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This paper focuses on a class of generalized nonlinear wave equations with doubly dispersive over equation whole lines. By employing the potential well theory, we classify the initial profile such that the solution blows up or globally exists.

We present an example of a three-degrees-of-freedom polynomial Hamilton function with a critical point characterized by indefinite quadratic part with a Morse index 2. This function generates a Hamiltonian system wherein all eigenvalues equal $ \pm \mathrm{i} $, but it lacks small-amplitude periodic solutions with a period $ \approx 2\pi. $

We studied the modified nonlinear Schrödinger equation $ \begin{equation} -\Delta u-\frac12\Delta(u^2)u = g(u)+h(x), \quad u\in H^1({\mathbb{R}}^N), \end{equation} $ where $ N\geq3 $, $ g\in C({\mathbb{R}}, {\mathbb{R}}) $ is a nonlinear function of Berestycki-Lions type, and $ h\not\equiv 0 $ is a nonnegative function. When $ \|h\|_{L^2({\mathbb{R}}^N)} $ is suitably small, we proved that (0.1) possesses at least two positive solutions by variational approach, one of which is a ground state while the other is of mountain pass type.

In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control ([1], [2]). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $. In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in [2]). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by ([3], [4]) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.

In this paper, we were concerned with the multiplicity of the large periodic solutions to a super-linear wave equation with a general variable coefficient. In general, the variable coefficient $ \rho(\cdot) $ needs to be satisfied $ \text{ess inf}\, \eta_\rho(\cdot) > 0 $ with $ \eta_\rho(\cdot) = \frac{1}{2}\frac{\rho''}{\rho}-\frac{1}{4}\big(\frac{\rho'}{\rho}\big)^2 $. Especially, the case $ \eta_\rho(\cdot) = 0 $ is presented as an open problem in [Trans. Amer. Math. 349: 2015-2048, 1997]. Here, without any restrictions on $ \eta_{\rho}(\cdot) $, we established the multiplicity of large periodic solutions for the Dirichlet-Neumann boundary condition and Dirichlet-Robin boundary condition when the period $ T = 2\pi\frac{2a-1}{b} $ with $ a, b \in \mathbb{N}^+ $. The key ingredient of the proof is the combination of the variational method and an approximation argument. Since the sign of $ \eta_\rho(\cdot) $ can change, our results can be applied to the classical wave equation.

In this paper, the Cauchy problem for the nonlinear Schrödinger system $ \begin{equation*} \begin{cases} i\partial_tu_1(x, t) = \Delta u_1(x, t)-|u_1(x, t)|^{p-1}u_1(x, t)-|u_2(x, t)|^{p-1}u_1(x, t), \\ i\partial_tu_2(x, t) = \Delta u_2(x, t)-|u_2(x, t)|^{p-1}u_2(x, t)-|u_1(x, t)|^{p-1}u_2(x, t), \end{cases} \end{equation*} $ was investigated in $ d $ space dimensions. For $ 1 < p\le 1+2/d $, the nonexistence of asymptotically free solutions for the nonlinear Schrödinger system was proved based on mathematical analysis and scattering theory methods. The novelty of this paper was to give the proof of pseudo-conformal identity on the nonlinear Schrödinger system. The present results improved and complemented these of Bisognin, Sepúlveda, and Vera(Appl. Numer. Math. 59(9)(2009): 2285–2302), in which they only proved the nonexistence of asymptotically free solutions when $ d = 1, \; p = 3 $.

In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system $ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $ where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.

We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities $ \begin{equation*} \left\{\begin{aligned} -\Delta_{\mathbb{G}}u-\mu_{1}\frac{\psi^{2}u}{\text{d}(z)^{2}} = \lambda_{1}\frac{\psi^{\alpha}|u|^{2^*(\alpha)-2}u}{\text{d}(z)^{\alpha}}+\beta p_{1}f(z)\frac{\psi^{\gamma}|u|^{p_{1}-2}u|v|^{p_{2}}}{\text{d}(z)^{\gamma}}\,\,\, \text{in } \mathbb{G},\\ -\Delta_{\mathbb{G}}v-\mu_{2}\frac{\psi^{2}v}{\text{d}(z)^{2}} = \lambda_{2}\frac{\psi^{\alpha}|v|^{2^*(\alpha)-2}v}{\text{d}(z)^{\alpha}}+\beta p_{2}f(z)\frac{\psi^{\gamma}|u|^{p_{1}}|v|^{p_{2}-2}v}{\text{d}(z)^{\gamma}}\,\,\, \text{in } \mathbb{G}, \end{aligned}\right. \end{equation*} $ where $ -\Delta_{\mathbb{G}} $ is the sub-Laplacian on Carnot group $ \mathbb{G} $, $ \mu_{1} $, $ \mu_{2}\in [0, \mu_{\mathbb{G}}) $, $ \alpha, \, \gamma\in (0, 2) $, $ \lambda_{1} $, $ \lambda_{2} $, $ \beta $, $ p_{1} $, $ p_{2} > 0 $ with $ 1 < p_{1}+p_{2} < 2 $, $ \text{d}(z) $ is the $ \Delta_{\mathbb{G}} $-gauge, $ \psi = |\nabla_{\mathbb{G}}\text{d}(z)| $, $ 2^*(\alpha): = \frac{2(Q-\alpha)}{Q-2} $ is the critical Sobolev-Hardy exponents, and $ \mu_{\mathbb{G}} = (\frac{Q-2}{2})^{2} $ is the best Hardy constant on $ \mathbb{G} $. By combining a variant of the symmetric mountain pass theorem with the genus theory, we prove the existence of infinitely many weak solutions whose energy tends to zero when $ \beta $ or $ \lambda_{1} $, $ \lambda_{2} $ belong to a suitable range.

In this work, by stochastic analyses, we study stochastic representation, well-posedness, and regularity of generalized time fractional Schrödinger equation $ \begin{equation*} \left\{\begin{aligned} \partial_t^wu(t,x)& = \mathcal{L} u(t,x)-\kappa(x)u(t,x),\; t\in(0,\infty),\; x\in \mathcal{X},\\ u(0,x)& = g(x),\; x\in \mathcal{X},\\ \end{aligned}\right. \end{equation*} $ where the potential $ \kappa $ is signed, $ \mathcal{X} $ is a Lusin space, $ \partial_t^w $ is a generalized time fractional derivative, and $ \mathcal{L} $ is infinitesimal generator in terms of semigroup induced by a symmetric Markov process $ X $. Our results are applicable to some typical physical models.

In this study, we consider a chaotic model in which fractional differential operators and the delay term are added. Using the Carathéodory existence-uniqueness theorem for this chaotic model modified with the Caputo fractional derivative, we show that the solution of the associated system exists and is unique. We consider the chaotic model with a delay term with Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives and present a numerical algorithm for these models. We then present the numerical solution of chaotic models with delay terms by using piecewise differential operators, where fractional, classical and stochastic processes can be used. We present the numerical solution of chaotic models with delay terms, as modified by using piecewise differential operators. The graphical representations of these models are simulated for different values of the fractional order.

We study the well-posedness and stability for a nonlinear Euler-Bernoulli beam equation modeling railway track deflections in the framework of input-to-state stability (ISS) theory. More specifically, in the presence of both distributed in-domain and boundary disturbances, we prove first the existence and uniqueness of a classical solution by using the technique of lifting and the semigroup method, and then establish the $ L^r $-integral input-to-state stability estimate for the solution whenever $ r\in [2, +\infty] $ by constructing a suitable Lyapunov functional with the aid of Sobolev-like inequalities, which are used to deal with the boundary terms. We provide an extensive extension of relevant work presented in the existing literature.

In this paper, we consider the existence and multiplicity of normalized solutions to the following pseudo-relativistic Schrödinger equations $ \begin{equation*} \left\{ \begin{array}{lll} \sqrt{-\Delta+m^2}u +\lambda u = \vartheta |u|^{p-2}v +|u|^{2^\sharp-2}v, & x\in \mathbb{R}^N, \ u>0, \\ \ \int_{{\mathbb{R}^N}}|u|^2dx = a^2, \end{array} \right. \end{equation*} $ where $ N\geq2, $ $ a, \vartheta, m > 0, $ $ \lambda $ is a real Lagrange parameter, $ 2 < p < 2^\sharp = \frac{2N}{N-1} $ and $ 2^\sharp $ is the critical Sobolev exponent. The operator $ \sqrt{-\Delta+m^2} $ is the fractional relativistic Schrödinger operator. Under appropriate assumptions, with the aid of truncation technique, concentration-compactness principle and genus theory, we show the existence and the multiplicity of normalized solutions for the above problem.

In this paper we consider a class of anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with nonlinear right-hand sides that are superlinear at $ \pm\infty $. We prove the existence of two nontrivial weak solutions to this kind of problem by applying an abstract critical point theorem under very general assumptions on the data without supposing the Ambrosetti-Rabinowitz condition.

In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.

In this paper, a fully-discrete alternating direction implicit (ADI) difference method is proposed for solving three-dimensional (3D) fractional subdiffusion equations with variable coefficients, whose solution presents a weak singularity at $ t = 0 $. The proposed method is established via the L1 scheme on graded mesh for the Caputo fractional derivative and central difference method for spatial derivative, and an ADI method is structured to change the 3D problem into three 1D problems. Using the modified Grönwall inequality we prove the stability and $ \alpha $-robust convergence. The results presented in numerical experiments are in accordance with the theoretical analysis.

Our main objective of this paper is to study the singular $ p $-Monge-Ampère problems: equations and systems of equations. New multiplicity results of nontrivial $ p $-convex radial solutions to a single equation involving $ p $-Monge-Ampère operator are first analyzed. Then, some new criteria of existence, nonexistence and multiplicity for nontrivial $ p $-convex radial solutions for a singular system of $ p $-Monge-Ampère equation are also established.

In the framework of the self-consistent Maxwell-Pauli theory, the non-linear Pauli equation is obtained. Stationary and nonstationary solutions of the nonlinear Pauli equation for the hydrogen atom are studied. We show that spontaneous emission and the related rearrangement of the internal structure of an atom, which is traditionally called a spontaneous transition, have a simple and natural description in the framework of classical field theory without any quantization and additional hypotheses. The behavior of the intrinsic magnetic moment (spin) of an EW in an external magnetic field is considered. We show that, according to the self-consistent Maxwell-Pauli theory, in a weak magnetic field, the intrinsic magnetic moment of an EW is always oriented parallel to the magnetic field strength vector, while in a strong magnetic field, depending on the initial orientation of the intrinsic magnetic moment, two orientations are realized: either parallel or antiparallel to the magnetic field strength vector.

This paper is concerned with the following first-order Hamiltonian system $ \begin{equation} \nonumber \dot{z} = \mathscr{J}H_{z}(t, z), \end{equation} $ where the Hamiltonian function $ H(t, z) = \frac{1}{2}Lz\cdot z+A(\epsilon t)G(|z|) $ and $ \epsilon > 0 $ is a small parameter. Under some natural conditions, we obtain a new existence result for ground state homoclinic orbits by applying variational methods. Moreover, the concentration behavior and exponential decay of these ground state homoclinic orbits are also investigated.

In this work, we consider an $ \alpha $-robust high-order numerical method for the time fractional nonlinear Korteweg-de Vries (KdV) equation. The time fractional derivatives are discretized by the L1 formula based on the graded meshes. For the spatial derivative, the nonlinear operator is defined to approximate the $ uu_x $, and two coupling equations are obtained by processing the $ u_{xxx} $ with the order reduction method. Finally, the nonlinear difference schemes with order ($ 2-\alpha $) in time and order $ 2 $ precision in space are obtained. This means that we can get a higher precision solution and improve the computational efficiency. The existence and uniqueness of numerical solutions for the proposed nonlinear difference scheme are proved theoretically. It is worth noting the unconditional stability and $ \alpha $-robust stability are also derived. Moreover, the optimal convergence result in the $ L_2 $ norms is attained. Finally, two numerical examples are given, which is consistent with the theoretical analysis.

It is known that V.A. Ilyin and E.I. Moiseev studied generalized nonlocal boundary value problems for the Sturm-Liouville equation, the nonlocal boundary conditions specified at the interior points of the interval under consideration. For such problems, uniqueness and existence theorems for a solution to the problem were proven. There are many difficulties in studying these generalized nonlocal boundary value problems for partial differential equations, especially in obtaining a priori estimates. Therefore, it is necessary to use new methods for solving generalized nonlocal problems (forward problems). As we know, it is not difficult to establish a connection between forward and inverse problems. Therefore, when solving generalized nonlocal boundary value problems for partial differential equations, reducing them to multipoint inverse problems is necessary. The first results in the direction belong to S.Z. Dzhamalov. In his works, he proposed and investigated multipoint inverse problems for some equations of mathematical physics. In this article, the authors studied the correctness of one linear two-point inverse problem for the multidimensional heat conduction equation. Using the methods of a priori estimates, Galerkin’s method, a sequence of approximations and contracting mappings, the unique solvability of the generalized solution of the linear two-point inverse problem for the multidimensional heat equation was proved.

The embedding theory of spaces of differentiable functions of many variables studies important connections and relationships between differential (smoothness) and metric properties of functions and has wide application in various branches of pure mathematics and its applications. Earlier, we obtained the embedding theorems of different metrics for Nikol’skii-Besov spaces with a dominant mixed smoothness and mixed metric, and anisotropic Lorentz spaces. In this work, we showed that the conditions for the parameters of spaces in the above theorems are unimprovable. To do this, we built the extreme functions included in the spaces from the left sides of the embeddings and not included in the “slightly narrowed” spaces from the spaces in the right parts of the embeddings.

In this paper, we studied the issues of integrability with the weight of the sum of series with respect to multiplicative systems, provided that the coefficients of the series are monotonic. The conditions for the weight function and the series’ coefficients are found; the sum of the series belongs to the weighted Lebesgue space Lp (1 < p < ∞). In addition, the case of p = 1 was considered. In this case, other conditions for the weighted integrability of the sum of the series under consideration are found. In the case of the generating sequence’s boundedness, the proved theorems imply an analogue of the well-known Hardy-Littlewood theorem on trigonometric series with monotone coefficients.

Equations and methods of classical mechanics are used to describe the dynamics of technical systems containing elements of various physical nature, planning and management tasks of production and economic objects. The direct use of known dynamics equations with indefinite multipliers leads to an increase in deviations from the constraint equations in the numerical solution. Common methods of constraint stabilization, known from publications, are not always effective. In the general formulation, the problem of constraint stabilization was considered as an inverse problem of dynamics and it requires the determination of Lagrange multipliers or control actions, in which holonomic and differential constraints are partial integrals of the equations of the dynamics of a closed system. The conditions of stability of the integral manifold determined by the constraint equations and stabilization of the constraint in the numerical solution of the dynamic equations were formulated.

In the article, the boundary value problem for the wave equation with a fractional time derivative and with initial conditions specified in the form of a fractional derivative in the Riemann-Liouville sense is solved. The definition domain of the desired function is the upper half-plane (x,t). To solve the problem, the Fourier transform with respect to the spatial variable was applied, then the Laplace transform with respect to the time variable was used. After applying the inverse Laplace transform, the solution to the transformed problem contains a two-parameter Mittag-Leffler function. Using the inverse Fourier transform, a solution to the problem was obtained in explicit form, which contains the Wright function. Next, we consider limiting cases of the fractional derivative’s order which is included in the equation of the problem.

The Ginzburg-Landau equation with rapidly oscillating terms in the equation and boundary conditions in a perforated domain was considered. Proof was given that the trajectory attractors of this equation converge weakly to the trajectory attractors of the homogenized Ginzburg-Landau equation. To do this, we use the approach from the articles and monographs of V.V. Chepyzhov and M.I. Vishik about trajectory attractors of evolutionary equations, and we also use homogenization methods that appeared at the end of the 20th century. First, we use asymptotic methods to construct asymptotics formally, and then we justify the form of the main terms of the asymptotic series using functional analysis and integral estimates. By defining the corresponding auxiliary function spaces with weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Then, we formulate the main theorems and prove them by using auxiliary lemmas. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized Ginzburg-Landau equation in the subcritical case, and they disappear in the supercritical case.

The work is devoted to the formulation and study of the solvability for a problem with missing conditions on the characteristic boundary of the domain and an analogue of the Frankl condition on the segment of the degeneracy for a hyperbolic equation. The difference between this problem and known local and nonlocal problems is that, firstly, a hyperbolic equation is taken with arbitrary positive power degeneracy and singular coefficients on the part of the boundary, and secondly, the characteristic boundary of the domain is arbitrarily divided into two pieces and the value of the desired function is set on the first piece, and the second piece is freed from the boundary condition and this missing Goursat condition is replaced by an analogue of the Frankl condition on the degeneracy segment, and the value of an unknown function on another characteristic boundary of the domain is also considered to be known. The conditions for the coefficients of the equation and the data of the formulated problem, ensuring the validity of the uniqueness theorem are found. The theorem of the existence of a solution to the problem is proved by reducing to the problem of solving a non-standard singular integral equation with a non-Fredholm integral operator in the non-characteristic part of the equation, the kernel of which has an isolated first-order singularity. Applying the Carleman regularization method to the received equation, the Wiener-Hopf integral equation is obtained. It is proved that the index of the Wiener-Hopf equation is zero, therefore it is uniquely reduced to the Fredholm integral equation of the second kind, the solvability of which follows from the uniqueness of the problem’s solution.

Previously, boundary control problems for the second order parabolic type equation in the bounded domain were studied. In this paper, a boundary control problem associated with a fourth-order parabolic equation in a bounded two-dimensional domain was considered. On the part of the considered domain’s boundary, the value of the solution with control function is given. Restrictions on the control are given in such a way that the average value of the solution in the considered domain gets a given value. By the method of separation of variables the given problem is reduced to a Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method and an estimate was found for the minimal time at which the given average temperature in the domain is reached.

As you know, the third order partial differential equation is one of the basic equations of wave theory. For example, in particular, a linearized Korteweg-de Vries type equation with variable coefficients models ion-acoustic waves into plasma and acoustic waves on a crystal lattice. In this paper, the properties of solutions of а class of the third order degenerate partial differential equations with variable coefficients given in a rectangle were studied. Sufficient conditions for the existence and uniqueness of a strong solution have been established. Note that the solution of the degenerate equation does not retain its smoothness, therefore, these difficulties in turn affect the coercive estimates.

In this work, we introduce a new concept of the stream function and derive the equation for the stream function in the three-dimensional case. To construct a basis in the space of solutions of the NavierStokes system, we solve an auxiliary spectral problem for the bi-Laplacian with Dirichlet conditions on the boundary. Then, using the formulas employed for introducing the stream function, we find a system of functions forming a basis in the space of solutions of the Navier-Stokes system. It is worth noting that this basis can be utilized for the approximate solution of direct and inverse problems for the Navier-Stokes system, both in its linearized and nonlinear forms. The main idea of this work can be summarized as follows: instead of changing the boundary conditions (which remain unchanged), we change the differential equations for the stream function with a spectral parameter. As a result, we obtain a spectral problem for the bi-Laplacian in the domain represented by a three-dimensional unit sphere, with Dirichlet conditions on the boundary of the domain. By solving this problem, we find a system of eigenfunctions forming a basis in the space of solutions to the Navier-Stokes equations. Importantly, the boundary conditions are preserved, and the continuity equation for the fluid is satisfied. It is also noteworthy that, for the three-dimensional case of the Navier-Stokes system, an analogue of the stream function was previously unknown.

In the paper the second boundary value problem in a rectangular domain for an inhomogeneous third-order partial differential equation with multiple characteristics with constant coefficients was considered. The uniqueness of the solution to the problem posed is proven by the method of energy integrals. A counterexample is constructed in case when the uniqueness theorem’s conditions are violated. Using the method of separation of variables, the solution to the problem is sought in the form of a product of two functions X(x) and Y (y). To determine Y (y), we obtain a second-order ordinary differential equation with two boundary conditions at the boundaries of the segment [0,q]. For this problem, the eigenvalues and the corresponding eigenfunctions are found for n = 0 and n ∈ N. To determine X(x), we obtain a third-order ordinary differential equation with three boundary conditions at the boundaries of the segment [0,p]. Using the Green’s function method, we constructed solution of the specified problem. A separate Green’s function for n = 0 and a separate Green’s function for the case when n is natural were constructed. The solution for X(x) is written in terms of the constructed Green’s function. After some transformations, an integral Fredholm equation of the second kind is obtained, the solution of which is written through the resolvent. Estimates for the resolvent and Green’s function are obtained. The uniform convergence of the solution and the possibility of its term-by-term differentiation under certain conditions on given functions are proven. When justifying the uniform convergence of the solution, the absence of a “small denominator” is proven.

In the paper spaces of periodic functions of several variables were considered, namely the Lorentz space L2,τ(Tm), the class of functions with bounded mixed fractional derivative Wr2,τ, 1 ≤ τ < ∞, and the order of the best M-term approximation of a function f ∈ Lp,τ(Tm) by trigonometric polynomials was studied. The article consists of an introduction, a main part, and a conclusion. In the introduction, basic concepts, definitions and necessary statements for the proof of the main results were considered. One can be found information about previous results on the mentioned topic. In the main part, exact-order estimates are established for the best M-term approximations of functions of the Sobolev class Wr2,τ1 in the norm of the space Lp,τ2(Tm) for various relations between the parameters p,τ1,τ2.

The class K of algebraic systems of signature σ is called a formula-definable class if there exists an algebraic system A of signature σ such that for any algebraic system B of signature σ it is B ∈ K if and only if Th(B) · Th(A) = Th(A). The paper shows that the formula-definable class of algebraic systems is idempotently formula-definable and is an axiomatizable class of algebraic systems. Any variety of algebraic systems is an idempotently formula-definite class. If the class K of all existentially closed algebraic systems of a theory T is formula-definable, then a theory of the class K is a model companion of the theory T. Also, in the paper the examples of some theories on the properties of formula-definability, pseudofiniteness and smoothly approximability of their model companion were discussed.

The work is related to the study of the model-theoretic properties of Jonsson theories, which, generally speaking, are not complete. In the article, on the Boolean of Jonsson subsets of the semantic model of some fixed Jonsson theory, the concept of the Jonsson closure operator Jcl was introduced, defining the J-pregeometry on these subsets, and some results were obtained describing this closure operator.

The study of boundary value problems for elliptic equations is of both theoretical and applied interest. A thorough study of model physical and spectral problems requires an explicit and effective representation of the problem solution. Integral representations of solutions of problems of differential equations are one of the main tools of mathematical physics. Currently, the integral representation of the Green function of classical problems for the Laplace equation for an arbitrary domain is obtained only in a two-dimensional domain by the Riemann conformal mapping method. Starting from the three-dimensional case, these classical problems are solved only for spherical sectors and for the regions lying between the faces of the hyperplane. The problem of constructing integral representations of general boundary value problems and studying their spectral problems remains relevant. In this work, using the boundary condition of the Newtonian (volume) potential and the spectral property of the potential of a simple layer, the Green function of the Dirichlet problem for the Laplace equation was constructed.

In the paper a novel boundary value problem for a third-order partial differential equation (PDE) of a parabolic-hyperbolic type, within a pentagonal domain consisting of both parabolic and hyperbolic regions was investigated. Such equations are pivotal in modeling complex physical phenomena across diverse fields such as physics, engineering, and finance due to their ability to encapsulate a wide range of dynamics through their mixed-type nature. By employing a constructive solution approach, we demonstrate the unique solvability of the posed problem. The significance of this study lies in its extension of the mathematical framework for understanding and solving higher-order mixed PDEs in complex geometrical domains, thus offering new avenues for theoretical and applied research in mathematical physics and related disciplines.

Professor Dulat Syzdykbekovich Dzhumabaev, Doctor of Physical and Mathematical Sciences, was a prominent scientist, a well-known specialist in the field of the qualitative theory of differential and integro-differential equations, the theory of nonlinear operator equations, numerical and approximate methods for solving boundary value problems.

For a spatial twisted central configuration of the Newtonian ($ 2N $+1)-body problem where $ 2N $ masses are at the vertices of two paralleled regular $ N $-polygons with distance $ h > 0 $, and the twist angle between the two regular $ N $-polygons is $ 0\leq\theta < 2\pi $, we study the sufficient and necessary conditions for the existence of the spatial twisted central configuration. Additionally, we obtain the uniqueness of the spatial twisted central configuration.

In this paper, we study the following quasilinear chemotaxis system $ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi \nabla \cdot (\varphi (u)\nabla v)-\xi \nabla \cdot (\psi(u)\nabla w)+f(u), \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta v-v+v_{1}^{\gamma_{1}}, \ 0 = \Delta v_{1}-v_{1}+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta w-w+w_{1}^{\gamma_{3}}, \ 0 = \Delta w_{1}-w_{1}+u^{\gamma_{4}}, \ &\ \ x\in \Omega, \ t>0, \end{array} \right. \end{equation*} $ in a smoothly bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \varphi(\varrho)\leq\varrho(\varrho+1)^{\theta-1}, $ $ \psi(\varrho)\leq\varrho(\varrho+1)^{l-1} $ and $ f(\varrho)\leq a \varrho-b\varrho^{s} $ for all $ \varrho\geq0, $ and the parameters satisfy $ a, b, \chi, \xi, \gamma_{2}, \gamma_{4} > 0, $ $ s > 1, $ $ \gamma_{1}, \gamma_{3}\geq1 $ and $ \theta, l\in \mathbb{R}. $ It has been proven that if $ s \geq\max\{ \gamma_{1}\gamma_{2}+\theta, \gamma_{3}\gamma_{4}+l\}, $ then the system has a nonnegative classical solution that is globally bounded. The boundedness condition obtained in this paper relies only on the power exponents of the system, which is independent of the coefficients of the system and space dimension $ n. $ In this work, we generalize the results established by previous researchers.

In this paper, we discuss the beingness conditions for algebraic Schouten solitons associated with Yano connections in the background of three-dimensional Lorentzian Lie groups. By transforming equations of algebraic Schouten solitons into algebraic equations, the existence conditions of solitons are found. In particular, we deduce some formulations for Yano connections and related Ricci operators. Furthermore, we find the detailed categorization for those algebraic Schouten solitons on three-dimensional Lorentzian Lie groups. The major results demonstrate that algebraic Schouten solitons related to Yano connections are present in $ G_{1} $, $ G_{2} $, $ G_{3} $, $ G_{5} $, $ G_{6} $ and $ G_{7} $, while they are not identifiable in $ G_{4} $.

In this paper, we study the nonlinear Kirchhoff equation $ \begin{align*} -\Big(a+b\int_{\mathbb{Z}^{3}}|\nabla u|^{2} d \mu\Big)\Delta u+V(x)u = f(u) \end{align*} $ on lattice graph $ \mathbb{Z}^3 $, where $ a, b > 0 $ are constants and $ V:\mathbb{Z}^{3}\rightarrow \mathbb{R} $ is a positive function. Under a Nehari-type condition and 4-superlinearity condition on $ f $, we use the Nehari method to prove the existence of ground-state solutions to the above equation when $ V $ is coercive. Moreover, we extend the result to noncompact cases in which $ V $ is a periodic function or a bounded potential well.

In this paper, we first prove that the space $ (X, \|\cdot\|) $ is separable if and only if for every $ \varepsilon\in $ $ (0, 1) $, there is a dense subset $ G $ of $ X^{*} $ and a $ w^{*} $-lower semicontinuous norm $ \|\cdot\|_{0} $ of $ X^{*} $ so that (1) the norm $ \|\cdot\|_{0} $ is Frechet differentiable at every point of $ G $ and $ d_{F}\|x^{*}\|_{0}\in X $ is a $ w^{*} $-strongly exposed point of $ B(X^{**}, \|\cdot\|_{0}) $ whenever $ x^{*}\in G $; (2) $ \left(1+\varepsilon^{2}\right) {\left\| {{x^{***}}} \right\|_0} \le \left\| {{x^{***}}} \right\| \le \left(1 + \varepsilon \right){\left\| {{x^{***}}} \right\|_0} $ for each $ x^{***}\in X^{***} $; (3) there exists $ \{x_{i}^{*}\}_{i = 1}^{\infty}\subset G $ such that ball-covering $ \{ B({x_{i}^{*}}, {r_i})\} _{i = 1}^\infty $ of $ (X^{*}, \|\cdot\|_{0}) $ is $ (1+\varepsilon)^{-1} $-off the origin and $ S(X^{*}, \|\cdot\|)\subset \cup_{i = 1}^{\infty}B({x_{i}^{*}}, {r_i}) $. Moreover, we also prove that if space $ X $ is weakly locally uniform convex, then the space $ X $ is separable if and only if $ X^{*} $ has the ball-covering property. As an application, we get that Orlicz sequence space $ l_{M} $ has the ball-covering property.

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $ \begin{cases} \Delta u = 0 \qquad &\text{in}~~ \Omega , \\ u = 0 &\text{on}~~ \Gamma_0 , \\ -\Delta_\Gamma u +\partial_\nu u = |u|^{p-2}u\qquad &\text{on}~~ \Gamma_1 , \end{cases} $ where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ ($ N\ge 2 $) with $ C^1 $ boundary $ \partial\Omega = \Gamma_0\cup\Gamma_1 $, $ \Gamma_0\cap\Gamma_1 = \emptyset $, $ \Gamma_1 $ being nonempty and relatively open on $ \Gamma $, $ \mathcal{H}^{N-1}(\Gamma_0) > 0 $ and $ p > 2 $ being subcritical with respect to Sobolev embedding on $ \partial\Omega $. We prove that the problem admits nontrivial solutions at the potential-well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.

We investigate the non-trivial solutions for a partial discrete Dirichlet nonlinear problem with $ p $-Laplacian by applying Ricceri's variational principle and a two non-zero critical points theorem. In addition, we identify open intervals of the parameter $ \lambda $ under appropriate constraints imposed on the nonlinear term. This allows us to ensure that the nonlinear problem has at least one or two non-trivial solutions.

In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.

In this paper, we consider the following Schrödinger-Poisson system $ \begin{equation*} \qquad \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u = |u|^{p-2}u+ \lambda K(x)|u|^{q-2}u\ \ \ &\ \rm in\; \mathbb{R}^{3}, \\ -\Delta \phi = u^2 \ \ \ &\ \rm in\; \mathbb{R}^{3}.\ \end{array} \right. \end{equation*} $ Under the weakly coercive assumption on $ V $ and an appropriate condition on $ K $, we investigate the cases when the nonlinearities are of concave-convex type, that is, $ 1 < q < 2 $ and $ 4 < p < 6 $. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that $ \lambda\in(-\infty, \lambda_*) $, where $ \lambda_* > 0 $ is a constant.

In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution.

In this paper, we study the global $ C^{\infty} $ regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary $ y = 0 $ for any $ x > 0 $.

In this paper, a boundary integral equation method is proposed for the fluid-solid interaction scattering problem, and a high-precision numerical method is developed. More specifically, by introducing the Helmholtz decomposition, the corresponding problem is transformed into a coupled boundary value problem for the Helmholtz equation. Based on the integral equation method, the coupled value problem is reduced to a system of three coupled hypersingular integral equations. Semi-discrete and fully-discrete collocation methods are proposed for the singular integral equations. The presented method is based on trigonometric interpolation and discretized singular operators applied to differentiated interpolation. The convergence of the method is verified by a numerical experiment.

In this paper, we consider the following discrete fractional $ p $-Laplacian equations: $ \begin{equation*} (-\Delta_{1})^{s}_{p}u(a)+V(a)|u(a)|^{p-2}u(a) = \lambda f(a, u(a)), \; \mbox{in}\ \mathbb{Z}, \end{equation*} $ where $ \lambda $ is the parameter and $ f(a, u(a)) $ satisfies no symmetry assumption. As a result, a specific positive parameter interval is determined by some requirements for the nonlinear term near zero, and then infinitely many homoclinic solutions are obtained by using a special version of Ricceri's variational principle.

February 24, 2024 to Doctor of Physical and Mathematical Sciences, Professor M.I. Ramazanov turned 75 years old. M.I. Ramazanov is a highly qualified specialist in the field of loaded partial differential equations, integral equations, and their applications to applied problems.