Search

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 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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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 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 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]

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]

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 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]

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]

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 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]

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 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]

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]

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]

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]

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]

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 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]

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 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]

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 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]

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 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]