ABSTRACT In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog

In this paper, we study the following modified quasilinear Schrödinger equation − Δ u + V ( x ) u − κ u Δ ( u 2 ) + q h 2 ( | x | ) | x | 2 ( 1 + κ u 2 ) u + q ∫ | x | + ∞ h ( s ) s ( 2 + κ u 2 ( s ) ) u 2 ( s ) d s u = | u | p − 2 u i n   R 2 , where κ > 0 , q>0, p>6 and V ∈ C ( R 2 , R ) . We develop a more direct and simpler approach to prove the existence of ground state solutions. Our method is

The blow-up of solutions to a coupled system of semilinear wave equations in the sub-critical and critical cases is investigated. The system contains scattering dampings and combined nonlinearities. The upper bound lifespan estimates of solutions which are related to the indexes of nonlinear terms are established by applying the functional method and iteration technique.

ABSTRACT In this paper, we consider the regularity criteria of 3D incompressible Boussinesq equations. By using the Littlewood–Paley decomposition technique to establishing a regularity criterion in terms of the one-directional derivative of velocity in Besov spaces. Our result improves some previous works (Liu Q. A regularity criterion for the Navier–Stokes equations in terms of one-directional derivative

ABSTRACT In the subcritical energy case, local well-posedness is established in the radial energy space for a class of fractional inhomogeneous Choquard equations. The best constant of a Gagliardo–Nirenberg type inequality is obtained. Moreover, a sharp threshold of global existence versus blow-up dichotomy is obtained for mass super-critical and energy subcritical solutions.

ABSTRACT The stationary Stokes equations for a generalized Newtonian fluid with nonlinear unilateral, and slip and leak boundary conditions are investigated. Boundary conditions include the generalized Clarke gradient and the convex subdifferential, and the variational formulation of the problem is the variational–hemivariational inequality for the velocity field. Existence and uniqueness result for

In this paper, we consider a nonlocal diffusion equation involving the fractional p ( x ) -Laplacian with nonlinearities of variable exponent type. Employing the subdifferential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the

This paper is concerned with the entire solutions defined in the whole space and for all time t ∈ R of a public good game model. The existence of the entire solutions is proved by super- and sub-solution method. Traveling fronts with possibly different wave speeds are used to construct the super- and sub-solutions, in particular, the traveling fronts of a variant of KPP equation are used to derive

ABSTRACT In this paper, an efficient semi-implicit difference scheme for solving the fractional nonlinear Schrödinger equation with wave operator are proposed and analyzed. The semi-implicit scheme involves three-time levels, is unconditionally stable and fourth-order accurate in space and second-order accurate in time. Furthermore, the unique solvability, unconditional stability and convergence of

We study an affine version of the Hardy inequality which is truly stronger than the usual Hardy inequality. We also set up a sharp version of the Hardy inequality under Sobolev–Lorentz norm.

In this paper, we are interested in the study of a porous-elastic system in one-dimensional open bounded domain and with dissipation due to microtemperatures and frictional damping acting on the first equation (displacement of the solid elastic material). Using the energy method, we prove the exponential decay for the energy.

ABSTRACT In this paper, we introduce the concept of ◊ α -measurability and establish the combined measure theory on time scales. Particularly, one can obtain Δ-measure theory and ∇-measure theory on time scales by letting α = 1 and α = 0 , respectively. Moreover, some criteria for ◊ α -measurability of a set are obtained and the notion of Lebesgue ◊ α -measurable functions is introduced and studied

ABSTRACT In this work we analyze the porous elastic system with a viscoelastic dissipative mechanism of Kelvin–Voigt type. We prove that the semigroup associated with the model is analytic if and only if the viscoelastic damping is present in both equations of the system. Otherwise, there exist lack of exponential stability, independently of any relationship between the coefficients. This result is

ABSTRACT In this paper, we study inverse source problems for the heat equation with an inverse-square potential localized on the boundary of a smooth bounded domain, and with a locally distributed observation. We first derive an improved Carleman estimate of that obtained by Cazacu [SIAM J Control Optim. 2014;52:2055–2089] and then use it to prove the Lipschitz stability result for the inverse source

We prove the long time orbital stability of the plane wave solutions to the nonlinear Schrödinger equation (NLS) in the defocusing ( λ = 1 ) or focusing ( λ = − 1 ) case, i u t = − Δ u + λ | u | 2 u , x ∈ T d , t ∈ R . More precisely, in a Gevrey space G σ := u :   |   ∥ u ∥ σ 2 = ∑ a ∈ Z d e 2 σ | a | | u a | 2 < ∞ for some positive constant σ, we show that solution with the initial datum in the 4

We consider the extinction regime in the spatial stochastic logistic model in R d (a.k.a. Bolker–Pacala–Dieckmann–Law model of spatial populations) using the first-order perturbation beyond the mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields

We consider a mean-field model of a polymer with a spherically symmetric finitely supported potential. We describe how the typical size of the polymer depends on the two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which goes to infinity.

In this paper, we consider an inverse conductive scattering problem for the Helmholtz equation from measurements on the outer boundary. We want to seek reconstruction algorithms to detect the number, location, size and shape of inhomogeneities within a body. This problem is nonlinear and ill-posed, thus we should consider regularization techniques in order to improve the corresponding approximation

In the present work we obtain the constants of motion for isoperimetric variational problems with time delay. We consider a constrained optimization problem where the Lagrangian function defining the functional depends on time delayed arguments. We prove the isoperimetric Euler–Lagrange and DuBois–Reymond type optimality conditions and, in order to investigate the constants of motion for this problem

In the paper, we characterize the polynomial stability of C 0 semigroup T ( t ) with generator A on Hilbert space H . Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for H . By the asymptotic relation of the real part and imaginary part of eigenvalues of A, we give the optimal decay rate of polynomial stability of T ( t ) . Moreover, we give the

In this paper, we prove the global strong solutions for the Cauchy problem of two-dimensional (2D) incompressible non-isothermal nematic liquid crystal flows, if the initial orientation satisfies a geometric condition. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states. When d is a constant vector and | d | = 1 , we also extend the corresponding result

In this paper, we consider a stochastic general epidemic model with relapse, graded cure and nonlinear incidence rate. We show the existence and uniqueness of a global positive solution for the model under some mild conditions. We also give some sufficient conditions for ensuring the existence of an ergodic stationary distribution for the model. Moreover, we analyse the extinction of the disease described

The paper is devoted to the duality of the Mayer problem for κ -th order viable differential inclusions with endpoint constraints, where κ is an arbitrary natural number. Thus, this paper for constructing the dual problems to viable differential inclusions of any order with endpoint constraints can make a great contribution to the modern development of optimal control theory. For this, using locally

In this paper, we study the energy decay for two one-dimensional thermoelastic Bresse-type systems in a bounded open interval under mixed homogeneous Dirichlet–Neumann boundary conditions and with two different kinds of dissipation working only on the vertical displacement and given by heat conduction of types I and III. The two systems are consisting of three wave equations (Bresse-type system) coupled

In this paper, we study the singular limit of three-dimensional nonisentropic compressible ideal MHD equations with horizontal boundary conditions. It is rigorously justified that, when the Mach number and Alfvén number are same and tend to zero, the solution of original equations converges to that of two-dimensional incompressible flow.

In this paper, by virtue of the Kakutani–Fan–Glicksberg fixed point theorem and a nonlinear scalarization function, we first obtain an existence theorem of general n-person noncooperative games problems with set payoffs. Then, we also obtain existence theorems of minimax regret equilibria and ϵ-minimax regret equilibria with scalar set payoffs. Some examples are given to illustrate our results.

In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection–reaction–diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank–Nicolson finite-difference formulation. In the stochastic

In this paper, using the subvariant functional method due to Favard [Favard J. Sur les équations différentielles linéaires á coefficients presque-périodiques. ActaMathematica. 1928;51(1):31–81.], we prove the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces provided the existence of at least one bounded solution on the right

This paper is concerned about a diffusive degenerate predator–prey model with Beddington–DeAngelis functional response subject to homogeneous Neumann boundary condition. First, the global bifurcation branches of positive stationary solutions are studied, which are quite different from those with different degeneracy or functional response. Second, the multiplicity and stability of positive stationary

In this paper, we investigate the convergence rates of the inverse heat transfer problem of simultaneously determining the space-dependent reaction coefficient and the initial temperature from some additional temperature observations. The strategy is to decouple the original problem into two inverse problems: (i) recover the reaction coefficient; (ii) determine the initial temperature with the estimated

In previous work, the synchronization of multi-links systems with Lévy noise (MLSLN) has not been fully investigated. Therefore, the synchronization of MLSLN is concerned via feedback discrete-time observations control. Therein, we provide a novel Lyapunov functional for MLSLN. Moreover, the upper bound of the state observations interval duration is obtained. By making use of Lyapunov functional method

For the Cauchy problem of the half-wave equation with L 2 -critical combined nonlinearities i ψ t = − Δ ψ − | ψ | 2 / N ψ − | ψ | p ψ , 0 < p < 2 N ( N ≥ 2 ) , ψ ( 0 , x ) = ψ 0 ( x ) , we obtain the existence of traveling wave solution of the form ψ ( t , x ) = e i t μ u v ( x − v t ) with 0 < | v | < 1 . If ∥ ψ 0 ∥ 2 < ∥ Q v ∥ 2 , where Q v is a solution of the equation − Δ u v + i ( v ⋅ ∇ ) u v

It is the purpose of this note, to obtain a scattering versus finite time blow-up dichotomy for a mass super-critical and energy sub-critical coupled Schrödinger system.

In this paper, we investigate the Tseng's extragradient algorithm for non-Lipschitzian variational inequalities with pseudomonotone vector fields on Hadamard manifolds. The convergence analysis of the proposed algorithm is discussed under mild assumptions. Two experiments are provided to illustrate the asymptotical behavior of the algorithm. The results presented in this paper generalize some known

The purpose of this paper is the study of the asymptotic modeling of a linearly thin elastic isotropic membrane shell which is subjected to frictionless contact with a rigid obstacle on part of the boundary. Using the formal asymptotic expansions method, we obtain a two dimensional Signorini membrane shell model, then we justify the result by convergence theorem.

The main concern of this paper is to investigate the asymptotic stability of stationary solution to the compressible Navier–Stokes–Poisson equations with the classical Boltzmann relation in a half line. We first show the unique existence of stationary solution with the aid of the stable manifold theory, and then prove that the stationary solution is time asymptotically stable under the small initial

Let Ω be a bounded domain of R N with smooth boundary, g 1 , g 2 : Ω → R and let A : D ( A ) → L 2 ( Ω ) be a self-adjoint operator defined on a dense subspace D ( A ) ⊂ L 2 ( Ω ) such that A has an orthonormal basis of eigenfunctions in L 2 ( Ω ) . For Y >0, giving the function f : Ω × [ 0 , Y ] → R , we consider the problem of finding a function u : Ω × [ 0 , Y ] → R such that − c ( y ) A u ( x

In this paper, we consider a class of p ( x ) -Laplacian problems of the form: ( − Δ ) p ( x ) u + a ( x ) | u | p ( x ) − 2 u = f ( x , u ) in    Ω , | ∇ u | p ( x ) − 2 ∂ u ∂ v + b ( x ) | u | q ( x ) − 2 u = g ( x , u ) on    ∂ Ω , where Ω ⊂ R N , N ≥ 2 is a bounded domain with Lipschitz boundary ∂ Ω , ( ∂ / ∂ v ) is outer unit normal derivative. The functions a, b, p, q, g and f are assumed to

Here, the existence of at least one positive radial solution of the Nonlinear nonhomogeneous Neumann problem − Δ H n u + R ( ξ ) u = a ( | ξ | H n ) | u | q − 2 u 1 + | u | q − p − b ( | ξ | H n ) | u | q − 2 u , ξ ∈ Ω is proved on the Heisenberg group H n , via variational principle, where Ω is the Korányi ball, a and b are nonnegative radial functions and R ( ξ ) holds in some suitable conditions

In this paper we study the blow-up analysis for some mean field equation on point vortices which is derived by C. Neri under a stochastic assumption. In particular, we derive some estimate which is the asymptotic behavior of blow-up solutions near the blow-up points. To obtain this result we shall employ the new scaling argument for blow-up solutions of Neri's mean field equation. Moreover, we also

In this paper we study the stability issue of the inverse problem of determining an inclusion contained in a stratified medium assuming the conductivity to be isotropic and variable coefficients. We prove a modulus of continuity of logarithmic type which, according to examples showed in [Di Cristo M, Rondi L. Examples of exponential instability for inverse inclusion and scattering problems. Inverse

This article is the second part of two works by the same authors on the same topic. Let ( X , d , μ ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors introduce the notion of paraproducts on X and obtain their boundedness from H p ( X ) × H q ( X ) into H r ( X ) for any given p , q , r ∈ ( n / ( n + η ) , ∞ ) satisfying 1 / r = 1 / p + 1 / q , where H

A new regularization method for an inverse source problem for a parabolic equation in a Banach space is proposed. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. Some numerical examples are presented for illustrating the efficiency of the method.

This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if we assume the existence of a super-solution. Second

In this paper, we present a nonconforming immersed finite element method for solving elliptic optimal control problems with interfaces. The immersed finite element space is constructed based on the rotated-Q1 nonconforming finite elements with the integral-value degrees of freedom. The method can overcome the difficulties encountered by conforming immersed finite element method. We derive error estimates

In this paper, by using the concentration–compactness principle of Lions for variable exponent spaces and the Mountain Pass Theorem, we obtain the existence of nontrivial solutions for a class of ( p 1 ( x ) , … , p n ( x ) ) -Kirchhoff-type potential systems type with critical exponent.

This paper investigates the asymptotic behavior of solutions for non-autonomous fractional stochastic reaction–diffusion equations with multiplicative noise in R . We prove the existence and uniqueness of the tempered pullback random attractor for the equations in L 2 ( R ) and obtain the finite fractal dimension for the pullback random attractor. Two main difficulties here are that the fractional

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces

In this article, we prove the existence of a solution for nonlinear system governed by ( p ( x ) , q ( x ) ) -Laplacian operators by using the concentration-compactness Principle for recovering the lack of compactness and some variable techniques.

This article is the first part of two works by the same authors on the same topic. Let ( X , d , μ ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish some basic estimates and a modified inhomogeneous discrete Calderón reproducing formula, which play essential roles in the proof of the boundedness of paraproducts in the second part of this work

In this paper, the fractional nonlinear Schrödinger equation system with periodic boundary conditions is considered. I establish an abstract infinite-dimensional KAM theorem and apply it to fractional nonlinear Schrödinger equation system with real Fourier multiplier. By establishing a block diagonal normal form, we prove the existence of a class of Whitney smooth family of small-amplitude quasi-periodic

The transmission of waves over a body composed of two different materials are considered in this paper. One component is a simple elastic part while the other is a viscoelastic component with time-varying delay and damping effect. Besides, one part of the boundary is controlled by a nonlinear memory source term while the other part of the boundary is clamped. Under the influence of time-varying delay

In this paper, we consider discrete semilinear wave equations u t t x , t = Δ ω u x , t + f u x , t , x , t ∈ S × 0 , + ∞ , u x , t = 0 , x , t ∈ ∂ S × 0 , + ∞ , u x , 0 = u 0 x , u t x , 0 = v 0 x , x ∈ S , where S is a network with boundary ∂ S . The main contribution of this work is to introduce a condition ( C ) α ∫ 0 u f s d s ≤ u f ( u ) + β u 2 + γ , u ∈ R , for some α > 2 and β , γ ≥ 0 with

Let Ω ∈ L s ( S n − 1 ) , with s ≥ 1 , be a homogeneous function of degree zero. In this paper, we establish the boundedness of the homogeneous fractional integral operator on weighted Herz spaces with variable exponent K ˙ p ( ⋅ ) α , q ( w ) .

This paper is dedicated to the study of micropolar fluids which is moving in the presence of a magnetic field in R 3 . We prove the global existence and uniqueness of strong solutions with initial data u 0 , w 0 , b 0 ∈ H σ 1 ( R 3 ) × H 1 ( R 3 ) × H σ 1 ( R 3 ) satisfying ∥ u 0 ∥ L 2 2 + ∥ w 0 ∥ L 2 2 + ∥ b 0 ∥ L 2 2 × ∥ ∇ u 0 ∥ L 2 2 + ∥ ∇ w 0 ∥ L 2 2 + ∥ ∇ b 0 ∥ L 2 2 + ∥ c u r l u 0 − 2 w 0 ∥

This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a reaction diffusion equation with special diffusion processes. It is shown, under certain conditions on the initial data, that the solutions to this problem blow up in finite time, by combining Hardy inequality, ‘moving’ potential well methods with some differential inequalities. Moreover, the upper

Synchronized stationary distribution and exponential synchronization analysis for stochastic memristor-based complex networks are investigated. It should be pointed out that aperiodically intermittent control is introduced to our models. Under the framework of the Lyapunov method and graph theory, the synchronized stationary distribution and the exponential synchronization of a class of complex networks

This paper deals with the blow-up phenomena of the solutions to an evolution heat equation with nonlinearities of variable exponent type. For a certain solution with positive initial energy, an upper bound for blow-up time is determined if the solutions blow-up.

The main aim of this paper is to obtain superconvergence result for a semilinear parabolic equation with 3-step backward differential formula Galerkin finite element method. The time-discrete system is established to split the error into the temporal error and spatial error. The initial two steps of the temporal error is dealt with by a new way to ensure the third-order accuracy of the scheme. Some

In this paper, a diffusive predator–prey model with Holling type II (Michaelis–Menten) functional response and a protection zone for prey is investigated. Dynamics and steady state solutions of the system are analyzed. We give the a priori estimates and obtain the nonexistence of nonconstant positive solution as the diffusion coefficients are large enough. Moreover, we demonstrate the existence and