We show the existence of positive minimal energy solutions for the equation −div(K(x)∇u)=K(x)f(u) in R2, where K(x)=exp(|x|2∕4) and f:R→R is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method.

We investigate the global well-posedness to the three dimensional chemotaxis-Navier–Stokes equations with a class of large initial data slowly varying in one direction. The proof uses the special coupling structure of the system and the localization technique in Fourier spaces.

A regularized method of moments based on the modified fundamental solution of the Helmholtz equation is proposed in this article. The regularized method of moments uses the origin intensity factor technique which is free of mesh and integration to deal with the singularity at origin of the basis function. Thus, the time-consuming singular integration can be avoided. In addition, the non-uniqueness

In this paper, solutions of a Novikov equation are discussed based on the bifurcation method of dynamical systems. Through establishing a Hamiltonian function, the existence of a smooth soliton solution and a periodic cuspon solution are established for the corresponding traveling wave system of the Novikov equation. Numerical results are carried out to illustrate the feasibility of the main results

Shadow systems are an approximation of reaction–diffusion-type models with the largest diffusion coefficient tending to infinity. They are often used to reduce the model and facilitate its analysis. In this paper we extend the already existing finite time interval analysis to a case of very long time intervals, i.e., time intervals scaled with the diffusion coefficient and tending to infinity for diffusion

We consider a quasilinear Keller–Segel system with density-dependent migration rates, coupled to the incompressible Stokes equations through transport and buoyancy. By means of an apparently novel approach based on certain conditional estimates for the taxis gradient and the fluid field, for diffusion rates asymptotically controllable by power-tape majorants and minorants a result on global existence

A new modified nonstandard finite difference method for solving one-dimensional autonomous differential equations is constructed and analyzed. It is based on the theta method and is both elementary stable and of second-order accuracy. A set of numerical simulations is also presented that supports the theoretical results.

In this paper, a family of arbitrarily high-order structure-preserving exponential Runge–Kutta methods are developed for the nonlinear Schrödinger equation by combining the scalar auxiliary variable approach with the exponential Runge–Kutta method. By introducing an auxiliary variable, we first transform the original model into an equivalent system which admits both mass and modified energy conservation

In this paper, a novel (3+1)-dimensional sin-Gorden and sinh-Gorden eqaution are derived. These two equations are derived for the first time from the extended (3+1) dimensional zero curvature equation, using the compatibility condition. Then the infinitesimal transformation of this equation is studied from the symmetry point of view. Meanwhile, it turns out that this equation can be reduced to the

This paper is concerned with Hilfer fractional stochastic delay differential equations with Poisson jumps. Under some assumptions, we obtain an averaging principle for the solution of the considered system. The required results are obtained based on fractional calculus, Doob’s martingale inequality and Cauchy–Schwarz inequality.

In this work, a modified regularized lattice Boltzmann model is constructed for convection–diffusion equation (CDE) with a source term. By modifying the nonequilibrium part of the total distribution function, the present model can recover the target CDE with no deviation term through the Chapman–Enskog analysis. We then test the present model by considering a CDE with a constant velocity. Based on

In this article, a new algorithm based on improved Legendre orthonormal basis for solving second-order BVPs is constructed. The method of this article is an improved Legendre orthogonal polynomial algorithm, and the ε-best approximate solution is constructed skillfully in the reproducing kernel space. Compared with other methods, this method has the advantages of high approximation, m-order convergence

In this paper, a localized version of the method of fundamental solutions (MFS) named as the localized MFS (LMFS) is applied to two-dimensional (2D) harmonic elastic wave problems. Due to the dense and ill-conditioned coefficient matrices, the traditional MFS is computationally expensive and time-consuming to solve the above-mentioned problems which require a large amount number of nodes to obtain

This study proposes a two-and-a-half-dimensional (2.5D) singular boundary method (SBM) to solve three dimensional (3D) acoustic problems with a constant cross section excited by harmonic point sources. The SBM expands the solution of a problem with a linear combination of fundamental solutions of the governing equation. The singular terms in the fundamental solutions are replaced by the origin intensity

In this paper, we construct and analyze a stochastic predator–prey system where apart from direct predation the prey population is affected by the fear induced from predators. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological interpretation

Depending on the range of Mach numbers, we consider the various solutions of piston problem for generalized Chaplygin Euler equations. Moreover, under the Mach number tends to infinity, the convergence of solutions and degeneration of equations are analyzed when the piston recedes from tube.

This paper develops a gradient descent (GD) method for solving a system of nonlinear equations with an explicit formulation. We theoretically prove that the GD method has linear convergence in general and, under certain conditions, is equivalent to Newton’s method locally with quadratic convergence. A stochastic version of the gradient descent is also proposed for solving large-scale systems of nonlinear

We study the effect of having a reflecting boundary condition in a superdiffusive model. Firstly it is described how the problem formulation is affected by this type of physical boundary and then it is shown how to implement an implicit numerical method to compute the numerical solutions. The consistency and stability analysis of the numerical method are discussed. In the end numerical experiments

In this paper, we prove the existence of a positive periodic solution for nonlinear differential equation with an indefinite singularity by using Green’s function and fixed point theorem in cones. The results are applicable to weak as well as strong singularities. At last, we give one example to show the application of the theorem.

By utilizing the Legendre polynomials, we first establish a broken reproducing kernel space which is provided with piecewise high order polynomials. Then a new efficient collocation method based on the reproducing kernel is proposed for solving the general 1-D interface problems. By considering a system of algebraic linear equations, we obtain an approximate solution. Our main contribution is that

Under investigation in this paper is the AB system, which is used to describe the baroclinic instability processes in the geophysical flows. With the generalized Darboux transformation, the Nth-order (N=1,2,3,…) rogue wave solutions of the AB system are obtained via the determinants. It is found that the rogue waves can be split into several small ones with the proper parameter selections, and moreover

An element-free Galerkin (EFG) method is presented for the numerical solution of the obstacle boundary value problem. To deal with the nonlinear inequality governing equations in the problem, a linearization technique is developed. Then, linear boundary value problems are formed and are solved by a stabilized EFG method. Convergence of the presented meshless method is discussed mathematically. Numerical

The paper aims to explore the existence of diverse lump and interaction solutions to the (1+1)-dimensional Sharma–Tasso–Olver–Burgers equation. Through the Cole–Hopf transformation with the nonzero seed solution, the remarkable richness of exact solutions are exhibited, including lump, lump-periodic, lump-multiple solitary wave and periodic-multiple solitary wave. Some specific interaction phenomena

In the paper, we revisit the uniform boundedness of the exact solution and then study pointwise error estimate for two kinds of conservative numerical discretization schemes. We show that both difference schemes share the conservative invariants similar to the original continuous model for all positive integers p. In particular, we prove that difference schemes are convergent in pointwise sense based

We present a new regularity criterion for suitable weak solutions of the one-dimensional surface growth initial-value problem in terms of mixed Lebesgue norms.

Calogero-Bogoyavlenskii-Schiff-type (CBS-type) equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Under investigation in this paper is a combined CBS-type equation. Lax pairs in the differential and matrix forms are derived respectively. Infinite conservation laws, which are different from those in the existing literatures, and n-fold Darboux transformation are

This paper proposes a fast and efficient spectral-Galerkin method for the nonlinear complex Ginzburg–Landau equation involving the fractional Laplacian in Rd. By employing the Fourier-like bi-orthogonal mapped Chebyshev function as basis functions, the fractional Laplacian can be fully diagonalized. Then for the resulting diagonalized semi-discrete system, an exponential time differencing scheme is

We consider a chemotaxis system with indirect signal production and rotational sensitivity: ∂tu=Δu−∇⋅(S(x,u,v,w)⋅∇v)+f(u),(x,t)∈Ω×(0,∞),∂tv=Δv−v+w,(x,t)∈Ω×(0,∞),∂tw=Δw−w+u,(x,t)∈Ω×(0,∞)in a smooth bounded domain Ω⊂RN(N≥2), where S(x,u,v,w)∈RN×N is a matrix-valued function, f is a smooth function and satisfies f(0)≥0 as well as f(s)≤κ−μsα for all s≥0, where κ≥0, μ>0 and α>1. For all sufficiently smooth

We present a new numerical method for solving a wide class of time fractional partial differential equations with a space operator of general form. The proposed method is based on the Fourier series expansion along the spatial coordinate which transforms the original time fractional partial differential equations into a sequence of multi-term fractional ordinary differential equations. The fractional

In this paper we discuss the classical problem of finding conditions that the entire coefficients A(z) and B(z) should satisfy to ensure all nontrivial solutions of f′′+A(z)f′+B(z)f=0 are of infinite order. Two different approaches are used. In the first approach the entire coefficient B(z) has dynamical property with a multiply connected Fatou component, and A(z) is extremal for Yang’s inequality

We continue the study of the regularizing effect of the interaction between the coefficient of the zero order term and the datum in the semilinear Dirichlet problem u∈W01,2(Ω):−div(M(x)∇u)+a(x)g(u)=f(x),where Ω is a bounded open set of RN, M is a bounded elliptic matrix, 0≤a(x)∈L1(Ω) and g is a continuous odd increasing function. Even if f(x) only belongs to L1(Ω), the assumption thereexistsQ∈(0,l

We consider a class of non-linear integro-differential equations describing microfluidic measurements. We prove that under reasonable conditions, solutions to these equations are convex functions of a flow-rate parameter ξ used in metrology. The key elements of our analysis are: (i) elevation of ξ to an independent variable through reformulation of the problem as a partial-differential equation (PDE);

In this paper, we are interested in the eigenproblem on the large and low-rank matrix S=ABH, where A,B∈ℂn×r are of full column rank and r≪n. To the best of our knowledge, there are no results on the relations between the Jordan decomposition and the Schur decomposition of BHA and those of ABH. Some known results are only on characteristic polynomials, elementary divisors, and Jordan blocks of ABH,

It is now known that, under a weaker condition than the Lipschitz one, the existence and uniqueness result for solutions of stochastic differential equations driven simultaneously by a fractional Brownian motion with Hurst parameter H>1∕2 and a standard Brownian motion is obtained. Therefore, it is quite natural to ask whether or when the averaging principle result also holds. In this paper we study

This paper presents the exact solutions of the homogeneous and the non-homogeneous fractional delay oscillation equations with order α∈(1,2). For non-homogeneous fractional delay oscillation equation, the Laplace transform is taken up as solving method in that it efficiently avoids some restrictions to the non-homogeneous term. Thereafter, we investigate the Hyers–Ulam stability of the fractional delay

The long wavelength limit for the pressureless two-fluid Euler–Poisson system in three-dimensional whole space case is discussed in the present paper. It is shown that the smooth solutions for the two-fluid Euler–Poisson system converge strongly to those of the Zakharov–Kuznetsov equation in an O(ε−3∕2) time interval by the Gardner–Morikawa transform and the elaborate energy method.

We study the blow-up result of a quasilinear von Karman equation of memory type with acoustic boundary conditions. For the asymptotic behavior of solution to the von Karman equation, many authors have been considered. We prove the finite time blow-up result of solution under suitable condition on the initial data.

We completely solve the problem of representing general solution to the cyclic bilinear system of difference equations in terms of a sequence naturally appearing in solvability of linear difference equations, by extending and complementing some previous results in this journal.

The two-parameter differential equation u′′(x)+λ(1−u)2−λε2(1−u)4=0 with the boundary condition u(−1)=u(1)=0 governs the steady-state solutions from a regularized MEMS model. We prove that there exist two constants εˆ(≈0.25458) and εˇ(≈0.29212) such that the bifurcation curve is S-shaped for 0<ε⩽εˆ and is strictly increasing for ε⩾εˇ in the λ,‖u‖∞-plane. This partly confirms the numerical simulations

This paper is concerned with the oscillatory behavior of solutions of the nth-order nonlinear differential equations with an advanced argument x(n)(t)+δq(t)xλ(g(t))=0,t≥t0>0, where n≥2 is even or odd and δ=±1. Here q(t)>0, g(t)≥t, and λ is the ratio of odd positive integers. The results are illustrated with an example.

In this paper, we present a fully discrete and structure-preserving scheme for the nonlinear fractional Schrödinger equations. The key is to introduce a scalar auxiliary variable and rewrite the equations as a new family of systems. The new systems are approximated by using the implicit midpoint rule, the repeated trapezoidal rule and the fractional centered difference method. It is shown the fully

The paper considers predator prey models with consuming resource and prey-taxis subject to homogeneous Neumann boundary conditions. For small prey-taxis coefficients, global in time solutions are proved in any space dimension. The homogeneous steady state remains global stability for weak attractive prey-taxis sensitivity, and pattern formation can be generated for repulsive prey-taxis sensitivity

In this letter, inspired by the idea of Ablowitz and Musslimani, we propose and investigate a discrete nonlocal reverse-space coupled Ablowitz–Ladik equation. Based on its Lax pair, we construct discrete nonlocal version of the generalized (m,N−m)-fold Darboux transformation for this system. Using the resulting Darboux transformation, we can obtain its different types of exact solutions including breathing-soliton

This paper generalizes the existing convergence result of Euler method for the overdamped generalized Langevin equation with fractional noise by Fang and Li (2020). In particular, several new techniques are developed.

The defocusing nonlinear Schrödinger (NLS) equation has no the modulational instability, and was not found to possess the rogue wave (RW) phenomenon up to now. In this paper, we firstly investigate some novel nonlinear wave structures in the defocusing NLS equation with real-valued time-dependent and time-independent potentials such that the stable new RWs and W-shaped solitons are found, respectively

This paper deals with a pseudo-parabolic equation involving p(x)-Laplacian, which has been studied in Di et al. (2017), Zhu et al. (2020) and Liao (2019). By the variational methods and embedding inequalities, we determine the complete classification of blow-up and global existence of weak solutions in some Sobolev space with variable exponent by covering all kinds of initial energy characterized with

This article is concerned with a stochastic different equation which describes the evolution of a species in the presence of Allee effects. By utilizing the Feller boundary classification criteria, it is testified that the model has a unique dynamical bifurcation point Δ with the following properties: if Δ<0, then the zero solution of the equation is stable; if Δ>0, then the equation possesses a unique

In this paper, we study the higher-order generalized nonlinear Schrödinger equation describing the propagation of ultrashort optical pulse in optical fibers. The Hirota’s bilinear method is employed to succinctly derive one-soliton and two-soliton solutions of the equation. Furthermore, the main characteristics of these solutions are graphically demonstrated. In particular, the influences of each parameters

When gun barrels are multiple fired, monitoring the non-smooth heat flux of the gun with the additional input data makes it possible to identify the corrosion of the gun. Based on this background, we study the identification of the non-smooth heat flux in a 1-dimensional heat transfer system. Aiming to overcome the non-smoothness of the unknown heat flux and to simplify the choice of regularization

In this paper, an efficient algorithm for recovering a density of a fourth-order Sturm–Liouville operator from two given spectra is investigated. Based on Lidskii’s theorem and Mercer’s theorem, we build a sequence of trace formulae which bridge explicitly the density and eigenvalues in terms of nonlinear Fredholm integral equations. Due to intrinsic difficulties on ill-posedness of an inverse spectral

In this short note, we prove interior regularity of weak solutions to a class of linear elliptic system with quadratic Jacobian structures in higher dimensions, extending the result of Hajlasz, Strzelecki and Zhong (2008) in dimension two.

The Poisson–Nernst–Planck (PNP) model characterizing the electro-diffusion process is widely used in ion channel simulations. The standard finite element method often fails to converge, due to complicated geometries, strong fixed charges and huge bias of electric potentials or ion concentrations. We propose a novel stabilized finite element method named SUPG–IP, which inherits the upwind characteristic

This paper mainly studies how to select reasonable stepsizes (regularization parameters) in Lanczos’ generalized derivatives. Four posterior selection strategies, in which two strategies need to know the noise level while the other two do not, are proposed with the convergence estimates of regularization solutions (Lanczos’ generalized derivatives). Numerical experiments show that the last three posterior

In this paper, we consider the Ambrosetti–Prodi type equation, that is, the nonlinearity f is asymptotically linear at −∞, while f is superlinear at +∞. Under weak conditions, we can obtain three nontrivial solutions by Morse theory.

We investigate the existence of a nontrivial nonnegative solution to (p,q)-Laplace equations involving two nonlinear terms, one grows as s with q

In this paper, we consider the impacts of the chemotaxis and delay on the dynamics of a diffusive predator–prey system with fear effect under the Neumann boundary conditions. Regarding chemotaxis coefficient and delay as bifurcation parameters, the existence of the codimension-two Turing–Hopf bifurcation is studied by analyzing the associated characteristic equation. We deduce that chemotaxis-driven

The effect of additive fading stochastic perturbations of the type of white noise and Poisson’s jumps on an exponentially p-stable zero solution of a stochastic delay differential equation is studying. Condition on the rate of perturbations fading is obtained by which the solution of the perturbed stochastic delay differential equation remains asymptotically p-trivial. The obtained conditions are formulated

This paper is devoted to investigate the global boundedness of a two-species predator–prey chemotaxis model with one signal. By establishing the uniform boundedness of L2-norm for prey and predator densities and the uniform boundedness of L4-norm for the signal gradient, we consider the global existence and boundedness of classical solutions to the model in a three-dimensional bounded domain.

We consider a free boundary model of epithelial cell migration with logistic growth and nonlinear diffusion induced by mechanical interactions. Using numerical simulations, phase plane and perturbation analysis, we find and analyse travelling wave solutions with negative, zero, and positive wavespeeds. Unlike classical travelling wave solutions of reaction–diffusion equations, the travelling wave solutions

This paper investigates the exponential ultimate boundedness for stochastic pantograph differential systems. With the aid of Lyapunov functions and algebraic inequality techniques, some pth moment exponential ultimate boundedness conditions are derived for the addressed systems.