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

In this paper, we study the spatial variable coefficient fractional convection–diffusion wave equation with the singe delay and multi-delay numerically when the exact solution satisfies a certain regularity. First, via the well-known exponential transformation, the delay problem can be greatly simplified, which allows us to use the variable coefficient four-order compact operator. Next, the numerical

We consider a reaction–diffusion equation in a domain consisting of two bulk regions connected via small channels periodically distributed within a thin layer. The height and the thickness of the channels are of order ε, and the equation inside the layer depends on the parameter ε and an additional parameter γ ∈ [ − 1 , 1 ) , which describes the size of the diffusion in the layer. We derive effective

In this paper, we first introduce the new fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces. Then, using the direct variational approach, we investigate the existence of a nontrivial weak solution for a class of fractional type problems with Dirichlet boundary data of the following form ( P a ) ( − Δ ) a ( x , . ) s u + a ˆ x ( | u | ) u = f ( x , u )

Fractional diffusion equations (FDEs) were shown to provide very competitive descriptions of challenging phenomena of anomalously diffusive transport or long-range interactions. However, FDEs introduce mathematical issues that are not common in the context of integer-order diffusion equations. For instance, the homogeneous Dirichlet boundary-value problems of linear elliptic FDEs with smooth data in

We provide a new analytical and computational study of the transmission eigenvalues with a conductive boundary condition. These eigenvalues are derived from the scalar inverse scattering problem for an inhomogeneous material with a conductive boundary condition. The goal is to study how these eigenvalues depend on the material parameters in order to estimate the refractive index. The analytical questions

In this article, we consider the stochastic tidal dynamics equations perturbed by multiplicative Gaussian noise and discuss some asymptotic behaviors. A central limit theorem and a moderate deviation principle are derived for such systems. The results are proved using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis.

A two-dimensional inclusion of core–shell structure is neutral to multiple uniform fields if and only if the core and the shell are concentric disks, provided that the conductivity of the matrix is isotropic. An inclusion is said to be neutral if upon its insertion the uniform field is not perturbed at all. In this paper, we consider inclusions of core–shell structure of general shape which are weakly

In this work, we first establish a local Carleman estimate for the ultrahyperbolic Schrödinger equation which arises in some theories of modern physics such as quantum mechanics and string theory. Next, we prove a Hölder stability estimate for the Cauchy Problem. In the proof, we introduce a suitable cut-off function and extend Cauchy data in a Sobolev space to reduce the problem to another problem

This paper deals with the initial-boundary value problem for the two-species chemotaxis-competition system with two signals ∂ t u 1 = Δ u 1 − χ 11 ∇ ⋅ ( u 1 ∇ v 1 ) − χ 12 ∇ ⋅ ( u 1 ∇ v 2 ) + μ 1 u 1 ( 1 − u 1 − a 1 u 2 ) , ∂ t u 2 = Δ u 2 − χ 21 ∇ ⋅ ( u 2 ∇ v 1 ) − χ 22 ∇ ⋅ ( u 2 ∇ v 2 ) + μ 2 u 2 ( 1 − u 2 − a 2 u 1 ) , 0 = Δ v 1 − λ 1 v 1 + α 11 u 1 + α 12 u 2 , 0 = Δ v 2 − λ 2 v 2 + α 21 u 1 +

This article concerns the following class of nonlocal nonhomogeneous elliptic problems: − ∫ Ω g ( x , u ) d x β div ⁡ ( a ( x ) ∇ u ) = ( g ( x , u ) ) α i n   Ω , u = 0 o n   ∂ Ω , where Ω is a bounded domain of R n ( n ≥ 1 ) , α, β ∈ R , a ( x ) is a matrix with variable coefficients and g : Ω × R → R is a positive function which are defined almost everywhere with respect to the variable x. By using

A system governed by a one-dimensional hyperbolic equation with a mixing transport term and both ends being general nonlinear boundary conditions is considered in this paper. By using the snap-back repeller theory, we rigorously prove that the system is chaotic in the sense of both Devaney and Li-Yorke when the system parameters satisfy certain conditions. Finally, numerical simulations are further

The aim of this paper is to study backward stochastic differential equation driven by a class of normal martingale. We prove the existence and uniqueness results of the solution under the stochastic Lipschitz condition by mean the martingale representation theorem. In addition, we show that the solution of these equations provides a viscosity solution of the associated system with partial differential

A new inclusion problem is introduced using generalized Cayley operator and we call it Cayley inclusion problem. We also study its corresponding resolvent equation problem. By using a generalized resolvent operator and generalized Yosida approximation operator, first we establish a fixed point formulation for Cayley inclusion problem. An algorithm is defined to find the solution of Cayley inclusion

The paper concerns a fractional Schrödinger–Poisson system involving Hardy potentials. By using Nehari manifold method, we prove the existence and asymptotic behaviour of ground state solution.

We provide an in-depth exploration of the mass-transport properties of Pollard's exact solution for a zonally propagating surface water-wave in infinite depth. Without resorting to approximations we discuss the Eulerian mass transport of this fully nonlinear, Lagrangian solution. We show that it has many commonalities with the linear, Eulerian wave-theory, and also find Pollard-like solutions in the

In this paper, we consider a cavity reconstruction problem for the interior acoustic scattering from limited-aperture measurements. To recover the shape of the cavity, the Bayesian inference technique is applied with the information of posterior distribution of the unknown object being explored in terms of the measured data. The posterior distribution provides us with sufficient knowledge about the

In this paper, we study the convergence of a coefficient inverse problem for the two dimensional semi-discrete damped wave equation. Based on the new discrete Carleman estimates for the semi-discrete wave operator, the uniform stability estimate is proved. Then we prove the convergence of the semi-discrete scheme with respect to the discretization grid step h>0.

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped σ-evolution models with different power nonlinearities. By assuming additional L m regularity on the initial data, with m ∈ [ 1 , 2 ) , we use ( L m ∩ L 2 ) − L 2 and L 2 − L 2 estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data

This paper is concerned with the global stability of traveling waves for non-monotone infinite-dimensional lattice differential equations with time delay. By establishing the boundedness estimate of the solution of the perturbation equation, we obtain that, for any initial perturbations around the traveling wave, the noncritical traveling waves are globally stable with the exponential convergence rate

This paper solves a question raised in the paper of Huong, Yao, and Yen [Optimal processes in a parametric optimal economic growth model. Taiwanese J Math. 2020. Available from: https://doi.org/10.11650/tjm/200203] about optimal economic growth problems with production functions and utility functions being all in the form of AK functions. By using a solution existence theorem from the paper of Huong [Solution

We study homogenization of a locally periodic two-scale dual-continuum system where each continuum interacts with the other. Equations for each continuum are written separately with interaction terms added. The homogenization limit depends strongly on the scale of this continuum interaction term with respect to the microscopic scale. In J. S. R. Park and V. H. Hoang, Hierarchical multiscale finite

In this paper, we introduce a Hausdorff-type distance relative to an ordering cone between two sets. We obtain some properties of the Hausdorff-type distance. In particular, we give a characterization of the Hausdorff-type distance. Moreover, we introduce the Clarke generalized directional derivative for set-valued mappings by using the nonlinear scalarizing function for l-type less order relation

Let T be an anisotropic Calderón-Zygmund operator and b ∈ L i p α , w ( R n , A ) with 0 < α < 1 and L i p α , w ( R n , A ) being an anisotropic weighted Lipschitz space. The goal of the paper is to give five boundedness theorems of the commutator [ b , T ] . Precisely, [ b , T ] is bounded from L w q ( R n ) to L w 1 − q r q r ( R n ) , where w ∈ A q r ( A ) , 1 / q r = 1 / q − α / n , 1 < q < n

We study a mean–variance acreage model, A = α ( E [ p ] , V [ p ] ) , where p is price at harvest time and E and V are the expectation and variance operators conditional on information known at planting time. Under the assumption that p = π ( A y ) where yield y is random and unknown at planting time, we will investigate the existence, uniqueness, and convergence of this fixed point problem as well

In this paper, the existence of the time-periodic solution to the time-dependent Ginzburg–Landau(TDGL) equations in the case of BCS–BEC crossover is established relying on the Leray–Schauder fixed point theorem.

We derive Lipschitz stability estimates for the Hausdorff distance of polygonal conductivity inclusions in terms of the Dirichlet-to-Neumann map.

We study a mixed discontinuous Galerkin (MDG) method for solving a time-dependent Darcy problem, which simulates an incompressible fluid such as water flowing in a rigid porous medium. The discretization of the unsteady Darcy problem relies on a backward Euler scheme for temporal variable and MDG method for spatial variables. Spatially semi-discrete and fully discrete schemes are analyzed. Existence

In this paper, we consider the problem of finding the source distribution for the linear inhomogeneous biparabolic equation when the measured output data is given in the form of the final overdetermination. The problem is severely ill-posed in the sense of Hadamard. First, we apply a general filter method to regularize the linear nonhomogeneous problem. Then, we give a regularized solution and consider

The paper deals with the optimal control of second-order viability problems for differential inclusions with endpoint constraint and duality. Based on the concept of infimal convolution and new approach to convex duality functions, we construct dual problems for discrete and differential inclusions and prove the duality results. It seems that the Euler–Lagrange type inclusions are ‘duality relations’

ABSTRACT We discuss a mathematical model for the equatorial current across the Pacific Ocean, obtained as a leading-order solution to the Navier-Stokes governing equations for geophysical flows in a rotating frame.

In this paper, we classify each pair of positive solutions of the following general integral system on half space R+n u(y)=∫R+n|x−y|νf(v)(x)dx,y∈∂R+nv(x)=∫∂R+n|x−y|νg(u)(y)dy,x∈R+n. The main technique we used is the method of moving sphere.

In this article, we consider the following new Kirchhoff-type problem: −a−b∫Ω|∇u|2dxΔu=|u|p−2,in Ω,u=0,on ∂Ω, where a and b are positive constants, Ω⊂RN is a bounded domain with C1 boundary ∂Ω, p∈[2,2∗) with 2∗=2N/(N−2) if N≥3, and 2∗=+∞ if N = 1, 2. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik–Schnirelman

To derive the increasing stability of the source term in the Helmholtz equation from local boundary data, we utilize sharp bounds of the analytic continuation for higher wave numbers, the Huygens' principle, and bounds in the lateral Cauchy problem for the wave equation.

We consider a class of severely degenerate elliptic operators and establish a Harnack inequality up to the boundary. As a consequence, we get smoothness of the weak solutions under very sharp assumptions regarding the lower order terms of the operator.

We deal with a class of magnetic pseudo-relativistic Hartree type equation (−i∇−A(x))2+m2u+W(x)u=(Iα∗|u|p)|u|p−2u,x∈RN, where N≥3, m>0, A:RN→RN is a continuous vector potential, W:RN→R is an external continuous scalar potential and Iα(x)=(cN,α/|x|N−α)(x≠ 0) is a convolution kernel, cN,α>0 is a positive constant, 2≤p<2N/(N−1), (N−1)p−N<α

The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential: ∂tu(x,t)−div(p(x)∇u(x,t))−(μ/|x|2)u(x,t)=f(x,t). Here μ is a real constant. It was proved in the paper of Goldstein and Zhang (2003) that the equation is well-posedness when 0≤μ≤p1(n−2)2/4, and in this paper, we mainly consider

This paper is concerned with the stabilization problem for nonlinear stochastic delay systems with Markovian switching by feedback control based on discrete-time state and mode observations. By constructing an efficient Lyapunov functionals, we establish the sufficient stabilization criteria not only in the sense of exponential stability (both the mean-square stability and the almost sure stability)

In this paper, we analyze acoustic wave propagation in anisotropic fluids and solids. By formulating the acoustic system as an evolution equation over a Hilbert space, we obtain global in time solutions when the associated material parameters are bounded and measurable. In particular, we prove well-posedness of a Cauchy problem for wave propagation in piezoelectric crystals. We then provide a stability

In this paper we investigate the unilateral problem for a plate equation with memory terms and lower order perturbation of p-Laplacian type u″+Δ2u−Δpu+∫0tg(t−s)Δu(s)ds+Δu′+f(u)=0in Ω×R+, where Ω is a bounded domain of Rn, g>0 is a memory kernel and f(u)is a nonlinear perturbation. Using the penalty and Faedo–Galerkin's methods, we establish results on existence and uniqueness of weak solutions.

In this paper, the two-dimensional incompressible chemotaxis fluid with logical source is studied as following: nt+u⋅∇n=Δn−∇⋅(n∇c)+n−n2,ct+u⋅∇c=Δc−nc,ut+u⋅∇u+∇P=Δu−n∇φ,∇⋅u=0. By taking advantage of a coupling structure of the equations and using a scale decomposition technique, the global existence and uniqueness of weak solutions to the above system for a large class of initial data is obtained.

In this research paper, we give some new results for the existence and uniqueness of doubly measure pseudo almost periodic (Or (μ,ν)-pap) solutions for some differential equations with reflection. We will use the Banach fixed-point theorem and some properties of pseudo almost periodic functions with measurements and we will discuss both linear and nonlinear cases. Finally, we finish with some applications

In this paper, we study the long-time behavior of the chemotaxis-Navier–Stokes system ∂tn+u⋅∇n=λΔn−∇⋅(χ(c)n∇c),∂tc+u⋅∇c=μΔc−f(c)n,∂tu+u⋅∇u+∇P=ζΔu−n∇φ,∇⋅u=0,t>0,x∈Ω posed in a strip domain Ω:=R2×[0,1]⊂R3. In Peng-Xiang (Math. Models Methods Appl. Sci., 28 (2018), 869-920), the authors have established the global existence of strong solutions to this system with non-flux boundary conditions for n and

In this paper, we address an inverse problem of reconstructing a space-dependent semilinear coefficient in the tumor growth model described by a system of semilinear partial differential equations (PDEs) with Dirichlet boundary condition using boundary-type measurement. We establish a new higher order weighted Carleman estimate for the given system with the help of Dirichlet boundary conditions. By

In this paper, we study the following fractional Choquard equation with critical or supercritical growth (−Δ)su+V(x)u=[|x|−μ∗F(u)]f(u)+λ[|x|−μ∗|u|p]p|u|p−2u,x∈RN, where 02s, 0<μ<2s and p≥2μ,s∗:=(2N−μ)/(N−2s). Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ>0 by the variational method.