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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 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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space. [ABSTRACT FROM AUTHOR]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

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]

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $ -\frac {\mu }{(1+t)^{\lambda }}n_iu_i(i=1, 2) $ − μ (1 + t) λ n i u i (i = 1 , 2) for $ -1 − 1 < λ < 1 , μ > 0. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the incompressible limit of the non-isentropic ideal magnetohydrodynamic equations with periodic boundary conditions and general initial data. Weak convergence results of the smooth solutions are established by filtering method in the torus. [ABSTRACT FROM AUTHOR]

In a bounded open domain $ \Omega \subset \mathbb {R}^{N} $ Ω ⊂ R N , where $ N\geq 2 $ N ≥ 2 , with Lipschitz boundary $ \partial \Omega $ ∂Ω , we consider the Dirichlet problem for the elliptic system given by \[ \left \{ \begin{array}{@{}ll} \displaystyle-\sum^{N}_{i=1}D_{i}\left(a_{i}(x,u(x),D_{i}u(x))\right)+F(x,u)=f(x), & x\quad \in \Omega,\\ u(x)=0, & x\quad \in \partial \Omega, \end{array} \right. \] { − ∑ i = 1 N D i (a i (x , u (x) , D i u (x))) + F (x , u) = f (x) , x ∈ Ω , u (x) = 0 , x ∈ ∂Ω , here, $ u:\Omega \rightarrow \mathbb {R}^{n} $ u : Ω → R n , $ n\geq 2 $ n ≥ 2 , represents a vector-valued function, $ D_{i}u=\frac {\partial u}{\partial x_{i}} $ D i u = ∂ u ∂ x i denotes the partial derivative of u with respect to $ x_i $ x i , and the vector fields $ a_i:\Omega \times \mathbb {R}^{n}\times \mathbb {R}^{n} \rightarrow \mathbb {R}^{n} $ a i : Ω × R n × R n → R n and $ F:\Omega \times \mathbb {R}^n\rightarrow \mathbb {R}^n $ F : Ω × R n → R n are Carathédory functions. In this paper, we focus on nonlinear degenerate anisotropic elliptic systems with variable growth and $ L^{m} $ L m data, where m is small. Specifically, we consider the case where the right-hand side term f belongs to $ L^{m}(\Omega ;\mathbb {R}^{n}) $ L m (Ω ; R n) with 1

This paper is devoted to studying a dynamic adhesive contact model where the body is composed of a viscoelastic material with long memory. The adhesion process is modeled by a bonding field on part of the contact surface and on which the contact and the friction are modeled by nonmonotone Clarke subdifferential conditions with adhesion. Meanwhile, on another part of the contact surface, the contact is comprised of a normal compliance contact condition while the memory effects of the obstacle are considered, the friction is modeled by a version of Coulomb's law of dry friction and the friction bound is dependent on the normal stress. The variational form of the model generates a system of a doubly-history dependent variational–hemivariational inequality as well as a differential equation. Then, on the basis of the proof of abstract variational–hemivariational inequality, the existence and uniqueness results of the contact problem are established. [ABSTRACT FROM AUTHOR]

We prove that if the total factor productivity A of an aggregative economy is right at the barrier $ \sigma +\lambda $ σ + λ , with σ being the growth rate of labor force and λ the real interest rate, then the unique policy to optimally control the economy is the same as the one for optimally controlling weak economies, where $ A \lt \sigma +\lambda $ A < σ + λ. This result gives a complete answer for the interesting open question raised by Vu Thi Huong in her recent paper [Optimal economic growth problems with high values of total factor productivity. Appl Anal. 2022;101:1315–1329]. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following Hartree system with nonlinear couplings: \[ \left\{ \begin{aligned} & -\varepsilon ^{2}\Delta u+V_{1}\left(x\right) u\\ & \quad =\dfrac{1}{\varepsilon^{N-\mu}} \left[\nu_{1}\left(\displaystyle\int_{\mathbb{R}^{N}} \dfrac{|u|^{p}}{|x-y|^{\mu}}\,\mathrm{d}y\right)|u|^{p-2}u \right.\\ & \qquad +\left. \beta \left(\displaystyle\int_{\mathbb{R}^{N}} \dfrac{|v|^{q}}{|x-y|^{\mu}}\,\mathrm{d}y\right)|u|^{q-2}u\right],\\ & -\varepsilon ^{2}\Delta v+V_{2}\left(x\right) v \\ & \quad =\dfrac{1}{\varepsilon^{N-\mu}} \left[\nu_{2}\left(\displaystyle\int_{\mathbb{R}^{N}} \dfrac{|v|^{p}}{|x-y|^{\mu}}\,\mathrm{d}y\right)|v|^{p-2}v \right. \\ & \qquad + \left. \beta\left(\displaystyle\int_{\mathbb{R}^{N}} \dfrac{|u|^{q}}{|x-y|^{\mu}}\,\mathrm{d}y\right)|v|^{q-2}v\right],\\ & u,v\in H^{1}(\mathbb{R}^{N}),\quad u,v \gt 0 \ {\rm in}\ \mathbb{R}^{N}, \end{aligned} \right. \] { − ε 2 Δ u + V 1 (x) u = 1 ε N − μ [ ν 1 (∫ R N | u | p | x − y | μ d y) | u | p − 2 u + β (∫ R N | v | q | x − y | μ d y) | u | q − 2 u ] , − ε 2 Δv + V 2 (x) v = 1 ε N − μ [ ν 2 (∫ R N | v | p | x − y | μ d y) | v | p − 2 v + β (∫ R N | u | q | x − y | μ d y) | v | q − 2 v ] , u , v ∈ H 1 (R N) , u , v > 0 in R N , where $ N\,{\geq}\,3 $ N ≥ 3 , $ 0\,{ \lt }\,\mu\,{ \lt }\,N $ 0 < μ < N , $ \frac {2N-\mu }{N}\,{\leq}\,p\,{\leq}\,\frac {2N-\mu }{N-2} $ 2 N − μ N ≤ p ≤ 2 N − μ N − 2 , $ \frac {2N-\mu }{N} \leq q \leq \min \{p, 2\} $ 2 N − μ N ≤ q ≤ min { p , 2 } , $ \nu _{1},\nu _{2} \gt 0 $ ν 1 , ν 2 > 0 , ε is a small parameter and $ \beta \lt 0 $ β < 0 is a coupling constant, and the potentials $ V_{1} $ V 1 and $ V_{2} $ V 2 have $ k_{1} $ k 1 and $ k_{2} $ k 2 isolated global minimum points, respectively. Using the Nehari manifold technique, the energy estimate method and the Lusternik–Schnirelmann theory, we find an interesting phenomenon that the problem possesses $ k_{1}k_{2} $ k 1 k 2 positive solutions when $ V_{1} $ V 1 and $ V_{2} $ V 2 do not have any common isolated global minimum points, and $ k_{1}k_{2}+m $ k 1 k 2 + m positive solutions when $ V_{1} $ V 1 and $ V_{2} $ V 2 have m common isolated global minimum points. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored. [ABSTRACT FROM AUTHOR]

In this paper, we classify the phase portraits of an SIR epidemic dynamics model. Depending on the values of the parameters, this model can exhibit seven different phase portraits. In particular, from a biological point of view we prove that the unique attractors of this model are one or two equilibrium points depending on the values of the parameters and from the phase portraits follow the basins of attraction of these equilibria. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method. [ABSTRACT FROM AUTHOR]

Parameter identification problems in PDEs are special class of nonlinear inverse problems which has many applications in science and technology. One such application is the Electrical Impedance Tomography (EIT) problem. Although many methods are available in literature to tackle nonlinear problems, the computation of Fréchet derivative is often a bottle neck for deriving the solution. Moreover, many assumptions are required to establish the convergence of such methods. In this paper, we propose a modified form of Levenberg–Marquardt scheme which does not require the knowledge of exact Fréchet derivative, instead, uses an approximate form of it and at the same time, no additional assumptions are required to establish the convergence of the scheme. We illustrate the theoretical result through numerical examples. In order to ensure that the proposed scheme can be applied to practical problems, we have applied the scheme to EIT problem and the reconstruction process clearly demonstrates that the method can be successfully applied to practical problems. [ABSTRACT FROM AUTHOR]

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of 'consistency' due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following problem: { − Δ u + u = Q ε | u | p − 2 u in R N , u (x) → 0 , as | x | → ∞ , where p ∈ (1 , 2) , Q ε ∈ L ∞ ( R N) and the self-focusing core supp Q ε + shrinks to a point as ε → 0. We show that there is a least energy solution u ε corresponding to Q ε and then investigate the limiting profile of concentration for u ε as ε → 0. The results are supplements to the works of [Ackermann and Szulkin. A concentration phenomenon for semilinear elliptic equations. Arch Ration Mech Anal. 2013;207(3):1075–1089; Fang and Wang. Limiting profile of solutions for Schrödinger equations with shrinking self-focusing core. Calc Var Partial Differential Equations. 2020;59(4):Paper No. 129, 18]. [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional singularly perturbed problem of the convection–diffusion type. The problem is solved numerically by the local discontinuous Galerkin (LDG) method on a Bakhvalov-type mesh. Here we propose new numerical fluxes and new penalty parameters in the LDG method and prove the supercloseness of the LDG method in an energy norm. Besides, a variant of the energy norm is proposed. It is proved that the method is convergent uniformly in the perturbation parameter with an improved order of k + 1 in the new norm (k is the degree of the piecewise polynomial used in the LDG method). [ABSTRACT FROM AUTHOR]

The paper presents an abstract theory regarding the problems with semilinear operator equations involving iterates of a strongly monotone symmetric linear operator. We obtain existence and localization results of positive solutions for such problems using Krasnosel'skiĭ's technique and abstract Harnack inequality. In particular, we obtain results for problems with semilinear poly-Laplace operators. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of exponential stabilization of a one-dimensional thermoelastic diffusion problem of type II for isotropic and homogeneous materials with frictional damping and time delay. The system of equations under consideration is a coupling of three hyperbolic equations involving three variables considering frictional damping and time delay on each variable. Using semigroup theory, we prove the well-posedness by the Lumer–Phillips theorem and the exponential stability by exploring the dissipative properties of the linear operator associated with the model through the Gearhart–Huang–Pruss theorem. [ABSTRACT FROM AUTHOR]

In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: \[ \delta\Delta^{2}u-\Delta u +u=|u|^{2\sigma}u, \quad u\in H^{2}(\mathbb{R}^{N}), \] δ Δ 2 u − Δu + u = | u | 2 σ u , u ∈ H 2 (R N) , where $ \delta \gt 0 $ δ > 0 is sufficiently small, $ 0 \lt \sigma \lt \frac {2}{(N-2)_+} $ 0 < σ < 2 (N − 2) + , $ \frac {2}{(N-2)_+}=\frac {2}{N-2} $ 2 (N − 2) + = 2 N − 2 for $ N\geq ~3 $ N ≥ 3 and $ \frac {2}{(N-2)_+}=+\infty $ 2 (N − 2) + = + ∞ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose $ P(x) $ P (x) and $ Q(x) $ Q (x) are two positive, radial and continuous functions satisfying that as $ r=|x|\rightarrow +\infty $ r = | x | → + ∞ , \[ P(r)=1+\frac{a_1}{r^{m_1}} +O\left(\frac{1}{r^{m_1+\theta_1}}\right),\quad Q(r)=1+\frac{a_2}{r^{m_2}} +O\left(\frac{1}{r^{m_2+\theta_2}}\right), \] P (r) = 1 + a 1 r m 1 + O (1 r m 1 + θ 1 ) , Q (r) = 1 + a 2 r m 2 + O (1 r m 2 + θ 2 ) , where $ a_1, a_2\in \mathbb {R} $ a 1 , a 2 ∈ R , $ m_1, m_2 \gt 1 $ m 1 , m 2 > 1 , $ \theta _1, \theta _2 \gt 0 $ θ 1 , θ 2 > 0. We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: \[ \delta\Delta^{2}u-\Delta u +P(x)u=Q(x)|u|^{2\sigma}u, \quad u\in H^{2}(\mathbb{R}^{N}). \] δ Δ 2 u − Δu + P (x) u = Q (x) | u | 2 σ u , u ∈ H 2 (R N). [ABSTRACT FROM AUTHOR]

This paper is consecrated to the asymptotic behavior of a coupled system consisting of the Kirchhoff and heat equations in a bounded domain. We establish that the problem is well-posed by using a Faedo–Galerkin scheme. Furthermore, the energy method combined with the multiplier techniques is used to show the exponential stability of the solutions to the system. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of weak solutions for full compressible magnetohydrodynamic flows in a 3D bounded domain, with non-homogeneous boundary condition for the velocity, absolute temperature and density on the inflow part, with Navier-type slip boundary condition for magnetic field. We show the weak–strong uniqueness principle as well as the global existence under a new notion of weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the 1D wave equation system \[ \begin{cases} u_{1tt} - u_{1xx} +\sigma u_1 +u_1u_2^2 = 0 \\ u_{2tt} - u_{2xx} +\mu u_2 +u_1^2 u_2 = 0 \end{cases} \] { u 1 tt − u 1 xx + σ u 1 + u 1 u 2 2 = 0 u 2 tt − u 2 xx + μ u 2 + u 1 2 u 2 = 0 under Dirichlet boundary conditions, where $ 0 \lt \sigma \in [\sigma _1,\sigma _2]\subset [0,1], $ 0 < σ ∈ [ σ 1 , σ 2 ] ⊂ [ 0 , 1 ] , $ 0 \lt \mu \in [\mu _1,\mu _2]\subset [0,1] $ 0 < μ ∈ [ μ 1 , μ 2 ] ⊂ [ 0 , 1 ] are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the nonlinear thermoelastic plates, with Fourier heat conduction, and consider a polynomial-type nonlinearity. We first develop a theoretical analysis of the corresponding linear system to derive time decay estimates in $ L^{\infty }(\mathbb {R}^n) $ L ∞ (R n) and $ H^s(\mathbb {R}^n) $ H s (R n). Then, using that set of decay estimates and controlling the nonlinearity, we prove the existence and uniqueness of local solutions with initial data $ (u(0),u_t(0),\theta (0))=(u_0,\Delta u_1,\Delta \theta _1) $ (u (0) , u t (0) , θ (0)) = (u 0 , Δ u 1 , Δ θ 1) , with $ u_0\in H^s $ u 0 ∈ H s , and $ u_1,\theta _1\in H^{s+1} $ u 1 , θ 1 ∈ H s + 1 , for $ s \gt \frac {n}{2}+1 $ s > n 2 + 1. [ABSTRACT FROM AUTHOR]

In this paper, we study the global strong solutions to the three-dimensional (3D) heat-conducting incompressible Navier–Stokes equations with density-temperature-dependent viscosity and heat-conducting coefficients in $ \mathbb {R}^3 $ R 3 . By using the t-weighted a priori estimates, we prove the global existence and exponential decay-in-time rates of strong solutions to the Cauchy problem when the $ L^{{3}/{2}} $ L 3 / 2 -norm of the initial density is suitably small. It should be noted that the velocity and absolute temperature can be large initially, and the initial density contains vacuum case. [ABSTRACT FROM AUTHOR]

The paper is concerned about the existence of solutions with prescribed $ L^2 $ L 2 -norm to the following Kirchhoff-type equation $$\begin{equation*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+(V+\lambda)u=|u|^{p-2}u+\mu|u|^{q-2}u\ {\rm in}\ \mathbb{R}^3, \end{equation*}$$ − (a + b ∫ R 3 | ∇u | 2) Δu + (V + λ) u = | u | p − 2 u + μ | u | q − 2 u in R 3 , where $ a,b\gt 0,2 \lt q \lt 14/3 \lt p\leqslant 6 $ a , b > 0 , 2 < q < 14 / 3 < p ⩽ 6 or $ 14/3 \lt q \lt p\leqslant 6 $ 14 / 3 < q < p ⩽ 6 , $ \mu \gt 0 $ μ > 0. Noting that 14/3 is the mass critical exponent, a Pohozaev constraint method is adopted in two cases. In the mass mixed critical case, i.e., $ 2 \lt q \lt 10/3,14/3 \lt p\leqslant 6 $ 2 < q < 10 / 3 , 14 / 3 < p ⩽ 6 , we get a normalized solution to above equation with small enough μ by Ekeland's variational principle. In the mass supercritical case, i.e., $ 14/3 \lt q \lt p\leqslant 6 $ 14 / 3 < q < p ⩽ 6 , we obtain a positive ground state normalized solution, and energy comparison argument is used in the Sobolev critical case. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of almost automorphic solutions in distribution to McKean–Vlasov stochastic differential equations are established provided the coeffcients satisfy some suitable conditions. The asymptotic stability of the unique almost automorphic distribution solution is discussed in the square-mean sense. Finally, some examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the following polyharmonic equation: \[ \begin{cases} -\Delta^m_{\dfrac{n}{m}} u=Q_\lambda(x) |u|^{q-1}u+\left(\displaystyle\int_{\Omega}\dfrac{F(u)}{|x-y|^\mu|y|^\alpha}\,{\rm d}y\right)\dfrac{f(u)}{|x|^\alpha} & \quad\mbox{in}\ \Omega,\\ u=\nabla u=\cdots=\nabla^{m-1} u=0 & \quad\mbox{on}\ \partial\Omega. \end{cases} \] { − Δ n m m u = Q λ (x) | u | q − 1 u + (∫ Ω F (u) | x − y | μ | y | α d y) f (u) | x | α in Ω , u = ∇ u = ⋯ = ∇ m − 1 u = 0 on ∂ Ω. Using the Nehari manifold and some analytical techniques, we show the existence of two weak solutions of the problem with respect to parameter λ. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimal time decay rates of the higher order spatial derivatives of global solution for the three-dimensional compressible Oldroyd-B model, which was derived independently by Barrett-Lu-Süli [Commun. Math. Sci., 15(5): 1265–1323, 2017] via micro–macro analysis of the compressible Navier–Stokes–Fokker–Planck system. We show that under $ H^3 $ H 3 framework, the third-order spatial derivatives of global small solution of the compressible Oldroyd-B model decay at the rate $ (1+t)^{-\frac 94} $ (1 + t) − 9 4 instead of $ (1+t)^{-\frac 74} $ (1 + t) − 7 4 stated by Wang-Wen [Math. Models Methods Appl. Sci., 30(1): 139–179, 2020]. This decay rate is optimal in the sense that it coincides with that of solutions to heat equation. [ABSTRACT FROM AUTHOR]

This paper focuses on the existence and approximate controllability of second-order non-autonomous impulsive stochastic neutral differential systems in Hilbert spaces. First, by using Schauder's fixed-point theorem, stochastic analysis theory, and evolution operators, a new set of sufficient conditions are formulated, and we prove the existence of mild solutions of second-order non-autonomous impulsive stochastic neutral differential systems. Further, the result is extended to study the approximate controllability results for second-order non-autonomous impulsive stochastic neutral differential systems. Then, a set of sufficient conditions are derived for the approximate controllability of second-order non-autonomous impulsive stochastic neutral differential system by assuming the associated linear system is approximately controllable. The results are obtained with the help of Krasnoselskii's fixed point theorem. Finally, an example is given to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial-boundary value problem for a logarithmic Lamé system with a time delay in a bounded domain. We prove the well-posedness of the system by utilizing the semigroup theory. Then, we prove the existence of global solutions by using the well-depth method. In addition, we establish an exponential stability decay result under appropriate assumptions on the weight of the time delay and that of frictional damping. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse coefficients problem of recovering a shear modulus μ and density ρ of a medium from the Neumann-to-Dirichlet map. This inverse problem is motivated by the reconstruction of mechanical properties of tissues in non-destructive testing. We prove Lipschitz stability results for any dimension $ d \geq 2 $ d ≥ 2 , provided that the parameters μ and ρ have upper and lower bounds and belong to a known finite dimensional subspace. The proofs rely on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, combined with the techniques of localized potentials. [ABSTRACT FROM AUTHOR]

The stability and large-time behavior problem on the magneto-micropolar equations has evoked a considerable interest in recent years. In this paper, we study the stability and exponential decay near magnetic hydrostatic equilibrium to the two-dimensional magneto-micropolar equations with partial dissipation in the domain $ \Omega =\mathbb {T}\times \mathbb {R} $ Ω = T × R . In particular, we takes advantage of the geometry of the domain $ \mathbb {T}\times \mathbb {R} $ T × R to divide u into zeroth mode and the nonzero modes, and obey a strong version of the Poincaré's inequality, which plays a crucial role in controlling the nonlinearity. Moreover, we find that the oscillation part of the solution decays exponentially to zero. Finally, our result mathematically verifies that the stabilization effect of a background magnetic field on magneto-micropolar fluids. [ABSTRACT FROM AUTHOR]

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) $$\begin{equation} \begin{cases} u_t= \Delta u - \chi \nabla \cdot (u \nabla v)+\xi \nabla \cdot (u \nabla w) & {\rm in}\ \Omega \times (0,T_{\max}),\\ \tau v_t=\Delta v-\varphi(t,v)+f(u) & {\rm in}\ \Omega \times (0,T_{\max}),\\ \tau w= \Delta w - \psi(t,w) + g(u) & {\rm in}\ \Omega \times (0,T_{\max}). \end{cases} \end{equation}$$ { u t = Δ u − χ ∇ ⋅ (u ∇ v) + ξ ∇ ⋅ (u ∇ w) in Ω × (0 , T max) , τ v t = Δ v − φ (t , v) + f (u) in Ω × (0 , T max) , τ w = Δ w − ψ (t , w) + g (u) in Ω × (0 , T max). In this problem, Ω is a bounded and smooth domain of $ \mathbb R^n $ R n , for $ n\geq 2 $ n ≥ 2 , $ \chi,\xi \gt 0 $ χ , ξ > 0 , $ f(u) $ f (u) , $ g(u) $ g (u) reasonably regular functions generalizing, respectively, the prototypes $ f(u)=\alpha u^k $ f (u) = α u k and $ g(u)= \gamma u^l $ g (u) = γ u l , for some $ k,l,\alpha,\gamma \gt 0 $ k , l , α , γ > 0 and all $ u\geq 0 $ u ≥ 0. Moreover, $ \varphi (t,v) $ φ (t , v) and $ \psi (t,w) $ ψ (t , w) have specific expressions, $ \tau \in \{0,1\} $ τ ∈ { 0 , 1 } and $ \Theta :=\chi \alpha -\xi \gamma $ Θ := χ α − ξ γ. Once for any sufficiently smooth $ u(x,0)=u_0(x)\geq 0 $ u (x , 0) = u 0 (x) ≥ 0 , $ \tau v(x,0)=\tau v_0(x)\geq 0 $ τ v (x , 0) = τ v 0 (x) ≥ 0 and $ \tau w(x,0)=\tau w_0(x)\geq 0, $ τ w (x , 0) = τ w 0 (x) ≥ 0 , the local well-posedness of problem ( $ \Diamond $ ◊) is ensured, and we establish for the classical solution $ (u,v,w) $ (u , v , w) defined in $ \Omega \times (0,T_{\max }) $ Ω × (0 , T max) that the life span is indeed $ T_{\max }=\infty $ T max = ∞ and u, v and w are uniformly bounded in $ \Omega \times (0,\infty) $ Ω × (0 , ∞) in the following cases: For $ \varphi (t,v)=\beta v $ φ (t , v) = β v , $ \beta \gt 0 $ β > 0 , $ \psi (t,w)=\delta w $ ψ (t , w) = δ w , $ \delta \gt 0 $ δ > 0 and $ \tau =0 $ τ = 0 , provided (I.1) k 0 , $ \psi (t,w)=\delta w $ ψ (t , w) = δ w , $ \delta \gt 0 $ δ > 0 and $ \tau =1 $ τ = 1 , whenever (II.1) $ l,k\in \left (0,\frac {1}{n}\right ] $ l , k ∈ (0 , 1 n ] ; (II.2) $ l\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ l ∈ (1 n , 1 n + 2 n 2 + 4) and $ k\in \left (0,\frac {1}{n}\right ] $ k ∈ (0 , 1 n ] , or $ k\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ k ∈ (1 n , 1 n + 2 n 2 + 4) and $ l\in \left (0,\frac {1}{n}\right ] $ l ∈ (0 , 1 n ] ; (II.3) $ l,k\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ l , k ∈ (1 n , 1 n + 2 n 2 + 4) . For $ \varphi (t,v)=\frac {1}{|\Omega |}\int _\Omega f(u) $ φ (t , v) = 1 | Ω | ∫ Ω f (u) and $ \psi (t,w)=\frac {1}{|\Omega |}\int _\Omega g(u) $ ψ (t , w) = 1 | Ω | ∫ Ω g (u) and $ \tau =0 $ τ = 0 , under the assumptions k

Pattern dynamics in reaction-diffusion modeling glycolysis are investigated. This paper includes linear stability analysis to deduce the threshold condition for Turing pattern formation and weakly nonlinear analysis to describe the time evolution of the pattern amplitude close to the instability threshold. We will assume that the emerging patterns do not have any spatial structure. We derive a criterion to find the region of parameters for which super/sub-critical Turing instability is possible. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the reduction of the model to the TH normal forms in order to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the degenerate Turing-Hopf bifurcation point. Our theoretical results are confirmed by numerical simulations. [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+1Fq 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+1Fq(−n,kn+α,…,kn+α+q−1q;kn+β,…,kn+β+q−1q;z)(n→∞), where n is a nonnegative integer, q is a positive integer and the constant parameters α and β are constrained by $ \alpha {<}\beta $ 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]