5,328 results
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2. Asymptotic distribution of the zeros of a certain family of generalized hypergeometric polynomials.
- Author
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Zhou, Jian-Rong, Li, Heng, and Xu, Yongzhi
- Subjects
JACOBI polynomials ,ASYMPTOTIC distribution ,POLYNOMIALS ,INTEGERS - Abstract
The primary aim of this paper is to investigate the asymptotic distribution of the zeros of certain classes of hypergeometric $ {}_{q+1}F_{q} $ q + 1 F q polynomials. We employ classical analytical techniques, including Watson's lemma and the method of steepest descent, to understand the asymptotic behavior of these polynomials: $$\begin{align*} & _{q+1}F_{q}\left(-n,kn+\alpha,\ldots, kn+\alpha+\frac{q-1}{q};kn+\beta,\ldots,kn+\beta+\frac{q-1}{q};z\right)\\ &\quad (n\rightarrow \infty), \end{align*} $$ q + 1 F q (− n , kn + α , ... , kn + α + q − 1 q ; kn + β , ... , kn + β + q − 1 q ; z) (n → ∞) , where n is a nonnegative integer, q is a positive integer and the constant parameters α and β are constrained by $ \alpha { α < β. By applying the general results established in this paper, we generate numerical evidence and graphical illustrations using Mathematica to show the clustering of zeros on certain curves. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. A modified Tseng's extragradient method for solving variational inequality problems.
- Author
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Peng, Jian-Wen, Qiu, Ying-Ming, and Shehu, Yekini
- Subjects
HILBERT space ,POINT set theory ,ALGORITHMS ,VARIATIONAL inequalities (Mathematics) - Abstract
In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. On initial-boundary value problem for the Burgers equation in nonlinearly degenerating domain.
- Author
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Jenaliyev, M. T. and Yergaliyev, M. G.
- Subjects
ORTHONORMAL basis ,GALERKIN methods - Abstract
In this paper, we study the solvability of one initial-boundary value problem for the Burgers equation with periodic boundary conditions in a nonlinearly degenerating domain. In this paper, we found an orthonormal basis for domains with time-varying boundaries. On this basis, we use the Faedo–Galerkin method to prove theorems about the unique solvability of the problem under consideration. We also present some numerical results in the form of graphs of solutions to the problem under study for various initial data. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. Semidiscrete numerical approximation for dynamic hemivariational inequalities with history-dependent operators.
- Author
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Li, Yujie, Cheng, Xiaoliang, and Xuan, Hailing
- Abstract
In this paper, we are concerned with a class of second-order hemivariational inequalities involving history-dependent operators. For the problem, we first derive a semidiscrete scheme by implicit Euler formula and prove its unique solvability. The existence and uniqueness of a solution to the inequality problem is given by Rothe method. As the core part of the paper, we propose a two-step semidiscrete approximation for the problem, provide its unique solvability and obtain its second-order error estimates. The two-step scheme is more accurate than the standard implicit Euler scheme. Finally, we apply the results to a dynamic frictionless contact problem with long memory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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6. The perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction.
- Author
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Zhang, Qingling and Wan, Youyan
- Subjects
RIEMANN-Hilbert problems ,SHOCK waves ,EULER equations ,CONSERVATION laws (Physics) ,CONSERVATION laws (Mathematics) - Abstract
In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. On first and second order multiobjective programming with interval-valued objective functions.
- Author
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Antczak, Tadeusz
- Subjects
SET-valued maps ,DIFFERENTIABLE functions ,DECISION making - Abstract
The growing use of optimization models to help decision making has created a demand for such tools that allow formulating and solving more models of real-world processes and systems related to human activity in which hypotheses are not verify in a way specific for classical optimization. One of the approaches for real-world extremum problems under uncertainty is interval-valued optimization. In this paper, a twice differentiable vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. In this paper, the first order necessary optimality conditions of Karush-Kuhn-Tucker type are proved for differentiable interval-valued vector optimization problems under the first order constraint qualification. If the interval-valued objective function is assumed to be twice weakly differentiable and constraints functions are assumed to be twice differentiable, then two types of second order necessary optimality conditions under two various constraint qualifications are proved for such smooth interval-valued vector optimization problems. Finally, in order to illustrate the Karush-Kuhn-Tucker type necessary optimality conditions established in the paper, an example of an interval-valued optimization is given. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. Mathematical problems of dynamical interaction of fluids and multiferroic solids.
- Author
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Chkadua, George and Natroshvili, David
- Subjects
DISTRIBUTION (Probability theory) ,SOUND pressure ,ELLIPTIC equations ,TEMPERATURE distribution ,SOUND wave scattering ,MULTIFERROIC materials - Abstract
In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region Ω
+ is immersed in an inviscid fluid occupying an unbounded domain Ω- = ℝ³ \ Ω+ +. In the solid region, we consider the generalized Green--Lindsay's model of the thermo-electro-magnetoelasticity theory. In this case, in the domain Ω+ we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain Ω- . The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundarytransmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ . We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727--9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electromagneto- elasticity theory are studied by the potential method and the theory of pseudodifferential equations. [ABSTRACT FROM AUTHOR]- Published
- 2023
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9. Orbital stability of the sum of N peakons for the mCH-Novikov equation.
- Author
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Wang, Jiajing, Deng, Tongjie, and Zhang, Kelei
- Subjects
CUBIC equations ,EQUATIONS ,ENERGY consumption ,SHALLOW-water equations - Abstract
This paper investigates a generalized Camassa–Holm equation with cubic nonlinearities (alias the mCH-Novikov equation), which is a generalization of some special equations. The mCH-Novikov equation possesses well-known peaked solitary waves that are called peakons. The peakons were proved orbital stable by Chen et al. in [Stability of peaked solitary waves for a class of cubic quasilinear shallow-water equations. Int Math Res Not. 2022;1–33]. We mainly prove the orbital stability of the multi-peakons in the mCH-Novikov equation. In this paper, it is proved that the sum of N fully decoupled peaks is orbitally stable in the energy space by using energy argument, combining the orbital stability of single peakons and local monotonicity of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
10. Scattering for a class of inhomogeneous generalized Hartree equations.
- Author
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Saanouni, Tarek and Peng, Congming
- Subjects
EQUATIONS ,CONSERVATION laws (Mathematics) ,BLOWING up (Algebraic geometry) ,NONLINEAR equations - Abstract
This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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11. BSDEs driven by fractional Brownian motion with time-delayed generators.
- Author
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Aidara, Sadibou and Sylla, Lamine
- Subjects
BROWNIAN motion ,STOCHASTIC differential equations ,STOCHASTIC integrals ,FRACTIONAL differential equations ,MOVING average process ,TIME perspective - Abstract
This paper deals with a class of backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1/2) with time-delayed generators. In this type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time-delay function, for instance, of the moving average type. We establish an existence and uniqueness result of solutions for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. The stochastic integral used throughout the paper is the divergence operator-type integral. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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12. Iterative approximation of common solution to variational inequality problems in Hadamard manifold.
- Author
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Oyewole, Olawale K., Abass, Hammed A., and Shehu, Yekini
- Subjects
PROBLEM solving ,EXTRAPOLATION ,GENERALIZATION ,ALGORITHMS - Abstract
The aim of this paper is to introduce a forward-backward-forward algorithm with inertial extrapolation to solve the problem of finding a common solution to the variational inequality problem (CSVIP) in a Hadamard manifold. Using a self-adaptive step size, we obtain convergence results under some standard conditions. Numerical examples are given to illustrate the theoretical analysis. Our result is a generalization and extension of previously announced results in this direction in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. Semi-analytic PINN methods for singularly perturbed boundary value problems.
- Author
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Gie, Gung-Min, Hong, Youngjoon, and Jung, Chang-Yeol
- Subjects
LINEAR differential equations ,SCIENCE education ,PARTIAL differential equations ,NONLINEAR differential equations ,BOUNDARY value problems - Abstract
In this paper, we propose a novel semi-analytic physics informed neural network (PINN) method for solving singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that shows great promise for finding approximate solutions to partial differential equations. PINNs have demonstrated impressive performance in solving a variety of differential equations, including time-dependent and multi-dimensional equations involving complex domain geometries. However, when it comes to stiff differential equations, neural networks in general struggle to capture the sharp transition of solutions, due to the spectral bias. To address this limitation, we develop a semi-analytic PINN approach, which is enriched by incorporating the so-called corrector functions obtained from boundary layer analysis. Our enriched PINN approach provides accurate predictions of solutions to singular perturbation problems. Our numerical experiments cover a wide range of singularly perturbed linear and nonlinear differential equations. Overall, our approach shows great potential for solving challenging problems in the field of partial differential equations and machine learning. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. Uniform attractors for 3D MHD equations with nonlinear damping.
- Author
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Song, Xiaoya
- Subjects
NONLINEAR equations ,EQUATIONS - Abstract
The aim of this paper is to investigate the uniform attractors for 3D MHD equations with nonlinear damping. Some uniform estimates for strong solutions are established. Furthermore, we prove that 3D damped equations have an $ (\mathbb {V},\mathbb {V}) $ (V , V) -uniform attractor and an $ (\mathbb {V},\mathbf{H}^{2}) $ (V , H 2) -uniform attractor, and the $ (\mathbb {V},\mathbb {V}) $ (V , V) -uniform attractor is actually the $ (\mathbb {V},\mathbf{H}^{2}) $ (V , H 2) -uniform attractor. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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15. Bloch-type theorems for meromorphic harmonic mappings.
- Author
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Liu, Ming-Sheng, Luo, Wen-Jie, and Ponnusamy, Saminathan
- Subjects
HARMONIC maps ,HOLOMORPHIC functions - Abstract
In this paper, we first derive a Landau-type theorem and a Bloch-type theorem for the class of meromorphic harmonic mappings. Then we establish two new versions of Bloch-type theorems for certain K-quasiregular meromorphic harmonic mappings in the unit disk. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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16. Variational approaches to Dirichlet gradient-type systems on the Sierpiński gasket.
- Author
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Ahmadi, Zahra, Lashkaripour, Rahmatollah, Heidarkhani, Shapour, and De Araujo, Anderson L. A.
- Subjects
CRITICAL point theory ,REACTION-diffusion equations ,FLUID flow ,NONLINEAR equations ,GASKETS - Abstract
In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. On anisotropic parabolic equation with nonstandard growth order.
- Author
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Zhan, Huashui
- Subjects
CAUCHY problem ,VISCOSITY solutions ,TRANSPORT equation ,NONLINEAR equations ,MATHEMATICS - Abstract
In this paper, the existence and the uniqueness of an evolutionary anisotropic $ p_i(x) $ p i (x) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the $ L^{\infty } $ L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the $ L^{\infty } $ L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic $ p(x)- $ p (x) − Laplace equation with convection term via $ L^{\infty }- $ L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. The existence of ground state normalized solution for Kirchhoff equation.
- Author
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He, Qihan and Lv, Zongyan
- Subjects
EQUATIONS - Abstract
In this paper, we study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities \[ -\left(a+b\displaystyle\int_{\mathbb{R}^N} |\nabla u|^2\right)\Delta u=\lambda u+|u|^{p-2}u+\mu |u|^{q-2}u \ \hbox{in}\ \mathbb{R}^N, \quad 1\leq N\leq 3 \] − (a + b ∫ R N | ∇u | 2) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass $ \int _{\mathbb {R}^N} u^2=c^2 $ ∫ R N u 2 = c 2 , where $ a{ \gt }0,b{ \gt }0,c{ \gt }0 $ a > 0 , b > 0 , c > 0 , $ \mu { \lt }0 $ μ < 0 , $ 2{ \lt }q\leq 2+\frac {8}{N}{ \lt }p{ \lt }2^{*} $ 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution $ (\tilde {u},\lambda _c)\in S_c \times \mathbb {R} $ (u ~ , λ c) ∈ S c × R , where $ S_c=\{ u\in H^1(\mathbb {R}^N): \int _{\mathbb {R}^N} u^2=c^2 \} $ S c = { u ∈ H 1 (R N) : ∫ R N u 2 = c 2 }. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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19. The shearlet transform and asymptotic behavior of Lizorkin distributions.
- Author
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Ferizi, Astrit and Saneva, Katerina Hadzi-Velkova
- Subjects
TAUBERIAN theorems ,ASYMPTOTIC distribution - Abstract
In this paper, we establish Abelian and Tauberian results that characterize the quasiasymptotic behavior of Lizorkin distributions via the asymptotic behavior of their shearlet transform. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. Second-order optimality conditions for the bilinear optimal control of a degenerate equation.
- Author
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Kenne, Cyrille, Djomegne, Landry, and Zongo, Pascal
- Subjects
EQUATIONS of state ,EQUATIONS - Abstract
The main purpose of this paper is the study of second-order optimality conditions for the bilinear control of a strongly degenerate parabolic equation. The equation is degenerate at the boundary of the spatial domain. The well-posedness of the state equation, as well as weak maximum principles are established. We prove some differentiability properties of the control-to-state operator and the existence of optimal solutions. Finally, we derive first- and second-order optimality conditions for the system. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. Global solutions to the bipolar non-isentropic Euler-Maxwell system in the Besov framework.
- Author
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Zhao, Shiqiang and Zhang, Kaijun
- Subjects
BESOV spaces ,LITTLEWOOD-Paley theory ,SYSTEMS theory ,SOBOLEV spaces ,SEPARATION of variables - Abstract
This paper investigates the bipolar non-isentropic compressible Euler-Maxwell system in $ \mathbb {R}^{3} $ R 3 and $ \mathbb {T}^{3} $ T 3 . For both problems, we establish the global existence of smooth solutions in the general Besov spaces, which covers the usual Sobolev spaces with higher regularity and the critical Besov space, when the initial perturbations around the constant states are small enough. As a byproduct, we obtain the large-time asymptotic behavior of the global solutions near the equilibrium state in the general Besov spaces with relatively lower regularity. The proof is based on the technical Fourier frequency-localization method developed through the Littlewood-Paley theory, but some new development and technique are proposed for treating the strong coupling and nonlinearity for the bipolar non-isentropic case. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Solutions for nonhomogeneous Kohn–Spencer Laplacian on Heisenberg group.
- Author
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Razani, Abdolrahman
- Subjects
LAPLACIAN operator - Abstract
In this paper, we study the existence of at least one bounded weak solution for Kohn–Spencer Laplacian with a weight depending on the solution and convection term of the form \[ -div_{\mathbb{H}^n}(\nu(\xi,u) |D_{\mathbb{H}^n}u|^{p-2}_{\mathbb{H}^n}D_{\mathbb{H}^n}u)=f(\xi,u,D_{\mathbb{H}^n} u) \] − di v H n (ν (ξ , u) | D H n u | H n p − 2 D H n u) = f (ξ , u , D H n u) in a bounded domain $ \Omega \subset \mathbb {H}^n $ Ω ⊂ H n . We show the set of solutions is uniformly bounded by a special Moser's iteration. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Analysis of a delayed spatiotemporal model of HBV infection with logistic growth.
- Author
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Foko, Séverin and Tadmon, Calvin
- Subjects
HEPATITIS B ,VIRUS diseases ,ENDEMIC diseases ,HEPATITIS B virus ,HOPF bifurcations ,BASIC reproduction number - Abstract
In this paper, following previous works of ours, we deal with a mathematical model of hepatitis B virus (HBV) infection. We assume spatial diffusion of free HBV particles, logistic growth for both healthy and infected hepatocytes, and use the standard incidence function for viral infection. Moreover, one time delay is introduced to account for actual virus production. Another time delay is used to account for virus maturation. The existence, uniqueness, positivity and boundedness of solutions are established. Analyzing the model qualitatively and using a Lyapunov functional, we establish the existence of a threshold $ \mathcal {T}_0 $ T 0 such that, if the basic reproduction number $ \mathcal {R}_0 $ R 0 verifies $ \mathcal {R}_0\leq \mathcal {T}_0{ \lt }1 $ R 0 ≤ T 0 < 1 , the infection-free equilibrium is globally asymptotically stable. When $ \mathcal {R}_0 $ R 0 is greater than one, we discuss the local asymptotic stability of the unique endemic equilibrium and the occurrence of a Hopf bifurcation. Also, when $ \mathcal {R}_0 $ R 0 is greater than one, the system is uniformly persistent, which means that the HBV infection is endemic. Finally, we carry out some relevant numerical simulations to clarify and interpret the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. Nonexistence of periodic peakon and peakon for a highly nonlinear shallow-water model.
- Author
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Liu, Yu and Liu, Xingxing
- Subjects
VORTEX motion ,CORIOLIS force ,EQUATIONS - Abstract
In this paper, we investigate the nonexistence of the periodic peakon and peakon for a highly nonlinear shallow-water model, which has been recently derived from the full governing equations for two dimensional flow with the Coriolis effect or with constant vorticity, under a larger scaling than the Camassa-Holm (CH) one. Note that the so obtained model not only has CH-type terms, but also exhibits cubic order nonlinearities. Thus it is interesting to study how the higher power nonlinear terms affect the existence of the (periodic) peakon. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. The backward problem for time-fractional evolution equations.
- Author
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Chorfi, S. E., Maniar, L., and Yamamoto, M.
- Subjects
HILBERT space - Abstract
In this paper, we consider the backward problem for fractional in time evolution equations $ \partial _t^\alpha u(t)= A u(t) $ ∂ t α u (t) = Au (t) with the Caputo derivative of order $ 0 0 < α ≤ 1 , where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag–Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Numerical treatment of time-fractional sub-diffusion equation using p-fractional linear multistep methods.
- Author
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Jangi Bahador, Nayier, Irandoust-Pakchin, Safar, and Abdi-Mazraeh, Somaiyeh
- Subjects
DIFFERENTIAL equations ,EQUATIONS ,MEMORANDUMS - Abstract
In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and $ \mathcal {Z} $ Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are $ \mathcal {O}(\tau ^{p}+h^2) $ O (τ p + h 2) for $ 1 \leq p \leq ~4 $ 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Anisotropic parabolic-elliptic systems with degenerate thermal conductivity.
- Author
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Khelifi, H.
- Subjects
SOBOLEV spaces ,THERMISTORS ,EQUATIONS ,THERMAL conductivity - Abstract
In this paper, we investigate the existence and regularity of a capacity solution for a coupled nonlinear anisotropic parabolic-elliptic system, where the elliptic component of the parabolic equation involves thermal conductivities $ a_{i}(u) $ a i (u) that satisfy $ \lim _{s\rightarrow +\infty }a_{i}(s)=0 $ lim s → + ∞ a i (s) = 0 for all $ i=1,\ldots,N $ i = 1 , ... , N. We work with anisotropic Sobolev spaces and use Schauder's fixed-point theorem to find weak solutions to approximate problems. In addition, we validate the convergence of a subsequence of approximate solutions to a capacity solution. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Decay estimates of the 3D magneto-micropolar system with applications to L3-strong solutions.
- Author
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Ye, Xiuping and Lin, Xueyun
- Subjects
EQUATIONS - Abstract
In this paper, we investigate the well-posedness and large time behavior of solutions to the 3D incompressible magneto-micropolar equations. By virtue of the $ L_p-L_q $ L p − L q estimate obtained through the spectral decomposition of the linearized magneto-micropolar equations, we show the existence and uniqueness of small $ L_3 $ L 3 -strong solutions of the equations with small initial data. Then basing on this result, we derive sharp time decay estimates of the $ L_3 $ L 3 -strong solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Analysis of a reaction–diffusion dengue model with vector bias on a growing domain.
- Author
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Wang, Jinliang, Qu, Hao, and Ji, Desheng
- Subjects
BASIC reproduction number ,DENGUE ,LYAPUNOV functions ,SPATIAL systems ,INFECTIOUS disease transmission ,FENITROTHION - Abstract
In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number $ \Re _0^{\rho } $ ℜ 0 ρ . The results show that (i) if $ \Re _0^{\rho } \gt 1 $ ℜ 0 ρ > 1 , the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when $ \Re _0^{\rho } \lt 1 $ ℜ 0 ρ < 1. Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. Reciprocity gap functional for potentials/sources with small-volume support for two elliptic equations.
- Author
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Granados, Govanni and Harris, Isaac
- Subjects
ELLIPTIC equations ,GREEN'S functions ,OPTICAL tomography ,HELMHOLTZ equation ,RECIPROCITY (Psychology) ,CIRCLE ,INVERSE scattering transform - Abstract
In this paper, we consider inverse shape problems coming from diffuse optical tomography and the Helmholtz equation. In both problems, our goal is to reconstruct small volume interior regions from measured data on the exterior surface of an object. In order to achieve this, we will derive an asymptotic expansion of the reciprocity gap functional associated with each problem. The reciprocity gap functional takes in the measured Cauchy data on the exterior surface of the object. In diffuse optical tomography, we prove that a MUSIC-type algorithm that does not require evaluating the Green's function can be used to recover the unknown subregions. This gives an analytically rigorous and computationally simple method for recovering the small volume regions. For the problem coming from inverse scattering, we recover the subregions of interest via a direct sampling method. The direct sampling method presented here allows us to accurately recover the small volume region from one pair of Cauchy data, requiring less data than many direct sampling methods. We also prove that the direct sampling method is stable with respect to noisy data. Numerical examples will be presented for both cases in two dimensions where the measurement surface is the unit circle. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. A singular Adams' inequality with logarithmic weights and applications.
- Author
-
Zhang, Shiqi
- Subjects
MATHEMATICS ,EQUATIONS - Abstract
In this paper, we consider a singular Adams' inequality with logarithmic weights in the unit ball of $ \mathbb {R}^4 $ R 4 . Our results extend the results of Zhu and Wang [Adams' inequality with logarithmic weights in $ \mathbb {R}^4 $ R 4 . Proc Amer Math Soc. 2021;149(8):3463–3472] on Adams' inequality with logarithmic weights to singular case. Then, we study the existence of solutions for some weighted mean field equations, relying on variational methods and the singular Adams' inequality with logarithmic weights we previously established. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. The critical Choquard equations with a Kirchhoff type perturbation in bounded domains.
- Author
-
Duan, Xueliang, Wu, Xiaofan, Wei, Gongming, and Yang, Haitao
- Subjects
EQUATIONS ,EIGENVALUES - Abstract
This paper deals with the following critical Choquard equation with a Kirchhoff type perturbation in bounded domains, \[ \begin{cases} -(1+b\|u\|^{2})\Delta u=\left(\int_{\Omega}\frac{u^{2}(y)}{|x-y|^{4}}\,{\rm d}y\right)u +\lambda u & {\rm in}\ \Omega,\\ u=0 & {\rm on}\ \partial\Omega, \end{cases} \] { − (1 + b ‖ u ‖ 2) Δu = (∫ Ω u 2 (y) | x − y | 4 d y) u + λu in Ω , u = 0 on ∂Ω , where $ \Omega \subset \mathbb {R}^{N}(N\geq 5) $ Ω ⊂ R N (N ≥ 5) is a smooth bounded domain and $ \|\cdot \| $ ‖ ⋅ ‖ is the standard norm of $ H_{0}^{1}(\Omega) $ H 0 1 (Ω). Under the suitable assumptions on the constant $ b\geq 0 $ b ≥ 0 , we prove the existence of solutions for $ 0 \lt \lambda \leq \lambda _{1} $ 0 < λ ≤ λ 1 , where $ \lambda _{1} \gt 0 $ λ 1 > 0 is the first eigenvalue of $ -\Delta $ − Δ on Ω. Moreover, we prove the multiplicity of solutions for $ \lambda \gt \lambda _{1} $ λ > λ 1 and b>0 in suitable intervals. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Volterra nonautonomous evolution inclusions: topological structure of solution sets and applications.
- Author
-
Yang-Yang Yu and Zhong-Xin Ma
- Subjects
FRECHET spaces - Abstract
This paper concerns with a class of Volterra nonautonomous evolution inclusions with time delay effect defined on a noncompact interval. It involves a family of (possibly unbounded) operators, which generates a noncompact evolution family. In the framework of Fréchet spaces, the topological structure of the solution set for evolution inclusion is considered. Moreover, we obtain geometric features of the corresponding solution map. Our results extend essentially the existing ones which do not involve Volterra integral in nonlinearity. We finally present a concrete example to illustrate the abstract results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. Some qualitative properties for the Kirchhoff total variation flow.
- Author
-
Boudjeriou, Tahir
- Subjects
EQUATIONS - Abstract
The first goal of this paper is to establish a result on the existence and uniqueness of solution to an initial-boundary value problem for parabolic equations of Kirchhoff type involving the 1-Laplace operator. The second goal is to discuss some qualitative properties, such as asymptotic behaviour and the extinction of solutions for the considered problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. Uniqueness and generic regularity of global weak conservative solutions to the Constantin-Lannes equation.
- Author
-
Yang, Li, Zhou, Shouming, and Yang, Hongying
- Subjects
SHALLOW-water equations ,WATER waves ,WATER depth ,EQUATIONS ,CONSERVATIVES - Abstract
This paper is devoted to the uniqueness of global-in-time conservative solutions and generic regularity for the shallow water waves of moderate amplitude equation (Constantin-Lannes equation). The Constantin-Lannes equation possible development of singularities in finite time, and beyond the occurrence of wave breaking, it exists global conservative solutions. In the present paper, we will prove the uniqueness of global-in-time conservative solutions for the Constantin-Lannes equation with general initial data u 0 ∈ H 1 (R) by analyzing the evolution of the quantities u and v = 2 arctan u x along each characteristic. Moreover, we consider that piecewise smooth solutions with only generic singularities are dense in the whole solution set by Thom's transversality Lemma. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Limiting dynamics of stochastic heat equations with memory on thin domains.
- Author
-
Shu, Ji, Li, Hui, Huang, Xin, and Zhang, Jian
- Subjects
ATTRACTORS (Mathematics) ,NOISE - Abstract
This paper is concerned with the limiting behavior of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an (n + 1) -dimensional narrow domain. We also establish the upper-semicontinuity of these attractors when a family of (n + 1) -dimensional thin domains collapses onto an n-dimensional domain. The main difficulty of this paper is the non-compactness of the generated RDS based on the fact that the memory term includes the whole past history of the phenomenon. To solve this, a splitting method is employed to prove the asymptotic compactness. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
37. Dynamical analysis on stochastic two-species models.
- Author
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Wang, Guangbin, Lv, Jingliang, and Zou, Xiaoling
- Subjects
STOCHASTIC analysis ,STOCHASTIC models ,GLOBAL asymptotic stability ,COMPUTER simulation - Abstract
In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. Multiplicity of solutions for a critical nonlinear Schrödinger–Kirchhoff-type equation.
- Author
-
Nie, Jianjun and Li, Quanqing
- Subjects
NONLINEAR equations ,MULTIPLICITY (Mathematics) ,CONTINUOUS functions ,EQUATIONS - Abstract
In this paper, we study the following critical nonlinear Schrödinger–Kirchhoff equation: ($P$) $$\begin{align*} \left \{ \begin{array}{@{}l@{}} \displaystyle -\left(a+b\int_{R^{N}}|\nabla u|^{2}\,{\rm d}x\right)\Delta u + V(x)u =P(x)|u|^{2^*-2}u+\mu|u|^{q-2}u, \ {\rm in}\ \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N) \end{array} \right. \end{align*}$$ { − (a + b ∫ R N | ∇u | 2 d x) Δu + V (x) u = P (x) | u | 2 ∗ − 2 u + μ | u | q − 2 u , in R N , u ∈ H 1 (R N) where $ a, b, \mu \gt 0 $ a , b , μ > 0 , $ N\geq 3 $ N ≥ 3 , $ \max \{2^*-1, 2\} \lt q \lt 2^*=\frac {2N}{N-2} $ max { 2 ∗ − 1 , 2 } < q < 2 ∗ = 2 N N − 2 , $ V(x) \gt 0 $ V (x) > 0 and $ P(x)\geq 0 $ P (x) ≥ 0 are two continuous functions. By using the variational method and truncation technique, we prove the multiplicity of solutions for Equation (P). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Unique local weak solutions of the non-resistive MHD equations in homogeneous Besov space.
- Author
-
Yuan, Baoquan and Ke, Xueli
- Subjects
BESOV spaces ,HOMOGENEOUS spaces ,EQUATIONS - Abstract
In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution $ (u,b) $ (u , b) of the non-resistive MHD equations for the initial data $ u_0\in \dot B^{\frac {d}{p}-1}_{p,1}(\mathbb {R}^{d}) $ u 0 ∈ B ˙ p , 1 d p − 1 (R d) and $ b_0\in \dot B^{\frac {d}{p}}_{p,1}(\mathbb {R}^{d}) $ b 0 ∈ B ˙ p , 1 d p (R d) with $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d. Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ from $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d yet. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Inertial subgradient extragradient method for solving pseudomonotone variational inequality problems in Banach spaces.
- Author
-
Peng, Zai-Yun, Peng, Zhi-Ying, Cai, Gang, and Li, Gao-Xi
- Subjects
BANACH spaces ,SUBGRADIENT methods ,VARIATIONAL inequalities (Mathematics) - Abstract
In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Blow-up of solutions to a scalar conservation law with nonlocal source arising in radiative gas.
- Author
-
Chen, Jian and Yang, Shaojie
- Subjects
CONSERVATION laws (Physics) ,LIFE spans ,GASES - Abstract
In this paper we consider a scalar conservation law with nonlocal source arising in radiative gas. We give a sufficient condition to assure that the solution blows up in a finite time. Moreover, the estimates of life span and blow-up rate are derived. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. On continuation criteria for the double-diffusive convection system in Vishik spaces.
- Author
-
Wu, Fan
- Subjects
NAVIER-Stokes equations - Abstract
This paper proves the continuation criteria of the strong solution for the 3D double-diffusive convection system involving the deformation tensor in Vishik spaces. As a bi-product, our theorem also improve some well-known results on conditional regularity for the particular case of classical Navier–Stokes equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Sample average approximation method for a class of stochastic vector variational inequalities.
- Author
-
Dong, Dan-dan, Liu, Jian-xun, and Tang, Guo-ji
- Subjects
CONSTRAINED optimization ,MEAN value theorems - Abstract
In the present paper, the expected-value (EV) reformulation of a class of stochastic vector variational inequalities (SVVI) is investigated. By using the regularized gap function, the EV reformulation of SVVI is transformed into a constrained optimization problem. Then a sample average approximation (SAA) method is proposed for solving the constrained optimization problem. Under suitable assumptions, the limiting behaviors of the optimal values and optimal solutions of the approximation problem are investigated. Finally, the rates of convergence in the different senses of optimal solutions for sample average approximation problem are discussed under the error bound condition. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Mean square stability of the split-step theta method for non-linear time-changed stochastic differential equations.
- Author
-
Wu, Dongxuan, Li, Zhi, Xu, Liping, and Peng, Chuanhui
- Subjects
DIFFUSION coefficients - Abstract
This paper investigates the split-step theta (SST) method to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the SST method is proved, and the SST method attains the classical 1 of convergence. In addition, the mean square stability of the time-changed stochastic differential equations is investigated. Two examples are presented to show the consistency of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Existence and feedback control for a class of nonlinear evolutionary equations.
- Author
-
Yin, Bin and Zeng, Biao
- Subjects
EVOLUTION equations ,NONLINEAR equations ,SURJECTIONS ,MONOTONE operators - Abstract
In the paper we provide systematic approaches to study existence and feedback control for a new evolutionary equation involving pseudomonotone operators. We first establish several existence results for the evolutionary equation by exploiting the Rothe method and using a surjectivity result for multivalued pseudomonotone operators. Then we show the existence of feasible pairs for the feedback control problem by assuming some sufficient conditions. Moreover, the existence of solutions to an evolutionary hemivariational inequality is presented. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms.
- Author
-
Xie, Changping and Fang, Shaomei
- Subjects
RELATIVISTIC quantum mechanics ,QUANTUM field theory ,NONLINEAR wave equations ,KLEIN-Gordon equation ,BOUNDARY value problems ,INITIAL value problems ,SINE-Gordon equation ,BLOWING up (Algebraic geometry) - Abstract
In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Global error estimates in zero-relaxation limit of Euler–Poisson system for ion dynamics.
- Author
-
Sheng, Han and Liu, Cunming
- Subjects
SYSTEM dynamics ,ELECTRIC potential ,LEAD time (Supply chain management) ,ADVECTION-diffusion equations - Abstract
The zero-relaxation limit of Euler–Poisson systems for ion dynamics in a slow time scale leads to drift-diffusion equations. This fact was justified in previous works. This paper concerns global error estimates between solutions of Euler–Poisson systems and that of drift-diffusion equations. In the proof, we use Sobolev inequalities and employ a multiplier related to the electric potential. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Approximation by Haar polynomials in variable exponent grand Lebesgue spaces.
- Author
-
Volosivets, S. S.
- Subjects
POLYNOMIAL approximation ,EXPONENTS - Abstract
In this paper, we give direct theorems on approximation by Haar and Walsh polynomials in variable exponent grand Lebesgue space. Also the degree of approximation by Borel, Abel-Poisson, Riesz-Zygmund and Euler linear means of Haar-Fourier series are treated in the above-cited space. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. A new result on averaging principle for Caputo-type fractional delay stochastic differential equations with Brownian motion.
- Author
-
Zou, Jing and Luo, Danfeng
- Subjects
STOCHASTIC differential equations ,DELAY differential equations ,EQUATIONS of motion ,BROWNIAN motion ,JENSEN'S inequality ,LAPLACE transformation - Abstract
In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Scattering of plane compressional waves by cylindrical inclusion in a poroelastic medium.
- Author
-
Lee, Doo-Sung
- Subjects
LONGITUDINAL waves ,PLANE wavefronts ,STRAINS & stresses (Mechanics) ,STRESS concentration ,INFINITE series (Mathematics) ,POROELASTICITY - Abstract
This paper deals with the three-dimensional analysis of stress distribution in a long circular cylinder imbedded in a poroelatic medium. The surface of the cylinder is subjected to a known pressure. The equations of the classical theory of elasticity are solved in terms of an infinite series. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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