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The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper. [ABSTRACT FROM AUTHOR]

In the present paper, a new fractional order predator–prey model is considered. The applied fractional operator is a generalized Atangana–Baleanu–Caputo (ABC) derivative, which does not require any restrictions on the initial conditions as in the case of classical ABC fractional derivatives. On the theoretical aspect, we prove the existence, uniqueness, and Ulam–Hyers stability results by using some fixed point theorems and nonlinear analysis techniques. The numerical aspect discusses the approximation solutions for the proposed model by applying the generalized scheme of the Adams–Bashforth technique. At the end, we explain the behavior of the solution to the studied model through graphical representations and numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a set of small stoves was designed which is used for alcohol-based fuel combustion. The research object is the atomization process of alcohol-based fuel in the stove. By combining numerical analysis and experiment, this paper investigated the influence of spray pressure on the atomization characteristics of alcohol-based fuel in the stove under the static environment. The results showed that as the increase of spray pressure, the atomization cone angle increased firstly and then decreased slightly and when the spray pressure was 0.8 MPa, the atomization cone angle reached the maximum value of 79.5°; the SMD (Sauter mean diameter) at the same position of the combustion chamber decreased slowly and the spray height increased slowly and both of the SMD and spray height changed slightly when the spray pressure was not less than 0.8 MPa. The experiment verified the correctness of the numerical analysis method, and the coincidence degree between both was more than 92%. This paper also investigated the influence of swirl structure parameters on the atomization characteristics of fuel in the stove under air distribution condition by using numerical analysis method. The results showed that the air central recirculation zone only generated in the stove combustion chamber when the swirl angle was not less than 30°; the minimum SMD and the maximum average velocity of all central recirculation zones sections were obtained when the combustion chamber with 12 swirl plates and 45° swirl angle, and the atomization characteristics of the fuel in this structure were better. Further research showed that when the combustion chamber with 6 swirl plates and 40° swirl angle, the SMD of all the central recirculation zone sections is the smallest and the average velocity was slightly smaller than the maximum value; and after comprehensive analysis, the atomization characteristics of the fuel in the stove with this structure are the best. These above research results will provide reference value for the design and development of alcohol-based fuel special stoves. [ABSTRACT FROM AUTHOR]

The effects of variation in humidity coupled with long-term loading give rise to dimensional changes and creep effects in wooden elements. Many wooden products such as cross-laminated timber (CLT) plates as well as many common structural details used in timber engineering are vulnerable to variations in moisture content (MC) as well as to creep effects. This paper addresses the long-term effects in the material modelling of timber by the finite element method (FEM), also considering the viscoelastic and mechanosorptive effects in wood. The model was calibrated using both relaxation tests and creep tests. The results from both long-term compression perpendicular- to-grain tests (relaxation and creep) performed on glulam (GL30c) from Norway spruce (Picea abies) with moisture control are presented in this paper. The material model considers the effect of loading and moisture changes. For realistic comparison, the pith location of each lamella was specified in the numerical analyses. Ultimately, a comparison between the numerical results and the experimental results has been provided, exhibiting an overall good estimation of timber response. [ABSTRACT FROM AUTHOR]

A list of forthcoming papers for the "Russian Journal of Numerical Analysis and Mathematical Modelling" is presented, including "Analysis of a Total Profitability Asymptotic Distribution for a Trade Algorithm," "A Fast Solving Method for Elliptic Problems in Domains With Re-Entrants Corners," and "Existence and Uniqueness of a Solution to the Primitive Equations With Stratification in the Large."

This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, Ulam’s-type stabilities are studied for a class of first-order impulsive differential equations with bounded variable delays on compact interval with finite number of impulses. Results of stability are proved via newly established integral inequality of Bellman–Grönwall–Bihari type with delay for discontinuous functions. Using this inequality for the first time and assumption of α $\alpha$ -H o ¨ $\ddot{o}$ lder’s condition instead of common Lipschitz condition is novelty of this paper. Moreover, solution is obtained in quasi–Banach spaces which is best suited for obtaining results under the assumptions of α $\alpha$ -H o ¨ $\ddot{o}$ lder’s condition. [ABSTRACT FROM AUTHOR]

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units. [ABSTRACT FROM AUTHOR]

Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry. [ABSTRACT FROM AUTHOR]

A new model for calculation of circulation in shallow water basins is created based on the shallow water equations taking into account the Coriolis force and quadratic friction on the bottom. Wind effects are taken into account as forcing. The main feature of the model is a new numerical method based on regularized shallow water equations allowing one to construct the simple and sufficiently accurate numerical algorithms possessing a number of advantages over existing methods. The paper provides a detailed description of all construction steps of the model. The developed model was implemented for the water area of the Azov Sea. The paper presents the modelling of extreme surges in March 2013 and September 2014, the results of calculations are compared with observation data of hydrometeorological stations in Taganrog and Yeysk. [ABSTRACT FROM AUTHOR]

Recently, Semenova [12] considered a derivative free iterative method for nonlinear ill-posed operator equations with a monotone operator. In this paper, a modified form of Semenova's method is considered providing simple convergence analysis under more realistic nonlinearity assumptions. The paper also provides a stopping rule for the iteration based on an a priori choice of the regularization parameter and also under the adaptive procedure considered by Pereverzev and Schock [11]. [ABSTRACT FROM AUTHOR]

This study deals with a narrow escape problem, a well-know difficult problem of evaluating the probability for a diffusing particle to reach a small part of a boundary far away from the starting position of the particle. A direct simulation of the diffusion trajectories would take an enormous computer simulation time. Instead, we use a different approach which drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift velocity directed to the target position. The method can be efficiently applied to solve narrow escape problems for domains of long extension in one direction which is the case in many practical problems in biology and chemistry. The algorithm is meshless both in space and time, and is well applied to solve high-dimensional problems in complicated domains. We present in this paper a detailed numerical analysis of the method for the case of a rectangular parallelepiped. Both stationary and transient diffusion problems are handled. [ABSTRACT FROM AUTHOR]

Changes in relative humidity of the ambient air, RH (%), cause wetting and drying of wood material, which results in non-uniform moisture contents or moisture gradients, and consequently in moisture-induced stresses and strains in the glued-laminated timber (glulam) members. The aim of the present paper is to perform a hygro-mechanical analysis to predict the mechanical behavior of glulam specimens exposed to two RH regimes, causing wetting from 50% to 90% RH and drying from 90% to 50% RH, and compare the numerical to the experimental results. The aims are also to quantitatively analyze the influence of characteristic material parameters required in the multi-Fickian moisture transport model and the mechanical model on moisture-induced strains and stresses in glulam specimens and to determine the possibility of cracking of the material by analyzing the maximum tensile stresses perpendicular to the grain. Accurate numerical predictions of moisture contents and moisture-induced strains are obtained in the glulam specimens during wetting and drying as compared to the experimental results. The influence of a particular characteristic material parameter on moisture-induced strains and stresses is characterized as significant, but not crucial when a rough numerical estimation of the mechanical behavior of the glulam beam exposed to RH changes is required. [ABSTRACT FROM AUTHOR]

In this paper, we present the concept of singularly perturbed vector field (SPVF) method and its application to spark ignition turbocharger engine. Given a mathematical/physical model, which consist of hidden multi-scale variables, the SPVF methods transfer (using the change of coordinates) and decompose such system to fast and slow subsystems. This decomposition enables one to apply different asymptotic methods such as the method of the integral manifold, homotopy analysis method, singular perturbation method, etc. The resulting subsystem enables us to understand better the complex system and to simplify the calculations. In addition, we investigated the stability of the equilibrium points of the model. This analysis has been done due to the SPVF method which reduces the complexity of the mathematical model. [ABSTRACT FROM AUTHOR]

We study piecewise synergetic systems originating from Biochemical Systems Theory. In the first part of the paper, the emphasis is put on practical calculations with such systems.We consider four examples: calculation of trajectories and steady states, solution of an optimization problem and a method of estimation of parameters (kinetic orders), all examples being biologically motivated. In the second part of the paper, we study convergence of solutions, in particularly, steady states, of a sequence of piecewise synergetic systems approximating an arbitrary compartment model. This convergence analysis is then applied to the optimization problem and the method of estimating sensitivities (kinetic orders) in a generic compartment model. In this paper we put forward arguments for the importance of the theoretical and numerical analysis of piecewise synergetic systems. [ABSTRACT FROM AUTHOR]

In the present paper, it is shown that if a prime ring R admits a generalized derivation f associated with a nonzero derivation d such that either f ⁢ ([ x m , y n ]) + [ x m , y n ] ∈ Z ⁢ (R) for all ⁢ x , y ∈ R f([x^{m},y^{n}])+[x^{m},y^{n}]\in Z(R)\quad\text{for all }x,y\in R or f ⁢ ([ x m , y n ]) - [ x m , y n ] ∈ Z ⁢ (R) for all ⁢ x , y ∈ R , f([x^{m},y^{n}])-[x^{m},y^{n}]\in Z(R)\quad\text{for all }x,y\in R, then R is commutative. We apply this purely ring theoretic result to obtain commutativity of Banach algebras and prove that if A is a prime Banach algebra which admits a continuous linear generalized derivation f associated with a nonzero continuous linear derivation d such that either f ⁢ ([ x m , y n ]) - [ x m , y n ] ∈ Z ⁢ (A) {f([x^{m},y^{n}])-[x^{m},y^{n}]\in Z(A)} or f ⁢ ([ x m , y n ]) + [ x m , y n ] ∈ Z ⁢ (A) {f([x^{m},y^{n}])+[x^{m},y^{n}]\in Z(A)} for an integer m = m ⁢ (x , y) > 1 {m=m(x,y)>1} and sufficiently many x , y {x,y} in A, then A is commutative. A similar result is obtained for a unital prime Banach algebra A which admits a nonzero continuous linear generalized derivation f associated with a continuous linear derivation d such that d ⁢ (Z ⁢ (A)) ≠ 0 {d(Z(A))\neq 0} satisfying either f ⁢ ((x ⁢ y) m) - x m ⁢ y m ∈ Z ⁢ (A) {f((xy)^{m})-x^{m}y^{m}\in Z(A)} or f ⁢ ((x ⁢ y) m) - y m ⁢ x m ∈ Z ⁢ (A) {f((xy)^{m})-y^{m}x^{m}\in Z(A)} for each x ∈ G 1 {x\in G_{1}} and y ∈ G 2 {y\in G_{2}} , where G 1 , G 2 {G_{1},G_{2}} are open sets in A and m = m ⁢ (x , y) > 1 {m=m(x,y)>1} is an integer: then A is commutative. [ABSTRACT FROM AUTHOR]

In this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form 1 + β ⁢ θ {1+\beta\,\theta} , where θ denotes the amount of transported mass and β > 0 {\beta>0} is a fixed parameter (notice that the limit case β = 0 {\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ( { ℱ ε } ε > 0 {\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}}) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of { ℱ ε } {\{\mathcal{F}_{\varepsilon}\}} as ε ↓ 0 {\varepsilon\downarrow 0}. Our functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method. [ABSTRACT FROM AUTHOR]

The paper analyzes the sensitivity of boundary crossing probabilities of the Brownian motion to perturbations of the boundary. The first- and second-order sensitivities, i.e. the directional derivatives of the probability, are derived. Except in cases where boundary crossing probabilities for the Brownian bridge are given in closed form, the sensitivities have to be computed numerically. We propose an efficient Monte Carlo procedure. [ABSTRACT FROM AUTHOR]

In this paper, we show that the abstract framework developed in [G. Pagès and C. Rey, Recursive computation of the invariant distribution of Markov and Feller processes, preprint 2017, https://arxiv.org/abs/1703.04557] and inspired by [D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 2002, 3, 367–405] can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean-reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of the fractional-order chaotic system corresponding to the original Chua system with the same nonlinearity. We place bounds on the fractional order to guarantee a chaotic behavior. In addition, we propose a one-dimensional adaptive synchronization strategy, whereby we assume knowledge of one of the states and reconstruct the rest. The proposed synchronization scheme is put to the test in a secure communication scenario based on the antipodal chaos shift keying modulation scheme. Throughout the analysis and examples, numerical results are presented to affirm the validity of the findings. [ABSTRACT FROM AUTHOR]

The weighted method of dependent trials or weighted method of correlated sampling (MCS) allows one to construct estimators for functionals based on the same Markov chain simultaneously for a given range of the problem parameters. Choosing an appropriate Markov chain, it is necessary to take into account additional conditions providing the finiteness of the computational cost of weighted MCS. In this paper we study the issue of finite computational cost of the method of correlated sampling (MCS) in application to evaluation of linear functionals of solutions to a set of systems of 2nd kind integral equations. A universal modification of the vector weighted MCS is constructed providing the branching of chain trajectory according to elements of matrix weights. It is proved that the computational cost of the constructed algorithm is bounded in the case the base functionals are also bounded. The results of numerical experiments using the modified weighted estimator are presented for some problems of the theory of radiation transfer subject to polarization. [ABSTRACT FROM AUTHOR]

In this paper, we study the theoretical analysis of a frictionless contact between two general elastic bodies in a stationary regime in a three-dimensional thin domain Ω ε {\Omega^{\varepsilon}} with Tresca friction law. Firstly, the problem statement and variational formulation are presented. We then obtain the estimates on displacement independently of the parameter ε. Finally, we obtain the main results concerning the limit of a weak problem and its uniqueness. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach; that is, to determine the space-dependent source term from a noisy final data. Based on the series expression of the solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions, and we provide the existence and uniqueness for the adjoint problem. Further, we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we investigate the backward heat problem (BHP) which is a classical ill-posed problem. Although there are many papers relating to the BHP in many domains, considering this problem in polar coordinates is still scarce. Therefore, we wish to deal with this problem associated with a space and time-dependent heat source in polar coordinates. By modifying the quasi-boundary value method, we propose the stable solution for the problem. Furthermore, under some initial assumptions, we get the Hölder type of error estimates between the exact solution and the approximated solution. Eventually, a numerical experiment is provided to prove the effectiveness and feasibility of our method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which 1 < p < n, n ≥ 3. In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called "strange term" in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles. [ABSTRACT FROM AUTHOR]

In this paper, we consider the analogous of the Hénon equation for the prescribed mean curvature problem in ℝN, both in the Euclidean and in the Minkowski spaces. Motivated by the studies of Ni and Serrin [W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Convegni Lincei 77 1985, 231–257], we have been interested in finding the relations between the growth of the potential and that of the local nonlinearity in order to prove the nonexistence of a radial ground state. We also present a partial result on the existence of a ground state solution in the Minkowski space. [ABSTRACT FROM AUTHOR]

Ultrafunctions are a particular class of functions defined on a hyperreal field ℝ ∗ ⊃ ℝ. They have been introduced and studied in some previous works [2, 6-7]. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence of solutions for some classes of singular systems involving the Δp(x) and Δq(x) Laplacian operators. The approach is based on bifurcation theory and the sub-supersolution method for systems of quasilinear equations involving singular terms. [ABSTRACT FROM AUTHOR]

Consider nonlinear Choquard equations { −Δu + u = (Iα * |u|p)|u|p−2⁢ u in⁢ ℝN, limx → ∞ ⁡u(x) = 0, where Iα denotes the Riesz potential and α ∈ (0, N). In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as α → 0 or α → N. This leads to the uniqueness and nondegeneracy of ground states when α is sufficiently close to 0 or close to N. [ABSTRACT FROM AUTHOR]

In this paper we explore Liouville-type theorems to solutions of PDEs involving the ϕ-Laplace operator in the setting of Orlicz–Sobolev spaces. Our results extend Liouville-type theorems that have been obtained recently. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the critical and subcritical Trudinger–Moser-type inequalities for functions in a fractional Sobolev space H1/2, 2 on the whole real line. We prove the relation between two inequalities and discuss the attainability of the suprema. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following critical nonlocal problem: { M ⁢ (∬ℝ2N |u(x) − u(y)|2/|x − y|N+2s⁢ 𝑑x 𝑑y) ⁢(− Δ)su = λ/uγ + u2s*−1 in ⁢ Ω, u > 0 in Ω, u = 0 in ⁢ ℝN∖Ω, where Ω is an open bounded subset of ℝ N with continuous boundary, dimension N > 2s with parameter s ∈ (0 , 1), 2*s = 2N /(N − 2s) is the fractional critical Sobolev exponent, λ > 0 is a real parameter, γ ∈ (0 , 1) and M models a Kirchhoff-type coefficient, while (−Δ)s is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions. [ABSTRACT FROM AUTHOR]

Motivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE Lu = ƒ(u) + h(x) on bounded smooth domains Ω ⊆ ℝn, where L is a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation, ƒ is a continuous non-decreasing function that satisfies the Keller–Osserman condition, while h is a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions on ƒ and h that would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions on ƒ and h for the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients of L to be bounded in Ω, but we still allow h to be unbounded on Ω. [ABSTRACT FROM AUTHOR]

We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal Lp- Lq regularity results for the non-homogeneous Stokes problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the following water wave model with a nonlocal viscous term: ut + ux + βuxxx + √ν/√π ⁢ ∂/∂t ⁢ ∫0t u(s)/√t−s ⁢ 𝑑 s + uux = νuxx, where 1/√π ∂/∂t ⁢ ∫0t u(s)/√t−s 𝑑 s is the Riemann–Liouville half-order derivative. We prove the well-posedness of this model using diffusive realization of the half-order derivative, and we discuss the asymptotic convergence of the solution. Also, we compare our mathematical results with those given in [5] and [14] for similar equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the p(x)-biharmonic operator: { Δ(|Δu|p(x)−2 Δu) = g(x) u − γ(x) ∓ λ ⁢ƒ(x, u) in ⁢ Ω, Δu = u = 0 on ⁢ ∂Ω, where Ω ⊂ ℝN (N ≥ 3) is a bounded domain with C2 boundary, λ is a positive parameter, γ : Ω ¯ → (0 , 1) is a continuous function, p ∈ C ⁢ (Ω ¯)} with 1 < p − := inf x ∈ Ω ⁡ p(x) ≤ p + := sup x ∈ Ω ⁡ p(x) < N/2, as usual, p*(x) = Np(x)/N−2p(x), g ∈ Lp*(x)/p*(x)+γ(x)−1(Ω), and ƒ(x,u) is assumed to satisfy assumptions (f1)–(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff type equation: −(1 + b ∫ℝN |∇u|2dx) Δu + u = a(x) ƒ(u) in ℝN, u ∈ H1 (ℝN), where N ≥ 3, b > 0 and ƒ(s) is asymptotically linear at infinity, that is, ƒ(s) ∼ O(s) as s → + ∞. By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on a(x). [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain Ω ⊂ ℝN, that is, we deal with −1/p−1 |∇u|2−p⁢ div(|∇⁡u|p−2∇u) = λ eu in Ω with u = 0 on ∂⁡Ω. For this problem we show that, for p ∈ [ 2, ∞ ], there exists a positive critical value λ* = λ* ⁢ (Ω, N, p) such that the following holds: • If λ < λ*, the problem admits a minimal positive solution wλ. • If λ > λ*, the problem admits no solution. Moreover, the branch of minimal solutions wλ is increasing with λ. In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution u = 0 with λ = 0, and for a small fixed λ we also obtain a continuum of solutions with p ∈ [ 2, ∞ ]. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of quasilinear elliptic equations involving the Sobolev critical exponent −εpΔpu − εpΔp(u2)u + V(x)|u|p−2u = h(u) + |u|2 p*−2u in ⁢ ℝN, where Δpu = div(|∇⁡u|p−2∇u) is the p-Laplace operator, p* = Np/N−p (N ≥ 3, N > p ≥ 2) is the usual Sobolev critical exponent, the potential V(x) is a continuous function, and the nonlinearity h(u) is a nonnegative function of C1 class. Under some suitable assumptions on V and h, we establish the existence, multiplicity and concentration behavior of solutions by using combing variational methods and the theory of the Ljusternik–Schnirelman category. [ABSTRACT FROM AUTHOR]

In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula. [ABSTRACT FROM AUTHOR]

Existence, uniqueness and continuous dependence results together with maximum principles represent key tools in the analysis of lattice reaction-diffusion equations. In this paper, we study these questions in full generality by considering nonautonomous reaction functions, possibly nonsymmetric diffusion and continuous, discrete or mixed time. First, we prove the local existence and global uniqueness of bounded solutions, as well as the continuous dependence of solutions on the underlying time structure and on initial conditions. Next, we obtain the weak maximum principle which enables us to get the global existence of solutions. Finally, we provide the strong maximum principle which exhibits an interesting dependence on the time structure. Our results are illustrated by the autonomous Fisher and Nagumo lattice equations and a nonautonomous logistic population model with a variable carrying capacity. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study higher-order (in space) Allen–Cahn and Cahn–Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor. We also give, for the Allen–Cahn models, numerical simulations which illustrate the effects of the higher-order terms and the anisotropy. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for converting boundary value problems of impulsive fractional differential equations to integral equations. Applications of this method are given to solve some types of anti-periodic boundary value problems for impulsive fractional differential equations. Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann–Liouville and Hadamard fractional derivatives with order q ∈ (0, 1), see Theorem 2, Theorem 4, Theorem 6 and Theorem 8. Then we obtain exact expression of piecewise continuous solutions of these fractional differential equations see Theorem 1, Theorem 2, Theorem 3 and Theorem 4. Finally, four classes of integral type anti-periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. See Theorems 4.1–4.4. We allow the nonlinearity p(t)ƒ(t,x) in fractional differential equations to be singular at t = 0, 1 and be involved a super-linear and sub-linear term. The analysis relies on Schaefer's fixed point theorem. In order to avoid misleading readers, we correct the results in [28] and [65]. We establish sufficient conditions for the existence of solutions of an anti-periodic boundary value problem for impulsive fractional differential equation. The results in [68] are complemented. The results in [81] are corrected. See Lemma 5.1, Lemma 5.7, Lemma 5.10 and Lemma 5.13. [ABSTRACT FROM AUTHOR]

As an essential extension of the well known case β ∈ (1/2, 1] to the hyper-dissipative case β ∈ (1, ∞), this paper establishes both well-posedness and ill-posedness (not only norm inflation but also indifferentiability of the solution map) for the mild solutions of the incompressible Navier–Stokes system with dissipation (−Δ)1/2 < β < ∞ through the generalized Carleson measure spaces of initial data that unify many diverse spaces, including the Q space (Q−s = − α)n, the BMO-Sobolev space ( (−Δ)−s/2⁢ BMO)n, the Lip-Sobolev space ( (− Δ)−s/2Lipα)n, and the Besov space (Bs∞, ∞ s)n. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the following problem: { (−Δ)p,βs⁢u = ƒ(x,u) in⁢ Ω, u = 0 in ⁢ ℝN∖Ω , where Ω is a smooth bounded domain of ℝN containing the origin, (−Δ)p,βs⁢u(x) := PV ⁢ ∫ℝN |u⁢(x) − u(y)|p−2(u(x) − u(y))/|x − y|N+p⁢s⁢ dy/|x|β|y|β with 0 ≤ β < N−p⁢s/2, 1 < p < N, s ∈ (0 , 1), and ps < N. The main purpose of this work is to prove the existence of a weak solution under some hypotheses on ƒ. In particular, we will consider two cases: (i) ƒ(x, σ) = ƒ(x); in this case we prove the existence of a weak solution, that is, in a suitable weighted fractional Sobolev space for all ƒ ∈ L1(Ω). In addition, if ƒ ⪈ 0, we show that the problem above has a unique entropy positive solution. (ii) ƒ(x, σ) = λσq + g(x), σ ≥ 0; in this case, according to the values of λ and q, we get the largest class of data g for which the problem above has a positive solution. [ABSTRACT FROM AUTHOR]

In the present paper we consider the first results of modelling the World Ocean biogeochemistry system within the framework of the Earth system model: a global atmosphere-ocean-ice-land-biogeochemistry model. It is based on the INMCM climate model (version INMCM39) coupled with the pelagic ecosystem model BFM. The horizontal resolution was relatively low: 2∘ × 2.5∘ for the 'longitude' and 'latitude' in transformed coordinates with the North Pole moved to land, 33 non-equidistant σ-horizons, 1 hour time step. We have taken into account 54 main rivers worldwide with run–off supplied by the atmosphere submodel. The setup includes nine plankton groups, 60 tracers in total. Some components sink with variable speed. We discuss challenges of coupling the BFM with the σ-coordinate ocean model. The presented results prove that the model output is realistic in comparison with the observed data, the numerical efficiency is high enough, and the coupled model may serve as a basis for further simulations of the long-term climate change. [ABSTRACT FROM AUTHOR]

Total variation (TV) based models have been used widely in multiplicative denoising problem. However, these models are always accompanied by an unsatisfactory effect named staircase due to the property of BV space. In this paper, we present two high-order variational models based on total generalized variation (TGV) for two kinds of multiplicative noises. The proposed models reduce the staircase while preserving the edges. In the meantime we develop an efficient algorithm which is called Prediction-Correction proximal alternative direction method of multipliers (PADMM) to solve our models. Moreover, we show the convergence of our algorithm under certain conditions. Numerical experiments demonstrate that our high-order models outperform the classical TV-based models in PSNR and SSIM values. [ABSTRACT FROM AUTHOR]

In this paper, we improve the famous Reid inequality related to linear operators. Some monotony results for positive operators are also established with a different approach from what is known in the existing literature. Lastly, Reid's (and Halmos-Reid's) inequalities are extended to unbounded operators. [ABSTRACT FROM AUTHOR]

In this paper, several sufficient conditions for oscillation of all solutions of fourth order functional differential equations of neutral type of the form (r (t) (y (t) + p (t) y (t − τ)) ″ ) ″ + q (t) G (y (t − σ)) = 0 $$\begin{array}{} \displaystyle \bigl(r(t)(y(t)+p(t)y(t-\tau))''\bigr)''+q(t)G\bigl(y(t-\sigma)\bigr)=0 \end{array}$$ are studied under the assumption ∫ 0 ∞ t r (t) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}{\rm d} t =\infty \end{array}$$ [ABSTRACT FROM AUTHOR]