Search

In this paper, we study the following fractional Choquard equation with weight (−Δ)su=1|x|N−α∗h(x)|u|ph(x)|u|p−2uinℝN,$$ {\left(-\Delta \right)}&#x0005E;su&#x0003D;\left(\frac{1}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{N-\alpha }}\ast h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;p\right)h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\kern0.5em \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}&#x0005E;N, $$where 02s,p>2,α>0$$ 02s,p>2,\alpha >0 $$ and h$$ h $$ is a positive weight function satisfying h(x)≥C|x|a$$ h(x)\ge C{\left&#x0007C;x\right&#x0007C;}&#x0005E;a $$ at infinity, for some a≥0$$ a\ge 0 $$. We establish, in this paper, a Liouville type theorem saying that if maxN−4s−2a,0<α

This special issue of the Journal Mathematical Methods in the Applied Sciences on the topic of Modelling, Analysis, and Applications is devoted to the development of mathematical modelling, analysis, and applications, from theoretical and numerical perspectives, involving different applied sciences and engineering. It collects invited contributions, from selected quality full papers submitted to the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMA2SC2022). The conference took place in Porto, Portugal in 27–29 June 2022 (https://www.isep.ipp.pt/Page/ViewPage/ICMASC). [ABSTRACT FROM AUTHOR]

This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics. [ABSTRACT FROM AUTHOR]

The sequence spaces ℓp(Cn)$$ {\ell}_p\left({C}&amp;#x0005E;n\right) $$, c0(Cn)$$ {c}_0\left({C}&amp;#x0005E;n\right) $$ and c(Cn)$$ c\left({C}&amp;#x0005E;n\right) $$ are introduced by Roopaei and Başar [8] as the domain of the Cesàro matrix Cn$$ {C}&amp;#x0005E;n $$ of order n$$ n $$ in the spaces ℓp$$ {\ell}_p $$, c0$$ {c}_0 $$ and c$$ c $$ of absolutely p$$ p $$‐summable, null and convergent sequences, respectively, where 0

This paper is concerned with the time‐decay rates of the strong solutions of the three‐dimensional nonisentropic compressible magnetohydrodynamic (MHD) system. First, motivated by Pu and Guo's result [Z. Angew. Math. Phys. 64 (2013) 519–538], we establish the existence result of a unique local‐in‐time strong solution for the MHD system. Then, we derive a priori estimates and use the continuity argument to obtain the global‐in‐time solution, where the initial perturbation is small in H2$$ {H}^2 $$‐norm. Finally, based on Fourier theory and the idea of cancelation of a low‐medium frequent part as in [Sci. China Math. 65 (2022) 1199–1228], we get the optimal time‐decay rates (including highest‐order derivatives) of strong solutions for nonisentropic MHD fluids when the boundedness of L1$$ {L}^1 $$‐norm of the initial perturbation is required. Our result is the first one concerning with the optimal decay estimates of the highest‐order derivatives of the nonisentropic MHD system. [ABSTRACT FROM AUTHOR]

Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third‐order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high‐order nonlinear conservative or non‐conservative oscillators. The current work sheds light on the approximate proposition to the damped third‐order nonlinear differential equations. The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the non‐perturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature. [ABSTRACT FROM AUTHOR]

In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order α(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and λ(x) of the Morrey space. Assumptions on the exponents were different depending on whether α(x)p(x)−n+λ(x)p(x) takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range 0⩽α(x)p(x)−n+λ(x)p(x)⩽1. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider equations involving a uniformly elliptic nonlocal operator Aβu(x)=CN,βP.V.∫ℝNa(x−y)(u(x)−u(y))|x−y|N+βdy,$$ {A}_{\beta }u(x)={C}_{N,\beta}\mathrm{P}.\mathrm{V}.\underset{{\mathbb{R}}^N}{\int}\frac{a\left(x-y\right)\left(u(x)-u(y)\right)}{{\left|x-y\right|}^{N+\beta }} dy, $$where the function a:ℝN↦ℝ$$ a:{\mathbb{R}}^N\mapsto \mathbb{R} $$ is uniformly bounded and radial decreasing. We establish some maximum principles for Aβ$$ {A}_{\beta } $$ in bounded and unbounded domains. Since there is no decay condition in the unbounded domain, we make use of an approximate method to estimate the singular integral to get the maximum principle. As applications of these principles, by carrying out the method of moving planes, we give the monotonicity of solutions to the semilinear equation in the coercive epigraph, which extends the result of Dipierro‐Soave‐Valdinoci [Math. Ann.2017, 369(3‐4): 1283–1326]. Moreover, we obtain the radial symmetry and monotonicity of solutions to the generalized Schrödinger equation in a weaker condition, which is the improvement of the result of Tang [Math. Methods Appl. Sci. 2017, 40(7): 2596–2609]. In addition, the maximum principle also plays an important role in acquiring monotonicity of solutions and a Liouville theorem. [ABSTRACT FROM AUTHOR]

We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. More precisely, we are referring respectively to the papers "Boundedness in a nonlinear attraction‐repulsion Keller–Segel system with production and consumption," by S. Frassu, C. van der Mee and G. Viglialoro [J. Math. Anal. Appl.504(2):125428, 2021] and "Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent," by S. Frassu and G. Viglialoro [Nonlinear Anal.213:112505, 2021]. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements, and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study. [ABSTRACT FROM AUTHOR]

In this paper, we consider the pseudo‐relativistic Hartree equation i∂tψ=−△+m2ψ−1|x|∗|ψ|2ψonℝ3and study travelling solitary waves of the form ψ(t, x) = eitμφ(x − v t) , where v∈ℝ3 denotes travelling velocity. Fröhlich, Jonsson and Lenzmann in [Comm. Math. Phys. 2007, 274:1‐30] proved that for |v|<1 there exists a critical constant Nc(v), such that the travelling waves exist if and only if 0 < N < Nc(v), where N denotes particle number. In this paper, we consider v=(β,0,0) with 0 < β < 1, and let Nc(β)=Nc(v)|v=(β,0,0). We find that Nc(β) is Lipschitz continuity with respect to β. Based on this fact, we then prove that the boosted ground states φβ with ‖φβ‖L22=(1−β)Nc(β) satisfy limβ→0+‖φβ‖H1/2→+∞. The explicit blow‐up profile and rate will be computed. [ABSTRACT FROM AUTHOR]

Historically, the minimality of surfaces is extremely important in mathematics, and the study of minimal surfaces is a central problem, which has been widely concerned by mathematicians. Meanwhile, the study of the shape and the properties of the production models is a great interest subject in economic analysis. The aim of this paper is to study the minimality of quasi‐sum production functions as graphs in a Euclidean space. We obtain minimal characterizations of quasi‐sum production functions with two or three factors as hypersurfaces in Euclidean spaces. As a result, our results also give a classification of minimal quasi‐sum hypersurfaces in dimensions two and three. [ABSTRACT FROM AUTHOR]

This paper deals with a Cauchy problem of the inhomogeneous parabolic equation ut=Δu+〈x〉γup+tσw(x) in ℝN×(0,T), where constants γ>0,p>1, and σ>−1. The Japanese brackets 〈x〉γ:=1+|x|2γ; w(≥,≢0) and the initial data are continuous functions in ℝN. We determine the Fujita exponent for global and non‐global solutions of the problem, depending strictly on N,γ and σ, which complete the ones for the nonnegative solutions in J. Math. Anal. Appl. 251 (2000) 624–648 for N=1,2. It is so interesting that the inhomogeneous term leads to the discontinuity of this critical exponent with respect to σ at zero. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcations of the following difference equations xn+1=axn+byne−xn,yn+1=cyn+dxne−yn,xn+1=ayn+bxne−yn,yn+1=cxn+dyne−xn,where a,b,c, and d are positive constants and the initial conditions x0 and y0 are positive numbers. Psarros, Papaschinopoulos, and Schinas (Math. Methods Appl. Sci., 2016, 39: 5216–5222) presented the semistability of the fixed point (0,0) when one eigenvalue is equal to 1 and the other eigenvalue has absolute value less than 1. In this paper, we consider another case: one eigenvalue is equal to −1. With the aid of the center manifold reduction theorem, we rigorously show that these two systems undergo flip bifurcations or generalized flip bifurcations. Moreover, the stability of the fixed point (0,0) and the existence of period‐two cycles are also given. [ABSTRACT FROM AUTHOR]

This paper deals with the parabolic–elliptic Keller–Segel system with density‐dependent sublinear sensitivity and logistic source, ut=Δu−χ∇·(u(u+1)α−1∇v)+λu−μuκ,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,where Ω:=BR(0)⊂Rn(n≥3) is a ball with some R>0 and χ>0, 0<α<1, λ∈R, μ>0, and κ>1. In the case α=1, Winkler (Z. Angew. Math. Phys.; 2018; 69: 40) discovered the condition for κ such that solutions blow up in finite time. The purpose of the present paper is to find conditions for α and κ such that there exist solutions that blow up in finite time in the case of weak‐chemotactic sensitivity, that is, in the case 0<α<1. [ABSTRACT FROM AUTHOR]

Distance distributions are a key building block in many subfields in mathematics, science and engineering. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random points each uniformly randomly distributed in respective arbitrary polygon regions. The proposed framework is based on measure theory and uses polar decomposition for simplifying and calculating the integrals to obtain closed form results. We validate our proposed framework by comparison with simulations and published closed form results in the literature for simple cases. We illustrate the versatility and advantage of the proposed framework by deriving closed form results for a case not yet reported in the literature. Finally, we also develop a Mathematica implementation of the proposed framework which allows a user to define any two arbitrary (concave or convex) polygons, with or without holes, which may be disjoint or overlap or coincide and determine the distance distribution numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and asymptotic properties of solutions to the following fractional Schrödinger equation: (−Δ)σu=λu+|u|q−2u+μIα∗|u|p|u|p−2uinℝNunder the normalized constraint ∫ℝNu2=a2,where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N), q ∈ (2 + 4σ/N, 2N/N − 2σ], p ∈ [2, 1 + 2σ + α/N), a > 0, μ > 0, Iα(x)=|x|α−N and λ∈ℝ appears as a Lagrange multiplier. In the Sobolev subcritical case q ∈ (2 + 4σ/N, 2N/N − 2σ), we show that the problem admits at least two solutions under suitable assumptions on α, a, and μ. In the Sobolev critical case q=2N/N−2σ, we prove that the problem possesses at least one ground state solution. Furthermore, we give some stability and asymptotic properties of the solutions. We mainly extend the results of S. Bhattarai published in 2017 on J. Differ. Equ. and B. H. Feng et al published in 2019 on J. Math. Phys. concerning the above problem from L2‐subcritical and L2‐critical setting to L2‐supercritical setting with respect to q, involving Sobolev critical case especially. [ABSTRACT FROM AUTHOR]

In this paper, we prove the upper semicontinuity of the strong global attractors 𝒜θ on the dissipative index θ in the topology of the stronger space for the Kirchhoff wave model with structural nonlinear damping: utt−ϕ(‖∇u‖2)△u+σ(‖∇u‖2)(−Δ)θut+f(u)=g(x), with θ ∈ [1/2, 1). It is continuation of the research in recent literatures1,2 where the upper semicontinuity of the strong attractor 𝒜θ on θ in the topology of natural energy space is obtained. This result improves and deepens those in recent literatures (Li and Yang in J Differ Equat. 2020; 268: 7741‐7743; Li, Yang and Ding in Appl. Math. Lett. 2020; 104:106258). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define a q‐analogue of a parametric generalization of Baskakov operators introduced by Aral and Erbay (Math. Commun. 24(1) (2019), 119‐131). We establish some local direct results for these operators by means of the modulus of continuity and the Peetre's K‐functional. Weighted approximation properties are also established. Next, we construct a bivariate case of the above q‐operators and determine the rate of convergence in terms of the moduli of continuity. A Voronovskaja‐type theorem is derived too. Further, we study the rate of approximation of Bögel continuous and Bögel differentiable functions by the associated generalized Boolean sum (GBS) operators with the aid of mixed modulus of smoothness. We illustrate the rate of approximation of the q‐operators, their bivariate, and the GBS cases by means of graphics and tables and show that the GBS operators provide better rate of convergence than the bivariate operators. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of the form −Δλu=v−Δλv=upinℝN,where p∈ℝ, and Δλ is a sub‐elliptic operator defined by Δλ=∑i=1N∂xiλi2∂xi.Under some general hypotheses of the functions λi, i=1,2,...,N, we first prove that the system has no positive super‐solution when p ≤ 1. In the case p > 1, we establish a Liouville type theorem for the class of stable positive solutions. This result is an extension of some result in Hajlaoui et al. (Pacific J Math. 2014;270(1):79–93) for the case of Laplace operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider 2‐D nonlinear wave equations with small initial data of noncompact support when the quadratic terms of the nonlinearity satisfy the null conditions but the cubic ones do not. Due to the blow‐up result for the corresponding problem with the initial data of compact support, which was first obtained by S. Alinhac (The null condition for quasilinear wave equations in two space dimensions II. Amer. J. Math, 123 [2001], no. 6, 1071‐1101), this means that it is impossible to prove the global solvability in time in general. Therefore, in this case, what we can expect is to obtain the same lower bound of the lifespan as in Alinhac's work. That is, we also show the almost global existence, and the lower bound of the lifespan is ec/ε2 for sufficiently small ε>0 and some constant c>0. To achieve this goal, the methods we adopt are the "ghost weight" technique and the modified weighted L∞‐ L∞ estimates of the solutions to the linear wave equations. In addition, an alternative proof of the global existence is given for a similar case in three dimensions, using only the L∞‐ L2 estimates of the "good derivatives." [ABSTRACT FROM AUTHOR]

Horváth and Kiss (Proc. Amer. Math. Soc., 2005) proved the upper bound estimate λnλm≤n2m2(n>m≥1) for Dirichlet eigenvalue ratios of the Schrödinger problem −y′′ + q(x)y  =  λy with nonnegative and single‐well potential q. In this paper, we prove that if q(x) is a nonpositive, continuous, and single‐barrier potential, then λnλm≥n2m2 for λn  >  λm≥ − 2q∗, where q∗=min{q(0),q(1)}. In particular, if q(x) satisfies the additional condition |q∗|≤π23, then λ1  >  0 and λnλm≥n2m2 for n  >  m  ≥  1. For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function. [ABSTRACT FROM AUTHOR]

Recently, we came across the paper entitled “Quantification of network structural dissimilarities” published by Nature Communications. In this paper, Schieber et al claimed to propose a novel technique for measuring the structural similarity of networks. The authors of this commentary are familiar with this special area of network analysis. After reading the paper, we doubt that the graph similarity (distance) measure due to Schieber et al is entirely novel. Instead, parts thereof are very similar to contributions we have authored. Unfortunately, they have not been cited nor acknowledged in Schieber et al. [ABSTRACT FROM AUTHOR]

The paper deals with some existence results for an elliptic equation of Kirchhoff‐type with changing sign data and a logarithmic nonlinearity by direct variational method, Galerkin approach, and sub‐super solutions method. Our results are natural extension of Boulaaras' work in (Math Methods Appl Sci; 41(13):5203‐5210). [ABSTRACT FROM AUTHOR]

Suggested by the max‐product sampling operators based on sinc‐Fejér kernels, in this paper, we introduce truncated max‐product Kantorovich operators based on generalized type kernels depending on two functions ϕ and ψ satisfying a set of suitable conditions. Pointwise convergence, quantitative uniform convergence in terms of the moduli of continuity, and quantitative Lp‐approximation results in terms of a K‐functional are obtained. Previous results in sampling and neural network approximation are recaptured, and new results for many concrete examples are obtained. [ABSTRACT FROM AUTHOR]

This paper establishes a regularity criterion for the strong solution to the Leray‐α‐MHD in Campanato‐Morrey space. We show that if the magnetic field b satisfies the following condition: ∫0T‖∇b(τ,·)‖M˙qp2p2p−3+‖D2b(τ,·)‖M˙qp2p2p−3dτ<∞, the strong solution (v,b) to the 3‐D Leray‐α MHD equation can be extended beyond T > 0. We also prove that if the magnetic field b satisfies ∫0T‖∇b(τ,·)‖M˙qppp−1dτ<∞, the strong solution (v,b) to the 2‐D Leray‐α MHD equation can be extended beyond T > 0, which extends the results in previous studies. [ABSTRACT FROM AUTHOR]

This paper shows that the following nonlinear boundary value problem: ( 1 + δ y ) y ′ ′ + 2 x y ′ = 0 , 0 < x < ∞ , y ( 0 ) = 0 , y ( ∞ ) = 1 , where δ≥ − 1 and y is defined as a modified error function, can be reduced to the boundary value problems for the generalized Emden‐Fowler equation and Abel differential equation of the second kind in the canonical form, which do not usually have closed‐form solutions. The existence and uniqueness of the solution are established by considering another contraction mapping. Highly accurate approximate solutions of this problem are also provided. [ABSTRACT FROM AUTHOR]

Capillary dynamics has been and is yet an important field of research, because of its very relevant role played as the core mechanism at the base of many applications. In this context, we are particularly interested in the liquid penetration inspection technique. Because of the obviously needed level of reliability involved with such a non-destructive test, this paper is devoted to study how the presence of an entrapped gas in a close-end capillary may affect the inspection outcome. This study is carried out through a one-dimensional ordinary differential model that despite its simplicity is able to point out quite well the capillary dynamics under the effect of an entrapped gas. The paper is divided into two main parts; the first starts from an introductory historical review of capillary flows modelling, goes on presenting the one-dimensional second order ordinary differential model, taking into account the presence of an entrapped gas and therefore ends by showing some numerical simulation results. The second part is devoted to the analytical study of the model by separating the initial transitory behaviour from the stationary one. Besides, these solutions are compared with the numerical ones, and finally, an expression is deduced for the threshold radius switching from a fully damped transitory to an oscillatory one. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper deals with the following chemotaxis system: [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed Hassell-Varley-type predator-prey model with harvesting on prey. By means of Mawhin's continuation theorem of coincidence degree theory, some new sufficient conditions are obtained for the existence of at least two positive almost periodic solutions for the aforementioned model. To the best of the author's knowledge, so far, the result of this paper is completely new. An example is employed to illustrate the result of this paper. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and non-existence of positive solutions for p-Kirchhoff type problems with a parameter on [ABSTRACT FROM AUTHOR]

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes-Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our ϵ-problem converges in L2 to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma-convergence of convolution sequences. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of determining the heat source, which depends only on spatial variable in one-dimensional heat equation in a bounded domain where data is given at some fixed time. A conditional stability result is given, and a quasi-boundary value regularization method is also provided. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward-backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data containing vacuum is investigated. If the initial data are of small energy but possibly large oscillations, we obtain the global well-posedness of classical solutions in the case of initially nonvacuum far fields. In particular, the smallness of the energy only depends on the [ABSTRACT FROM AUTHOR]

In this paper, a modified characteristics finite element method for the time dependent Navier-Stokes/Darcy problem with the Beavers-Joseph-Saffman interface condition is presented. In this method, the Navier-Stokes/Darcy equation is decoupled into two equations, one is the Navier-Stokes equation, the other is the Darcy equation, and the Navier-Stokes equation is solved by the modified characteristics finite element method. The theory analysis shows that this method has a good convergence property. In order to show the effect of our method, some numerical results was presented. The numerical results show that this method is highly efficient. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]