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In this paper, we study optimality conditions as well as sensitivity analysis of parametric convex minimax programming problems. The main tools used here are the formulas for computing the subdifferentials of maximum functions under some suitable regularity conditions. More precisely, we use these tools to study optimality conditions for the problem under consideration. These results are then applied to obtain optimality conditions for multiobjective optimization problems, which constitute the first part of the paper. In the second part, we derive formulas for computing the subdifferential in the sense of convex analysis and singular subdifferential of the optimal value function for the problem in question. [ABSTRACT FROM AUTHOR]

This paper focuses on approximating stochastic delay differential equations with delayed impulses using Euler-Maruyama-type approximations. One key difference from previous literature is that the impulsive perturbations considered in this paper are past-dependent. Additionally, both the time delays in the stochastic delay differential equations and in the impulsive functions are functions of time. We establish the mean square convergence of the Euler-Maruyama approximations under a local Lipschitz condition and a linear growth condition. Furthermore, we determine the order of convergence under a global Lipschitz condition and provide an illustrative example. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

In this paper, we consider the estimation of the weighted star discrepancy. A better weighted probabilistic star discrepancy bound than the use of plain Monte Carlo point sets is provided in terms of convergence order, i.e. the convergence order of the weighted probabilistic bound is improved from $ O(N^{-\frac {1}{2}}) $ O (N − 1 2 ) to $ O(N^{-\frac {1}{2}-\frac {1}{2d}}\cdot \ln ^{{1}/{2}}{N}) $ O (N − 1 2 − 1 2 d ⋅ ln 1 / 2 ⁡ N). [ABSTRACT FROM AUTHOR]

In this paper, we prove a stronger type of Heisenberg–Pauli–Weyl uncertainty principle for ambiguity function. An example is used to demonstrate the obtained result, and related uncertainty principles for short-time Fourier transform and Wigner distribution are given, respectively. In addition, we introduce the so-called $ \mathbf {K} $ K -ambiguity function, and obtain the related Heisenberg–Pauli–Weyl uncertainty principle. [ABSTRACT FROM AUTHOR]

This paper considers three thermoelastic laminated beam systems with Kelvin-Voigt damping and heat flow described by Fourier's law. First, we show that the system is globally well-posed using the linear operator semigroup approach. The main results are exponential and polynomial stability when the systems are fully and partially damped. We establish the exponential stability of a fully damped system and show the lack of exponential stability for two partially damped systems. Furthermore, we demonstrate its polynomial decay with rate $ t^{-\frac {1}{2}} $ t − 1 2 using frequency domain approach due to Borichev and Tomilov. [ABSTRACT FROM AUTHOR]

In this paper, we study the homogenization of quasilinear elliptic problems with a jump on the interface between the two components proportional to the flux depending on a real parameter γ. The data is presumed to be an $ L^1 $ L 1 function and consequently, we consider the notion of renormalized solutions. In the homogenization process, we employ the periodic unfolding method. The homogenized problem was afterwards identified to be a quasilinear elliptic problem with $ L^1 $ L 1 data for the case that the parameter γ is between $ -1 $ − 1 and 1 and for the case where γ is less than $ -1 $ − 1. [ABSTRACT FROM AUTHOR]

The typical treatment for tumor involves a combination of radiation therapy and chemotherapy. In this study, we explore a mathematical model based on proliferation and diffusion, considering the impacts of both radiotherapeutic and chemotherapeutic treatments. The mathematical model is formulated as a system of partial differential equations, accompanied by initial, boundary, and free boundary conditions. Existence and uniqueness of this mathematical problem are obtained by developing an iteration algorithm in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with a singular biharmonic equations with p-Laplacian and Hardy potential. Using Mountain Pass Theorem and Fountain Theorem with Cerami condition, we obtain the existence and multiplicity of sign-changing high-energy solutions under some suitable conditions on the nonlinear term $ f(x,u) $ f (x , u). In addition, by applying an abstract critical point theorem of Kajikiya in [A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370], a sequence of sign-changing solutions converging to zero for the biharmonic equations with sublinear nonlinearities is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: \[ \left\{ \begin{aligned} & \partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+};\cr & u(0,x)=f(x),\ x\in\mathbb R^{n}. \end{aligned}\right. \] { ∂ t β u (t , x) = − ν (− Δ) α / 2 u (t , x) , (t , x) ∈ R + 1 + n ; u (0 , x) = f (x) , x ∈ R n. Due to the time-fractional derivative, the semigroup property is invalid for the solutions $ u(\cdot,\cdot) $ u (⋅ , ⋅) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of $ u(\cdot,\cdot) $ u (⋅ , ⋅) , which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time–space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations. [ABSTRACT FROM AUTHOR]

In this paper, the problem of acoustic wave scattering by obstacles in the two-dimensional inhomogeneous medium is considered. When the refractive index is a binary function, a hybrid method for reconstructing multiple impenetrable obstacles is proposed. The obstacle number is predicted based on the measured far-field data by using the neural network method. And then, it is used as the prior information of Bayesian method, the shape and position parameters of multiple obstacles are reconstructed using Markov chain Monte Carlo algorithm. The results of numerical experiments show that this method can effectively reconstruct the number, shape and position of obstacles in the inhomogeneous medium. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff type problem with critical growth: \[ \left\{ \begin{array}{@{}l} -\left(a+b\displaystyle\int_{\mathbb R^N} |\nabla u|^2\,\textrm{d}x \right)\Delta u=\lambda k(x)|u|^{q-2}u +\mu|u|^{2^*-2}u,\\ x \in\mathbb R^N,\quad u\in D^{1,2}(\mathbb R^N),\\ \end{array} \right. \] { − (a + b ∫ R N | ∇u | 2 d x) Δu = λk (x) | u | q − 2 u + μ | u | 2 ∗ − 2 u , x ∈ R N , u ∈ D 1 , 2 (R N) , where $ N \geq 3 $ N ≥ 3 , $ a\geq 0 $ a ≥ 0 , b>0, 1 0 , $ 2^*=\frac {2N}{N-2} $ 2 ∗ = 2 N N − 2 is the critical Sobolev exponent and $ k(x) \in L^{2^{\ast }/(2^{\ast }-q)}(\mathbb {R}^N) $ k (x) ∈ L 2 ∗ / (2 ∗ − q) (R N) is a positive function. Using variational methods and the concentration-compactness principle, we demonstrate the existence and multiplicity of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial–boundary value problem (IBVP) of a coupled Cahn–Hilliard-phase-field-crystal (CH-PFC) system subject to homogeneous Neumann boundary conditions in one spatial dimension. This CH-PFC model couples the composition field of a diffusing species with the crystallographic and can be used to model the diffusion-induced grain boundary migration in crystalline materials. Under suitable assumptions on the coefficients and initial data, we prove that the IBVP possesses a global weak solution. Our existence proof, which contributes to the verification of the model, is only valid in one space dimension. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of solutions to a stationary neutron transport equation involving elastic and inelastic collision operators in $ L^p $ L p - espaces $ (1\leq p \lt \infty) $ (1 ≤ p < ∞). For $ 1 \lt p \lt \infty $ 1 < p < ∞ , we use the Krasnosel'slii fixed point theorem and the compactness which involved by the averaging result for neutron transport equation. For p = 1, our approach is different, it uses the measure of weak noncompactness of De Blasi, the concepts of Dunford–Pettis operators together with a recent version of Krasnosel'skii's fixed point theorem involving ws -compact and ww -compact operators and the weak compactness. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of reaction–diffusion equations arising from the competition of two competing species for a single limited nutrient with flocculation in an unstirred chemostat. By the conservation principle, we reduce the dimension of the system by eliminating the equation for the nutrient. Then the global structure of the reduced system is studied by the bifurcation theory in its feasible domain. Finally, we use numerical simulation to verify and supplement our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are paid attention to the following logarithmic Choquard equation $$\begin{align*} -\Delta u+u= \left[ \mathcal{I}_2*F(u) \right]f(u), \quad x\in \mathbb{R}^{2}, \end{align*}$$ − Δ u + u = [ I 2 ∗ F (u) ] f (u) , x ∈ R 2 , where $ \mathcal {I}_2=\frac {1}{2\pi } \ln \frac {1}{|x|} $ I 2 = 1 2 π ln ⁡ 1 | x | is the Newtonian kernel in dimension 2, $ F(u)=\int _0^uf(t) \,\mathrm {d}t $ F (u) = ∫ 0 u f (t) d t and $ f\in \mathcal {C}^1(\mathbb {R}, \mathbb {R}) $ f ∈ C 1 (R , R) is critical exponential growth with respect to Trudinger–Moser. Based on a variational framework proposed in Liu et al. (J Differ Equ 2022;328), we obtain a positive finite energy solution of the above logarithmic Choquard equation via the variational methods. Furthermore, we use another approach to investigate the planar logarithmic Choquard equations involving critical exponential growth. [ABSTRACT FROM AUTHOR]

The reaction-diffusion equations have been studied in various aspects of nature, such as heat transfer and biological dynamic modelling. In this paper, we study a nonlocal reaction-diffusion model with time delay associated with the density and diffusion behaviour of biological populations. Based on our previous conclusions and numerical strategy of the direct problem, we continue to study the inverse problem about the diffusion coefficient as well as the parameters in the birth function, and also perform numerical simulations with error analysis. In addition, we further generalize the constant diffusion coefficient to the position-dependent diffusion rate, which is intended to improve the generalizability to multiple organisms diffusion in nature. [ABSTRACT FROM AUTHOR]

This paper addresses the electromagnetic inverse scattering problem of determining the location and shape of anisotropic objects from near-field Cauchy data. We investigate both cases involving the Helmholtz equation and Maxwell's equations for this inverse problem. Our study focuses on developing efficient imaging functionals that enable a fast and stable recovery of the anisotropic object. The implementation of the imaging functionals is simple and avoids the need to solve an ill-posed problem. The resolution analysis of the imaging functionals is conducted using the Green representation formula. Furthermore, we establish stability estimates for these imaging functionals when noise is present in the data. To illustrate the effectiveness of the methods, we present numerical examples showcasing their performance. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic plate system with a boundary frictional damping having a variable exponent $ m(x) $ m (x). We investigate the competition between the two types of damping and their effects on the energy decay rates. Using the multiplier method, we establish explicit decay results depending on both m and the relaxation function g. Our results address both the optimality and generality and improve earlier related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the phenomenon of synchronization in nonlinearly coupled thermoelastic waves is investigated. The existence of a finite-dimensional smooth global attractor $ \mathcal {A}_\alpha $ A α is proven in the natural energy space, where α is the coupling parameter. It is also proved that the coupled thermoelastic wave system can be reduced to a single thermoelastic wave when α tends to infinity. Subsequently, the upper-semicontinuity of their attractors is proven as $ \alpha \to \infty $ α → ∞. These results are ultimately employed to establish the synchronization phenomena for coupled thermoelastic waves. To the best of our knowledge, the synchronization of thermoelastic waves has not been considered before. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-species reaction–diffusion competition model with Gompertz growth, where the intrinsic growth rates and carrying capacities of environments are heterogeneous. At firstly, assuming two competing species only admit different diffusive rates, we show that 'slower diffuser prevails', which is consistent with the well-known result in Dockery J, Hutson V, Mischaikow K, Pernarowski M. [The evolution of slow dispersal rates: a reaction–diffusion model. J Math Biol. 1998;37(1):61–83; Hastings A. Can spatial variation alone lead to selection for dispersal? Theor Popul Biol. 1983;24:244–251]. Then, for the "weak competition" case, we establish a prior estimate, which combined with the theory of monotone dynamical system and spectral analysis implies that the model admits a unique coexistence steady state, which is globally asymptotically stable. Finally, for the "strong–weak competition" case, we give the expression of critical competition intensity and the weak competitor will be wiped out. [ABSTRACT FROM AUTHOR]

In this paper, we consider a semilinear coupled Timoshenko beam and an elastodynamic system posed in an inhomogeneous one-dimensional medium subject to localized damping mechanisms acting in the three equations. We show uniform decay rates for the energy of the solutions of such a problem without assuming any restrictions on the non-constant coefficients. To establish these results, refined arguments of the Microlocal Analysis Theory are applied. [ABSTRACT FROM AUTHOR]

We derive an exponential decay rate of energy for the cubic Schrödinger equation on a dissipative waveguide with a damping term that is effective locally on a neighborhood at the boundary and at infinity. In this paper, we give a new example for which the geometric control condition is sufficient to obtain an exponential decay of energy Schrödinger equation. The proof is based on Strichartz's estimates and the smoothing effects argument associated with the linear Schrödinger equation(LS). [ABSTRACT FROM AUTHOR]

In this paper, we mainly concern with the system of the coupled nonlinear Schrödinger equation with initial data in critical spaces in dimension 3. We first obtain the global well-posedness of the solution to the coupled nonlinear Schrödinger equation with initial data in a critical space $ W^{\frac {11}{7}, \frac 76}({\mathbb {R}}^3) $ W 11 7 , 7 6 ( R 3). The key is to derive a uniform bound of a modified energy $ \mathcal {E}(t) $ E (t) based on a decomposition for the solution as in Dodson [Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. Rev Mat Iberoam. 2022;38:1087–1100]. In addition, inspired by Dodson [Scattering for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. International mathematics research notices; 2023], we make a new decomposition for the solution and show the scattering to the coupled nonlinear Schrödinger equation with radially symmetric initial data in a critical Besov space $ B_{1,1}^2({\mathbb {R}}^3) $ B 1 , 1 2 ( R 3). [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the hydrodynamic limit of Boltzmann equations for a gas mixture containing two kinds of particles, where one is assumed to be near a vacuum and the other is near a Maxwellian equilibrium state. By the method of Hilbert expansion, we could derive the compressible Euler systems together with the terms of Boltzmann equations according to the different orders of the Knudsen number. Next, we truncate the expansion and establish the uniform estimates of the remainder term by using $ L^2 \,- \, L^{\infty } $ L 2 − L ∞ estimate. [ABSTRACT FROM AUTHOR]

In this paper, we mainly devote to investigate the classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities. Utilizing the celebrated approach of classification of traveling wave solution which was proposed by Jonatan Lenells [Traveling wave solutions of the Camassa-Holm equation. J Diff Equ. 2005;217(2):393–430]. We show a result concerns the regularity of traveling waves and then classify all traveling wave solutions. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]