spaces are studied. The equations contain several fractional derivatives of Gerasimov – Caputo in the linear and nonlinear part. The sectoriality condition for a pencil of operators at derivatives in the linear part is used.

The asymptotic estimation of the density function at infinity, proved earlier for the case of the one-factor model, is generalized to the case of the multidimentional Heston model. The proof is based on the affinity of the Heston model, the Mellin transform, and the evaluation of the obtained integrals using the pass method.

We consider initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition and nonnegative initial datum. We find conditions which guarantee global existence of solutions as well blow-up of solutions in finite time.

This work is devoted to the critical conditions problem for an autocatalytic combustion model, taking into account the consumption of the reagent and oxidizer. To model critical phenomena, the asymptotic methods and technique for gluing of invariant manifolds are used.

A mathematical model is presented and optimization of a betavoltaic element with a silicon carbide film activated by the carbon-14 isotope is carried out. Special attention is paid to differential equations describing non-equilibrium injection processes and the dynamics of current densities in the SiC/Si heterojunction. By solving a system of equations, it was possible to determine the dependences of the parameters on the specific activity and distribution of isotopes. Optimization has led to an increase in the efficiency and specific power of the element.

The paper studies the solvability of the spectral nonlocal Ionkin-Samarsky problem for an elliptic equation of the second order. Some properties of eigenvalues are given for elliptic problems with non-local Samarsky-Ionkin conditions. The classical method of separating variables was used for the study.

In this paper we consider the solvability of the analog of the first initial boundary value problem for the quasihydrodynamic system of equations in the shallow water approximation. Under certain conditions on the data, it is shown that there exists a single regular solution of the problem locally in time.

Using the theory of fractional powers of the sectorial operator, the existence of a unique solution to an incomplete Cauchy-type problem for a quasi-linear differential equation in Banach space resolved with respect to the highest Riemann–Liouville derivative is proved.

The paper is devoted to the study of a laser diode model with optoelectronic feedback, described by a singularly perturbed system. The change of stability of the invariant manifold of the system, which can proceed according to different scenarios, is established.

The report presents results on the solvability of non-local problems with integral conditions with respect to the selected variable t for differential equations ∂2 ∂t2 + a(t) u + b(t)u = f(x, t) ( ) ( is the Laplace operator in spatial variables x1, . . . , xn). The essence of the results is to find sufficient conditions for the existence and uniqueness of regular solutions (i.e. solutions that have all derivatives generalized according to S. L. Sobolev, included in the equation ( )).

The work is devoted to the Poincare problem in the analytical theory of differential equations. Namely the constructions of asymptotic of solutions of ordinary differential equations with holomorphic or meromorphic coefficients in the vicinity of irregular points.The paper provides the general view of the asymptotic of solutions of differential equations with meromorphic coefficients in the neighborhood of irregular points.