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.

We study, in this article, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two- or three-dimensional bounded domain. The model consists of the Navier–Stokes equations for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. These equations are motivated by the dynamic of binary fluids under the influence of stochastic external forces. We prove the existence

We consider a parametric nonlinear elliptic problem driven by the sum of a p-Laplacian and of a q-Laplacian (a (p,q)-equation) with a singular and (p−1)-superlinear reaction and a Robin boundary condition with (q−1)-sublinear boundary term (q

ABSTRACT We study the inverse problem in Optical Tomography of determining the optical properties of a medium Ω⊂Rn, with n≥ 3, under the so-called diffusion approximation. We consider the time-harmonic case where Ω is probed with an input field that is modulated with a fixed harmonic frequency ω=k/c, where c is the speed of light and k is the wave number. We prove a result of Lipschitz stability of

A composite specimen, made of two slabs and an interface A is heated through one of its sides S, in order to evaluate the thermal conductance H of A. The direct model consists of a system of Initial Boundary Value Problems completed by suitable transmission conditions. Thanks to the properties of multilayer diffusion, we reduce the problem to the slab between A and S only. In this case evaluating the

The main objective of this paper is to provide various estimates of the first eigenvalue of the p-Laplacian operator on a closed oriented n-dimensional C-totally real submanifold Σn in a simply connected Sasakian space form M~2m+1(κ) with a constant ϕ-sectional curvature κ. As applications, we generalize the Reilly-type inequality for the Laplacian [Chen and Wei. Reilly-type inequalities for p-Laplacian

In this paper, we mainly study the Cauchy problem of a generalized Camassa–Holm equation. We prove the existence of global weak solutions for this generalized Camassa–Holm equation by the viscous approximation method.

The paper studies a system of two singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory, with general source terms and nonlocal boundary condition. We prove the existence of a global solution to the problem using the potential-well theory. Furthermore, we construct a Lyapunov functional and use it together with the perturbed energy method to prove

This paper considers the scattering of time-harmonic acoustic plane waves by a mixed scatterer. Such a scatterer is given as the union of a penetrable homogeneous medium with compact support and an open arc. Firstly, the well-posedness of the direct problem is established by the boundary integral method. Then, we study the factorization method for recovering the shape of the mixed scatterer. Furthermore

We consider an interior inverse scattering problem of reconstructing the shape of an elastic cavity. We prove a reciprocity relation for the scattered elastic field and a uniqueness theorem for the inverse problem. Then we employ the decomposition method to determine the boundary of the cavity and present some convergence results. Numerical examples are provided to show the viability of the method

This paper is concerned with a stochastic multigroup S-DI-A model for the transmission of HIV with general incidence rate. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a proper stochastic Lyapunov function. The stationary distribution implies that the diseases will be persistent in the long

In this paper, we investigate the energy decay for solutions of the weakly coupled dissipative Schrödinger system. Among the m-coupled equations, only one equation is directly damped. Under some assumptions about the damping and the coupling terms, it is shown that sufficiently smooth solutions of the system decay logarithmically with mixed boundary conditions, including the coupling of the Schrödinger

This work deals with the blow-up criterion of local smooth solution to the 3D Hall-magnetohydrodynamics equations in Besov spaces B˙∞,∞0.

The paper deals with a numerical method for solving a monotone variational inequality problem in a Hilbert space. The algorithm is inspired by Popov's modified extragradient method and the Bregman projection with a simple stepsize rule. Applying Bregman projection allows the algorithm to be more flexible in computations when choosing a projection. The stepsizes, which vary from step to step, are found

We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical

Enforcing Gauss' law in numerical simulations of Maxwell's equations, specially in plasma physics, is a well-known key issue. Although the set of Maxwell's equations used for simulations do not include divergence laws, augmented systems which incorporates Gauss' law while retaining an overall hyperbolic form are frequently used to cope with charge conservation defects. Focusing on bounded domains and

The paper studies the well-posedness and the longtime behaviors of the fractional damping wave equation on R3 with critical nonlinearity, we firstly prove the well-posedness of weak solutions via an established Strichartz estimate; Secondly, by the uniformly tail estimate and the energy identity, we prove that the corresponding semigroup possesses a global attractor in the energy space H1(R3)×L2(R3)

Our objective with this paper is to discuss multi-switching problems, arising as variational inequalities, that models decision under uncertainty. We prove general existence theory through monotone scheme and discuss iterative methods for numerical results. We also connect the recently developed models for asset bubbles (which is a non-local problem) to switching problems with two possible switching

We are interested in the asymptotic behavior of ground states for a class of quasilinear elliptic equations in R3 when the nonlinear term has H1-critical growth. In the previous result [Adachi et al. Asymptotic property of ground states for a class of quasilinear Schrödinger equation with H1-critical growth. Calc Var Partial Differential Equations. 2019;58(3). Art. 88, 29 pp.], it was shown that, after

In this paper, we establish the existence and uniqueness of invariant measures of the 3D stochastic magnetohydrodynamic-α model (MHD-α) driven by degenerate additive noise. We firstly study the Feller property of solutions and establish the existence of invariant measures by utilizing the classical Krylov–Bogoliubov theorem. Then, we prove the uniqueness of invariant measures for the corresponding

An initial-boundary value problem of Saut–Temam type for the modified Zakharov–Kuznetsov equation posed on a strip is considered. Critical power in nonlinearity has been studied. The results on existence, uniqueness and asymptotic behavior of solutions are presented.

In this paper, we consider a problem of a three-dimensional free surface flow over an obstacle lying on the bottom of an infinite channel. The flow is supercitical, irrotational, stationary and the fluid is ideal and incompressible. We linearize the problem, then by a Fourier expansion of the velocity potential, we reduce the three-dimensional problem to a sequence of two-dimensional problems for the

We consider a nonlinear shallow water equation. An inequality condition for wave breaking has been established for the Cauchy problem with periodic domain. Numerical computation confirms the result, as breaking happens for waves with a chosen initial profile satisfying the theoretical condition for wave breaking.

We study the asymptotic behavior of Bresse system with non-dissipative kernel memory acting only in the equation of longitudinal displacement. We show that the exponential stability depends on conditions regarding the decay rate of the kernel and a new relationship between the coefficients of the system. Moreover, this new condition on the constants of the system is used to prove strong stability and

Considered herein is the initial value problem for the periodic two-component Novikov system. It is shown that the solution map of this problem is not uniformly continuous in Besov spaces B2,rs(T)×B2,rs−1(T) with s>3/2, 1≤p<∞. Furthermore, the non-uniform continuous dependence on initial data is established in Besov space B2,13/2(T)×B2,11/2(T). Based on the well-posedness result and the lifespan for

In the paper, we consider a coefficient inverse problem for the heat equation in a degenerating angular domain. It has been shown that the inverse problem for the homogeneous heat equation with homogeneous boundary conditions has a nontrivial solution up to a constant factor consistent with the integral condition. Moreover, the solution of the considered inverse problem is found in explicit form. In

ABSTRACT In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, imaging and engineering. We show the regularizing properties of the new method, as well as the

A blow-up analysis for a nonlocal reaction-diffusion system with time-dependent coefficients is investigated under null Dirichlet boundary conditions. Based on the Kaplan's method, comparison principle and modified differential inequality technique, we establish a blow-up criteria and derive the bounds for the blow-up time under the appropriate measures in whole-dimensional space.

In this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in Rn with initial data in some critical Besov spaces. Under the assumptions that initial density is close to the equilibrium state and the smallness on the initial direction field and nonlinear smallness on the linearized velocity, we obtain the global existence of solutions by fully

This paper concerns the mathematical analysis of a mathematical model for price formation. We take a large number of rational buyers and vendors in the market who are trading the same good into consideration. Each buyer or vendor will choose his optimal strategy to buy or sell goods. Since markets seldom stabilize, our model mimics the real market behavior. We introduce three models. All of them are

This paper is concerned with the stability of traveling wave fronts of a delayed Belousov–Zhabotinskii model with spatial diffusion. The existence and comparison theorem of solutions of the corresponding Cauchy problem in a weight space are established for the system on R by appealing to the theories of semigroup and abstract functional differential equations. By means of comparison principle and the

Let L={L(t,x),t≥0,x∈R} be the local time of a G-Brownian motion B on a sublinear expectation space (Ω,H,Eˆ). In this paper, we show that the local time L is a rough path of roughness p quasi-surely for any 2

This paper presents asymptotic formulas in the case of the following two problems for the Pucci's extremal operators M±. It is considered the solution uε(x) of −ε2M±∇2uε+uε=0 in Ω such that uε=1 on Γ. Here, Ω⊂RN is a domain (not necessarily bounded) and Γ is its boundary. It is also considered v(x,t) the solution of vt−M±∇2v=0 in Ω×(0,∞), v = 1 on Γ×(0,∞) and v = 0 on Ω×{0}. In the spirit of their

We consider two final value problem of a stochastic strongly damped wave equation driven by white noise (Section 3) and fractional noise (Section 4). We show that a stochastic integral in the solution is not stable and the problem is not well posed. To regularize the problem in two cases of noise, we apply the Fourier truncated method to control the frequency in such a way that it depends on the noisy

We study the local behavior of a solution to a generalized non-stationary Stokes system with singular coefficients in Rn with n≥ 2. One of our main results is a bound on the vanishing order of a non-trivial solution u satisfying the generalized non-stationary Stokes system, which is a quantitative version of the (strong) unique continuation property for u. Different from the previously known results

This paper studies the controllability of a coupled system for a class of impulsive fractional integro-differential inclusions involving φ-Hilfer fractional derivative and subject to coupled nonlocal integral initial conditions in the case of convex set-valued maps. Some auxiliary conditions are introduced in order to apply a fixed point theorem due to Bohnenblust–Karlin. An illustrative example is

In this paper, we study the retrieval of water vapor profiles from simulated FORUM measurements. We show that the bias towards the a-priori introduced by the Optimal Estimation technique can be reduced by using larger errors for the a-priori. Reducing the strength of the a-priori may, however, cause unphysical oscillations in the resulting profiles because of the ill-conditioning of the retrieval problem

This paper concerns with a reaction–diffusion susceptible-vaccinated-infectious-recovered model with a fixed latent period. The model is formulated as a non-local and time-delayed reaction–diffusion model due to the fact that an individual infected by the disease in one place may not stay at the same space in the domain due to the movement of human during the incubation period. We then derive the basic

In this paper, we investigate a semi-discrete finite-element approximation of nonlocal hyperbolic problem. A priori error estimate for the semi-discrete scheme is derived. A fully discrete scheme based on backward difference method is constructed. We discuss the existence-uniqueness of the solution for fully discrete problem. In order to linearize the nonlinear fully discrete problem, we use Newton's

We prove the existence and uniqueness of a classical solution to a multidimensional non-potential stochastic Burgers equation with Hölder continuous initial data. Our motivation is the adhesion model in the theory of formation of the large-scale structure of the universe. Importantly, we drop the assumption on the potentiality of the velocity flow that has been questioned in physics literature.

We consider nonnegative solutions to the Cauchy problem for a degenerate parabolic system of the form ut=div⁡(vα1∇um1),(x,t)∈ST=RN×(0,T)vt=div⁡(uα2∇vm2),(x,t)∈STu(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈RN,N≥1. Under the suitable assumptions on the parameters of nonlinearities and initial data, we obtained optimal decay estimates of a solution for a large time. Moreover, the phenomena of extinction in finite

This paper is devoted to investigating regularity criteria for the nematic liquid crystal system in terms of one derivative component of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution (u,d) can be extended beyond T, provided that one derivative component of the pressure ∂x3P and gradient of the orientation field satisfy ∂x3P∈Ls(0,T;Lq(R3))

The present paper is dedicated to the optimal time-decay estimates of global strong solutions near constant equilibrium (away from vacuum) to the compressible magnetohydrodynamic (MHD) equations in the Lp critical Besov spaces. In which we claim a new low-frequency assumption that plays a key role in the large-time behavior of solutions. Precisely, we exhibit that if the low frequencies of initial

We derive upper and lower estimates of the area of unknown defects in the form of either cavities or rigid inclusions in Mindlin–Reissner elastic plates in terms of the difference δW of the works exerted by boundary loads on the defected and on the reference plate. It turns out that the upper estimates depend linearly on δW, whereas the lower ones depend quadratically on δW. These results continue

In this article, the Ekeland variational principle, mountain pass theorem and Trudinger–Moser inequality are applied to establish the existence and multiplicity of solutions for the following nonhomogeneous Kirchhoff-type elliptic problem: −m∫Ω|∇u|2dxΔu=f(x,u)+ϵh(x)in Ω;u=0on ∂Ω, where Ω is a smooth bounded domain in R2, m:R+→R+ is a Kirchhoff function, f:Ω×R→R is a continuous function, h∈W01,2(Ω)∗=W−1

We consider a semilinear differential variational inequality P in reflexive Banach spaces, governed by a set of constraints K. We associate to P a sequence of problems {Pn} where, for each n∈N, Pn is a differential variational inequality governed by a set of constraints Kn and a penalty parameter ρn. We use a result in [Liu ZH, Zeng SD. Penalty method for a class of differential variational inequalities