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The objective of this paper is to examine the vanishing viscosity limit of the nonhomogeneous incompressible NS systerm, subject to the no-slip boundary condition. By adopting Kato's approach of constructing an artificial boundary layer [1], within a smooth and bounded domain designated as $ \Omega\subseteq\mathbb{R}^{2} $, we derive a sufficient condition for the convergence to occur uniformly in time within the energy space $ L^{2}(\Omega) $.

In this paper we will introduce a discrete version of systems obtained by modifications of the Euler-Poincaré equations when we add a special type of dissipative force, so that the equations of motion can be described using the metriplectic formalism. The metriplectic representation of the dynamics allows us to describe the conservation of energy, as well as to guarantee entropy production. For deriving the discrete equations we use discrete gradients to numerically simulate the evolution of the continuous metriplectic equations preserving their main properties: preservation of energy and correct entropy production rate.

In this paper, we were concerned with the existence of the solution related to the discontinuous differential equation, which corresponded to the stratification phenomenon in the Antarctic Circumpolar Current (ACC). By considering the piecewise vorticity function, we demonstrated the existence of solution corresponding to the discontinuous differential equation using Green's function, fixed point theory, and topological degree theory. This primarily included cases with piecewise constant vorticity, piecewise linear vorticity, and piecewise nonlinear vorticity. Additionally, we provided some examples to verify our results.

Let $ L = -\Delta+|x|^2 $ be the Hermite operator on $ \mathbb R^{d} $, where $ \Delta $ is the Laplacian on $ \mathbb R^{d} $. In this paper, we will consider the Hardy-Sobolev spaces of higher order associated with $ L $. We also give some new characterizations of the Hardy spaces associated with $ L $.

In this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal constant in the mixed Sobolev inequality, which we show is never achieved. Furthermore, by using variational methods, we provide an existence and nonexistence theory for both linear and superlinear perturbation cases.

The following chemotaxis system has been considered: \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} v_{t} = \Delta v-\xi \nabla\cdot(v \nabla w_{1})+\chi \nabla\cdot(v \nabla w_{2})+\lambda v-\mu v^{\kappa},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] w_{1t} = \Delta w_{1}-w_{1}+w^{\kappa_{1}}, \ 0 = \Delta w-w+v^{\kappa_{2}}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w_{2}-w_{2}+v^{\kappa_{3}}, \ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $\end{document} under the boundary conditions of $ \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w_{1}}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = \frac{\partial{w_{2}}}{\partial{\nu}} $ on $ \partial \Omega, $ where $ \Omega $ was a bounded smooth domain of $ \mathbb{R}^{n}(n\geq 1), \; \nu $ was the normal vector of $ \partial\Omega, $ and the parameters were $ \lambda, \mu, \xi, \chi, \kappa_{1}, \; \kappa_{2}, \kappa_{3} > 0, $ and $ \kappa > 1. $ In this paper, we showed that if either $ \kappa_{1}\kappa_{2} < \max\{\frac{2}{n}, \kappa_{3}, \kappa-1\} $ or $ \kappa_{1}\kappa_{2} = \max\{\frac{2}{n}, \kappa_{3}, \kappa-1\} $ with the coefficients and initial data satisfying appropriate conditions, then the system possessed a global classical solution. Furthermore, we also have studied the convergence of solutions to a special case of the above system with $ \kappa = \delta+1, \kappa_{1} = 1, \kappa_{2} = \kappa_{3} = \delta $ for $ \delta > 0. $ It has been proven that if $ \mu > 0 $ is large enough, then the corresponding classical solutions exponentially converged to $ ((\frac{\lambda}{\mu})^{\frac{1}{\delta}}, \frac{\lambda}{\mu}, \frac{\lambda}{\mu}, \frac{\lambda}{\mu}), $ where the convergence rate could be formally expressed by the parameters of the system.

Following the ideas about analogies, mathematical, qualitative, and structural, introduced by Mihailo Petrović in Elements of Mathematical Phenomenology (Serbian Royal Academy, Belgrade, 1911), in this paper we present our research results focused on analogies of fractional-type oscillation models between mechanical and electrical oscillators, with a finite number of degrees of freedom of oscillation. In addition to reviewing basic results, we investigate new constitutive relations and generalizations of the energy dissipation function, mechanical dissipative element of fractional type, and electrical resistor of dissipative fractional type. Those constitutive relations are expressed by means of the fractional order differential operator. By applying the Laplace transformation and the power series expansions, we determine and graphically present the approximate analytical solutions for eigen oscillations of the fractional type, as well as for forced oscillations, using a convolution integral. Tables with elements of mathematical phenomenology and analogies of oscillatory mechanical and electrical systems of fractional type are shown, as well as the principal fractional-type eigen-modes for a class of discrete mechanical or electrical oscillators, when these fractional-type modes are independent and there is no interaction between them. A number of theorems on the properties of independent modes of fractional type and the energy analysis of the discrete systems are also given.

Let $ 0 < \alpha < mn $ and $ 0 < r, q < \infty $. In this paper, we obtain the boundedness of some multilinear operators by establishing pointwise inequalities and applying extrapolation methods on tent spaces $ T_{r}^{q}(\mathbb{R}_{+}^{n+1}) $, where these multilinear operators include multilinear Hardy–Littlewood maximal operator $ \mathcal{M} $, multilinear fractional maximal operator $ \mathcal{M}_{\alpha} $, multilinear Calderón–Zygmund operator $ \mathcal{T} $, and multilinear fractional integral operator $ \mathcal{I}_{\alpha} $. Therefore, the results of Auscher and Prisuelos–Arribas [Math. Z. 286 (2017), 1575–1604] are extended to the general case.

In this article, we consider a nonlinear viscoelastic hyperbolic problem with variable exponents. By using the Faedo$ - $Galerkin method and the contraction mapping principle, we obtain the existence of weak solutions under suitable assumptions on the variable exponents $ m(x) $ and $ p(x) $. Then we prove that a solution blows up in finite time with positive initial energy as well as nonpositive initial energy.

The notion of hybrid quantum-classical control system was introduced as a control dynamical system which combined classical and quantum degrees of freedom. Classical and quantum objects were combined within a geometrical description of both types of systems. We also considered the notion of hybrid quantum-classical controllability by means of the usual definitions of geometric control theory, and we discussed how the different concepts associated to quantum controllability are lost in the hybrid context because of the nonlinearity of the dynamics. We also considered several examples of physically relevant problems, such as the spin-boson model or the notion of hybrid spline.

In this paper, we studied the existence of multiple normalized solutions to the following Kirchhoff type equation:$ \begin{equation*} \begin{cases} -\left(a\varepsilon^2+b\varepsilon\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+V(x)u = \mu u+f(u) & {\rm{in}}\;\mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx = m\varepsilon^3 , u\in H^1(\mathbb{R}^3) , \end{cases} \end{equation*} $where $ a $, $ b $, $ m > 0 $, $ \varepsilon $ is a small positive parameter, $ V $ is a nonnegative continuous function, $ f $ is a continuous function with $ L^2 $-subcritical growth and $ \mu\in\mathbb{R} $ will arise as a Lagrange multiplier. Under the suitable assumptions on $ V $ and $ f $, the existence of multiple normalized solutions was obtained by using minimization techniques and the Lusternik-Schnirelmann theory. We pointed out that the number of normalized solutions was related to the topological richness of the set where the potential $ V $ attained its minimum value.

We considered a viscous incompressible fluid flow in a varying bounded domain consisting of branching thin cylindrical tubes whose axes are line segments that form a network of pre-fractal curves constituting an approximation of the Sierpinski gasket. We supposed that the fluid flow is driven by volumic forces and governed by Stokes equations with boundary conditions for the velocity and the pressure on the wall of the tubes and inner continuity conditions for the normal velocity on the interfaces between the junction zones and the rest of the pipes. We constructed local perturbations, related to boundary layers in the junction zones, from solutions of Leray problems in semi-infinite cylinders representing the rescaled junctions. Using -convergence methods, we studied the asymptotic behavior of the fluid as the radius of the tubes tends to zero and the sequence of the pre-fractal curves converges in the Hausdorff metric to the Sierpinski gasket. Based on the constructed local perturbations, we derived, according to a critical parameter related to a typical Reynolds number of the flow in the junction zones, three effective flow models in the Sierpinski gasket, consisting of a singular Brinkman flow, a singular Darcy flow, and a flow with constant velocity.

In this study, the initial-boundary value problem for cylindrically symmetric Navier-Stokes equations was considered with temperature-dependent viscosity and heat conductivity. Firstly, we established the existence and uniqueness of a strong solution when the viscosity and heat conductivity were both power functions of temperature. Moreover, the large-time behavior of the strong solution was obtained with large initial data, since all of the estimates in this paper were independent of time. It is worth noting that we identified the relationship between the initial data and the power of the temperature in the viscosity for the first time.

In this paper, the following critical Kirchhoff-type elliptic equation involving a logarithmic-type perturbation$ -\Big(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x\Big)\Delta u = \lambda|u|^{q-2}u\ln |u|^2+\mu|u|^2u $is considered in a bounded domain in $ \mathbb{R}^{4} $. One of the main obstructions one encounters when looking for weak solutions to Kirchhoff problems in high dimensions is that the boundedness of the $ (PS) $ sequence is hard to obtain. By combining a result by Jeanjean [27] with the mountain pass lemma and Brézis–Lieb's lemma, it is proved that either the norm of the sequence of approximation solutions goes to infinity or the problem admits a nontrivial weak solution, under some appropriate assumptions on $ a $, $ b $, $ \lambda $, and $ \mu $.

In the present paper, we study an anisotropic $ \overset{\rightarrow }{p}(\cdot) $-Laplace equation with combined effects of variable singular and sublinear nonlinearities. Using the Ekeland's variational principle and a constrained minimization, we show the existence of a positive solution for the case where the variable singularity $ \beta(x) $ assumes its values in the interval $ (1, \infty) $.

An existing approach for deriving analytical expressions for slip-flow properties of Stokes flow is generalised and applied to a range of micro and nanoscale applications. The technique, which exploits the reciprocal theorem, can generate first-order predictions of the impact of Navier or Maxwell slip boundary conditions on surface moments of the traction force (e.g. on drag and torque). This article brings dedicated focus to the technique, generalises it to predict first-order slip effects on arbitrary moments of the surface traction, numerically verifies the technique on a number of cases and applies the method to a range of micro and nano-scale applications. Applications include predicting: the drag on translating spheres with varying slip length; the efficiency of a micro journal bearing; the speed of a self-propelled particle (a ‘squirmer’); and the pressure drop required to drive flow through long, straight micro/nano channels. Certain general results are also obtained. For example, for low-slip Stokes flow: any surface distribution of positive slip length will reduce the drag on any translating particle; and any perimetric distribution of positive slip length will reduce the pressure loss through a straight channel flow of arbitrary cross-section.

Marine propellers are studied in design and off-design modes of operation like crashback, where the propeller rotates in reverse while the vehicle is in forward motion. Past experiments (Jessup et al., Proceedings of the 25th Symposium on Naval Hydrodynamics, St John's, Canada, 2004; Proceedings of the 26th Symposium on Naval Hydrodynamics, Rome, Italy, 2006) studied the marine propeller David Taylor Model Basin 4381 in the open-jet test section of the 36-inch variable pressure water tunnel (VPWT). In crashback, a significant discrepancy with unclear sources exists between the mean propeller loads from the VPWT and open-water towing tank (OW) experiments (Ebert et al., 2007 ONR Propulsor S & T Program Review, October, 2007). We perform large-eddy simulation at $Re=561\,000$ and advance ratios $J=-0.50$ and $-0.82$ with the VPWT geometry included, contrasting to the unconfined (OW) case at those same $J$ and $Re=480\,000$. We identify and delineate the water tunnel interference effects responsible, and demonstrate that these effects resemble those of a symmetric solid model or bluff body. Solid blockage due to jet expansion and nozzle blockage due to proximity to the tunnel nozzle are identified as the primary interference effects. Their impact varies with the advance ratio $J$ and strengthens for higher magnitudes of $J$. The effective length scale to assess the severity of interference effects is found to be larger than the vortex ring diameter.

The sloping boundaries of stratified aquatic systems, such as lakes, are crucial environmental dynamic zones. While the role of sloping boundaries as energy dissipation hotspots is well established, their contribution to triggering large-scale motions has received less attention. This review delves into the development of thermally driven cross-shore flows on sloping boundaries under weak wind conditions. We specifically examine ‘thermal siphons’ (TS), a dynamical process that occurs when local free convection transforms into a horizontal circulation over sloping boundaries. Thermal siphons result from bathymetrically induced temperature (i.e. density) gradients when a lake experiences a uniform surface buoyancy flux, also known as differential cooling or heating. In the most common case of differential cooling of waters above the temperature of maximum density, TS lead to an overturning circulation characterised by a downslope density current and a surface return flow within a convective environment. Field observations, laboratory experiments and high-fidelity simulations of TS provide insights into their temporal occurrence, formation mechanisms, water transport dynamics and cross-shore pathways, addressing pivotal questions from an aquatic system perspective. Fluid mechanics is a fundamental tool in addressing such environmental questions and thereby serves as the central theme in this review.

The article is devoted to the study of a singularly perturbed initial problem for a linear differential equation with a piecewise constant argument second-order for a small parameter. This paper is considered the asymptotic expansion of the solution to the Cauchy problem for singularly perturbed differential equations with piecewise-constant argument. The initial value problem for first order linear differential equations with piecewise-constant argument was obtained that determined the regular members. The Cauchy problems for linear nonhomogeneous differential equations with a constant coefficient were obtained, which determined the boundary layer terms. An asymptotic estimate for the remainder term of the solution of the Cauchy problem was obtained. Using the remainder term, we construct a uniform asymptotic solution with accuracy O(εN+1) on the θi ≤ t ≤ θi+1, i = 0, p segment of the singularly perturbed Cauchy problem with a piecewise constant argument.

In the present paper sets related to invariant Riemannian metrics of positive sectional and (or) Ricci curvature on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the influence of the normalized Ricci flow. For invariant Riemannian metrics of the Wallach spaces which admit positive sectional curvature and belong to a given invariant surface of the normalized Ricci flow equation we establish that they form a set bounded by three connected and pairwise disjoint regular space curves such that each of them approaches two others asymptotically at infinity. Analogously, for all generalized Wallach spaces with coincided parameters the set of Riemannian metrics which belong to the invariant surface of the normalized Ricci flow and admit positive Ricci curvature is bounded by three space curves each consisting of exactly two connected components as regular curves. Mutual intersections and asymptotical behaviors of these components are studied as well. We also establish that curves corresponding to Ka¨hler metrics of spaces under consideration form separatrices of saddles of a three-dimensional system of nonlinear autonomous ordinary differential equations obtained from the normalized Ricci flow equation.

First of all, we have to note that in this article, we introduced the new concepts of relations between Jonsson theories in the class of cosemanticness for some considered Jonsson spectrum. All consideration of this new approach was done under sufficiently important class of Jonsson theories, which we called as normal Jonsson theories class. The main result, that we obtained, describes the model-theoretical properties of syntactical and semantical similarities inside the fixed cosemanticness class. For all new concepts in the article, we provided classical samples. The main result of this paper is considering normal Jonsson theories class by similarity to some fixed class of polygons (S-acts).

The issues of utilizing the central type to analyze the theoretical and model properties of the idea of heredity were examined in this research, taking into account both theories and the Jonsson spectrum. Finding solutions to issues related to the enriching language for the fixed Jonsson theory is associated with the problems of heredity of Jonsson theory. Another feature of Jonsson theories was described in the presented article. That is, the conclusion concerning J-non-multidimensional theories was presented in this study. The connection between J-P-stable theories and J-non-multidimensional theories was also characterired. In addition, the main result in the article was considered for the class of semantic pairs.

In this paper, a characterization of essential pseudospectra of bounded linear operators on ultrametric Banach spaces over a spherically complete field was given and the notions of pseudospectra and condition pseudospectra of the direct sum of linear operators on ultrametric Banach spaces were introduced. In particular, we proved that the pseudospectra of the direct sum of bounded linear operators associated with various ε are nested sets and that the intersection of all the pseudospectra of bounded linear operators is the spectrum of the direct sum of bounded linear operators in the direct sum of ultrametric Banach spaces. In addition, many results were proved about them and examples were given.

Local and nonlocal boundary value problems (LNBVPs) related to fourth-order differential equations (FODEs) were explored. To tackle these problems numerically, we introduce novel compact four-step difference schemes (DSs) that achieve eighth-order of approximation. These DSs are derived from a novel Taylor series expansion involving five points. The theoretical foundations of these DSs are validated through extensive numerical experiments, demonstrating their effectiveness and precision.

Expansion formulas associated with the multidimensional Lauricella hypergeometric functions are wellestablished and extensively utilized. However, the recurrence relations inherit in these formulas add extra complexities to their use. A thorough analysis of the characteristics of these expansion formulas shows that they can be simplified and converted into a more convenient form. This paper presents new recurrence free decomposition formulas, which are employed to solve boundary value problems.

This article is about Minkowski difference of sets, which is one of the Minkowski operators. The necessary and sufficient conditions for the existence of the Minkowski difference of given regular polygons in the plane were derived. The method of finding the Minkowski difference of given regular tetrahedrons in the Euclidean space R3 was explained. At the end of the article, the obtained results were summarized and a geometric method for finding the Minkowski difference of the convex set M and compact set N given in Rn was shown. The theory of foliations was applied to find the Minkowski difference of sets. New geometric concepts such as “dense embedding” and “completely dense embedding” were introduced. An important geometric property of the Minkowski operator was introduced and proved as a theorem.

In this paper, we study precise and exact traveling wave solutions of the conformable differential nonlinear Schrödinger equation. Then, we transform the given equation into an integer order differential equation by utilizing the wave transformation and the characteristics of the conformable derivative. To extract optical soliton solutions, we divide the wave profile into amplitude and phase components. Further, we introduce a new extension of a modified Jacobi elliptic functions method to the conformable differential nonlinear Schrödinger equation with group velocity dispersion and coefficients of second-order spatiotemporal dispersion.

Singularly perturbed integro-differential equations with degenerate kernels are considered. It is shown that in the linear case these problems are always uniquely solvable with continuous coefficients, while nonlinear problems either have no real solutions at all or have several of them. For linear problems, the results of Bobojanova are refined; in particular, necessary and sufficient conditions are given for the existence of a finite limit of their solutions as the small parameter tends to zero and sufficient conditions under which the passage to the limit to the solution of the degenerate equation is possible.

In this article, the existence and uniqueness of solutions for non-linear fractional differential equation with Tempered Ψ−Caputo derivative with three-point boundary conditions were studied. The existence and uniqueness of the solution were proved by applying the Banach contraction mapping principle and Schaefer’s fixed point theorem.

In this paper, we derive some new fixed point results in C∗-algebra valued fuzzy metric space with the help of subadditive altering distance function with respect to a t-norm. Our results generalize some existing fixed point results in the literature. A common fixed point result is also derived for a pair of mappings on complete C∗-algebra valued fuzzy metric space. The results are supported by suitable examples along with the graphical demonstration of the used conditions. As application, we establish the solvability of a second order boundary value problem. Moreover, the results are also applied in control theory to study the possibility of optimally controlling the solution of an ordinary differential equation in dynamic programming.

This article investigates a fourth-order differential equation defined in a Hilbert space, with an unbounded intermediate coefficient and potential. The key distinction from previous research lies in the fact that the intermediate term of the equation does not obey to the differential operator formed by its extreme terms. The study establishes that the generalized solution to the equation is maximally regular, if the intermediate coefficient satisfies an additional condition of slow oscillation. A corresponding coercive estimate is obtained, with the constant explicitly expressed in terms of the coefficients’ conditions. Fourth-order differential equations appear in various models describing transverse vibrations of homogeneous beams or plates, viscous flows, bending waves, and etc. Boundary value problems for such equations have been addressed in numerous works, and the results obtained have been extended to cases with smooth variable coefficients. The smoothness conditions imposed on the coefficients in this study are necessary for the existence of the adjoint operator. One notable feature of the results is that the constraints only apply to the coefficients themselves; no conditions are placed on their derivatives. Secondly, the coefficient of the lowest order in the equation may be zero, moreover, it may not be unbounded from below.

Cesàro polynomials have been extended in various ways and applied in diverse areas. In this paper, we aim to introduce a multivariable and multiparameter generalization of Cesàro polynomials. Then we explore several generating functions, an addition formula, a differential-recurrence relation, a multiple integral formula for this extended Cesàro polynomial, as well as a multiple integral formula whose kernel is this extended Cesàro polynomial. Also we present several bilinear and bilateral generating functions for this extended Ces`aro polynomial, two of whose examples are demonstrated.

It depends on our information, non-polynomial spline functions have not been applied for solving Volterra- Fredholm integral equations of the second kind yet. In this paper, we want to use such functions for finding approximation solutions of Volterra-Fredholm integral equations. In our approach, the coefficients of the non-polynomial spline were found by solving a system of linear equations. Then, these functions were utilized to reduce the fredholm integral equations to the solution of algebraic equations. Analysis of convergences investigated. Finally, three examples were presented to show the effectiveness of the method. This was done with the help of a computer program that used the Python code program version 3.9.

This work investigates the structure of punctual numberings for families of punctually enumerable sets with respect to primitive recursively reducibility. We say that a numbering of a certain family is primitive recursively reducible to another numeration of the same family if there exists a primitive recursively procedure (an algorithm not employing unbounded search) mapping the numbers of objects in the first numbering to the numbers of the same objects in the second numbering. This study was motivated by the work of Bazhenov, Mustafa, and Ospichev on punctual Rogers semilattices for families of primitive recursively enumerable functions. The concept of punctually enumerable sets was introduced in the paper, and it was proven that not all recursively enumerable sets are punctually enumerable, but in all m-degrees, recursively enumerable sets include punctually enumerable sets. For two-element families of punctual sets, it was demonstrated that punctual Rogers semilattices can be of at least three types: (1) one-element family, (2) isomorphic to the upper semilattice of recursively enumerable sets with respect to primitive recursively m-reducibility, (3) without the greatest element. It was also proven that the set of all punctually enumerable sets does not have a punctual numbering, and punctual families with a Friedberg numbering do not have the least numbering.

In this paper, we studied the existence of normalized solutions to the following Kirchhoff equation with a perturbation:$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda u = |u|^{p-2} u+h(x)\left |u\right |^{q-2}u, \quad \text{ in } \mathbb{R}^{N}, \\ &\int_{\mathbb{R}^{N}}\left|u\right|^{2}dx = c, \quad u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right. $where $ 1\le N\le 3, a, b, c > 0, 1\leq q < 2 $, $ \lambda \in \mathbb{R} $. We treated three cases:(i) When $ 2 < p < 2+\frac{4}{N}, h(x)\ge0 $, we obtained the existence of a global constraint minimizer.(ii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\ge0 $, we proved the existence of a mountain pass solution.(iii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\leq0 $, we established the existence of a bound state solution.

If a triangular Lie algebra acts on a smooth manifold, it induces a Poisson bracket on it. In case this Poisson structure is actually symplectic, we show that this already implies the existence of a flat connection on any vector bundle over the manifold the Lie algebra acts on, in particular the tangent bundle. This implies, among other things, that $ \mathbb{C}P^n $ and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.

In this article, we study a double phase variable exponents problem with mixed boundary value conditions of the form$ \left\lbrace \begin{aligned} D(u) +\vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2}u & = f(x,u) \ \ \ \ \text{ in } \Omega,\\ u& = 0 \quad \quad \quad \ \text{ on } \Lambda _1, \\ \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu & = g(x,u) \quad \ \text{ on } \Lambda _2 . \end{aligned} \right. $First of all, using the mountain pass theorem, we establish that this problem admits at least one nontrivial weak solution without assuming the Ambrosetti–Rabinowitz condition. In addition, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain theorem with the Cerami condition.

In this article, we mainly study the global existence of multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb type potential$ \begin{equation*} -\Delta u+V(\epsilon x) u = \lambda (I_\alpha * |u|^p)|u|^{p-1}+u \log u^2 \text { in } \mathbb{R}^3, \end{equation*} $where $ u \in H^1(\mathbb{R}^3) $, $ \epsilon > 0 $, $ V $ is a continuous function with a global minimum, and Coulomb type energies with $ 0 < \alpha < 3 $ and $ p \geq 1 $. We explore the existence of local positive solutions without the functional having to be a combination of a $ C^1 $ functional and a convex semicontinuous functional, as is required in the global case.

In this paper, we considered a singular parabolic -Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.

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. $

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.

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.

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 the article the spectrum and resolvent of the so-called multichannel systems with nonzero internal energies were investigated. The spectrum and resolvent of multichannel Sturm-Liouville systems with non-zero internal energies mi2 and general boundary conditions were investigated. These systems describe the propagation of partial waves in the theory of quantum physics. The importance of studying the spectral characteristics of these systems is presented in the well-known books of the theory of quantum physics. The finiteness of the number of eigenvalues was proved, the multiplicity of positive eigenvalues was investigated, and as well as the resolvent kernel of the system was found.

In this study, the time-dependent source identification problem for the two-dimensional neutron transport equation was studied. For the approximate solution of this problem a first order of accuracy difference scheme was presented. Stability estimates for the solution of these differential and difference problems were established. Numerical results were given.

This paper is devoted to a new type of boundary-value problems for Sturm-Liouville equations defined on three disjoint intervals (−π,−π+d),(−π+d,π−d) and (π−d,π) together with eigenparameter dependent boundary conditions and with additional transmission conditions specified at the common end points −π+d and π−d, where 0

We study a source identification boundary value problem for a parabolic partial differential equation with multi-point Neumann type boundary condition. Stability estimates for the solution of the overdetermined mixed BVP for multi-dimensional parabolic equation were established. The first and second order of accuracy difference schemes for the approximate solution of this problem were proposed. Stability estimates for both difference schemes were obtained. The result of numerical illustration in test example was given.

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

In the article the problem of synthesizing uniformly distributed optimal control for nonlinear optimization of oscillatory processes described by integro-differential partial differential equations with the Volterra integral operator was explored. The study was conducted according to the Bellman-Egorov scheme and an algorithm for constructing a uniformly distributed optimal control in the form of a functional from the state of the controlled process was developed. Sufficient conditions for the solvability of the synthesis problem in nonlinear optimization were established.

The theory of m-convex (m − cv) functions is a new direction in the real geometry. In this work, by using the connection m − cv functions with strongly m-subharmonic (shm) functions and using well-known and rich properties of shm functions, we show a number of important properties of the class of m−cv functions, in particular, we study Hessians Hk(u), k = 1, 2, ..., n − m +1, in the class of bounded m − cv functions.