Nonconstant Solutions Research Articles

On constant solutions of SU(2) Yang-Mills equations with arbitrary current in Euclidean space ℝn

In this paper, we present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space ℝn of arbitrary finite dimension n. Using the invariance of the Yang-Mills equations under the orthogonal transformations of coordinates and gauge invariance, we choose a specific system of coordinates and a specific gauge fixing for each constant current and obtain all constant solutions of the Yang-Mills equations in this system of coordinates with this gauge fixing, and then in the original system of coordinates with the original gauge fixing. We use the singular value decomposition method and the method of two-sheeted covering of orthogonal group by spin group to do this. We prove that the number (0, 1, or 2) of constant solutions of the Yang-Mills equations in terms of the strength of the Yang-Mills field depends on the singular values of the matrix of current. The explicit form of all solutions and the invariant F2 can always be written using singular values of this matrix. The relevance of the study is explained by the fact that the Yang-Mills equations describe electroweak interactions in the case of the Lie group SU(2). Non-constant solutions of the Yang-Mills equations can be considered in the form of series of perturbation theory. The results of this paper are new and can be used to solve some problems in particle physics, in particular, to describe physical vacuum and to fully understand a quantum gauge theory.

On static manifolds and related critical spaces with zero radial Weyl curvature

The aim of this paper is to study compact Riemannian manifolds $$(M,\,g)$$ that admit a non-constant solution to the system of equations $$\begin{aligned} -\Delta f\, g+Hess f-fRic=\mu Ric+\lambda g, \end{aligned}$$where Ric is the Ricci tensor of g whereas $$\mu $$ and $$\lambda $$ are two real parameters. More precisely, under assumption that $$(M,\,g)$$ has zero radial Weyl curvature, this means that the interior product of $$\nabla f$$ with the Weyl tensor W is zero, we shall provide the complete classification for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.

Spatio-temporal coexistence in the cross-diffusion competition system

<p style='text-indent:20px;'>We study a two component cross-diffusion competition system which describes the population dynamics between two biological species. Since the cross-diffusion competition system possesses the so-called population pressure effects, a variety of solution behaviors can be exhibited compared with the classical diffusion competition system. In particular, we discuss on the existence of spatially non-constant time periodic solutions. Applying the center manifold theory and the standard normal form theory, the cross-diffusion competition system is reduced to a two dimensional dynamical system around a doubly degenerate point. As a result, we show the existence of stable time periodic solutions in the system. This means spatio-temporal coexistence between two biological species.

Local and global bifurcation of steady states to a general Brusselator model

In this paper, we consider the local and global bifurcation of nonnegative nonconstant solutions of a general Brusselator model{−d1△u=a−(b+1)f(u)+u2v,x∈Ω,−d2△v=bf(u)−u2v,x∈Ω,∂u∂n=∂v∂n=0,x∈∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -d_{1}\\triangle u=a-(b+1)f(u)+u^{2}v, & x\\in \\varOmega , \\\\ -d_{2}\\triangle v=bf(u)-u^{2}v, & x\\in \\varOmega , \\\\ \\frac{\\partial u}{\\partial n}=\\frac{\\partial v}{\\partial n}=0, & x\\in \\partial \\varOmega , \\end{cases} $$\\end{document} where d_{1},d_{2},a>0 are fixed parameters with d_{2}>d_{1}, b>0 is a bifurcation parameter; fin C([0,infty ) ,[0,infty )) is a strictly increasing function and f'(f^{-1}(a))in (0,infty ). Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of nonconstant solutions under the condition that frac{f(s)}{s^{2}} is nonincreasing in (0,infty ).

Nonconstant positive radial solutions for Neumann problem involving the mean extrinsic curvature operator

Let B be the unit ball in RN with N≥2. Let f∈C1([0,∞),R), f(0)=0, f(β)=β for some β∈(0,∞), f(s)<sfors∈(0,β),f(s)>sfors∈(β,∞) and f′(β)>λkr, where λkr is the k-th radial eigenvalue of −Δ+I in the unit ball with Neumann boundary condition. We use the unilateral global bifurcation theorem to show the existence of nonconstant, positive radial solutions of the quasilinear Neumann problem−div(∇u1−|∇u|2)+u=f(u)inB,∂νu=0on∂B.

A Paneitz–Branson type equation with Neumann boundary conditions

Abstract We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

Existence and metastability of non-constant steady states in a Keller-Segel model with density-suppressed motility

We are concerned with stationary solutions of a Keller-SegelModel with density-suppressed motility and without cell proliferation. we establish the existence and the analytical approximation of non-constant stationary solutions by applying the phase plane analysis and bifurcation analysis. We show that the one-step solutions is stable and two or more-step solutions are always unstable. Then we further show that two or more-step solutions possess metastability. Our analytical results are corroborated by direct simulations of the underlying system.

Qualitative analysis of a diffusive Crowley–Martin predator–prey model: the role of nonlinear predator harvesting

In this paper, we discuss a diffusive predator–prey system with mutually interfering predator and nonlinear harvesting in predator with Crowley–Martin functional response. The mathematical analysis of the system starts with the existence and uniqueness of solution of the system using $$C_0$$ semigroup. The analysis reflects that the upper bound of rate of predator harvesting for the coexistence of the species can be guaranteed. In addition, we establish the existence and nonexistence of non-constant positive steady state. Explicit conditions on predator harvesting are obtained for local and global stability of interior equilibrium and also for the existence and nonexistence of non-constant steady-state solution. We also investigate conditions for Turing instabilities of the diffusive system analytically. Our results show that the effort of harvesting (g) provides a threshold value for existence of non-constant positive stationary solution. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the system exhibits interesting patterns. Some biological implications of obtained theoretical results have also been discussed.

Spatiotemporal Patterns in a Diffusive Predator-Prey Model with Prey Social Behavior

Our aim in this paper is to investigate the behavior of pattern formation for a predator-prey model with social behavior and spatial diffusion. Firstly, we give some solution behavior where the non-existence of a non-constant steady state solution has been proved for some values of the diffusion coefficients. On the other hand, by using the Leray-Schauder degree theory the existence of the non-constant steady-state solution has been proved under a suitable conditions on the diffusion coefficients.

Bifurcation Analysis of a Diffusive Predator–Prey Model with Bazykin Functional Response

This paper deals with a diffusive predator–prey model with Bazykin functional response. The parameter regions for the stability and instability of the unique constant steady state are derived. The Turing (diffusion-driven) instability which induces spatial inhomogeneous patterns, the existence of time-periodic orbits which produce temporal inhomogeneous patterns, the existence and nonexistence of nonconstant steady state positive solutions are proved. Numerical simulations are presented to verify and illustrate the theoretical results.

Existence of nonconstant periodic solutions for p(t)-Laplacian Hamiltonian system

The purpose of this paper is to consider the existence of periodic solutions for the p(t)-Laplacian Hamiltonian system. Some results are obtained by using the least action principle and the minimax methods.

Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator–prey model with predator harvesting

The ratio-dependent predator–prey model exhibits rich dynamics due to the singularity of the origin. Harvesting in a ratio-dependent predator–prey model is relatively an important research project from both ecological and mathematical points of view. In this paper, we study the temporal, spatial and spatiotemporal dynamics of a ratio-dependent predator–prey diffusive model where the predator population harvest at catch-per-unit-effort hypothesis. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction–diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibit Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the existence and non-existence of positive non-constant steady-state solutions are established. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.

Half-space theorems for the Allen–Cahn equation and related problems

Abstract In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in {\mathbb{R}^{n}} , whose level set {\{u=0\}} is contained in a half-space. If {n\leq 3} , we prove that the solution must be one-dimensional. In dimension {n\geq 4} , we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

Dynamics analysis of a reaction–diffusion system with Beddington–DeAngelis functional response and strong Allee effect

In this paper, the dynamics of a diffusive predator–prey model with modified Leslie–Gower term and strong Allee effect on prey under homogeneous Neumann boundary condition is considered. Firstly, we obtain the qualitative properties of the system including the existence of the global positive solution and the local and global asymptotical stability of the constant equilibria. In addition, we investigate a priori estimate and the nonexistence of nonconstant positive steady state solutions. Finally, we establish the existence and local structure of steady state patterns and time-periodic patterns for the system.

Solutions of fixed period in the nonlinear wave equation on networks

The wave equation on network is defined by \(\partial _{tt}u=\Delta _{\mathfrak {G}}u+g(u)\), where \(u\in {\mathbb {R}}^{n}\) and the graph Laplacian \(\Delta _{\mathfrak {G}}\) is an operator on functions on n vertices. We suppose that \(g:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is an odd continuous function that satisfies \(g(0)=0\), \(Dg(0)=0\) and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations \(\Gamma \), using a \(\Gamma \)-equivariant topological invariant we prove the existence of multiple non-constant p-periodic solutions characterized by their symmetries.

Dynamics of Modified Leslie-Gower Harvested Predator-Prey Model with Beddington-DeAngelis Functional Response

This paper considers a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and nonlinear prey harvesting strategy subject to the zero-flux boundary conditions. To understand the dynamics of the considered system, we derive sufficient conditions for permanence analysis, local stability, global stability and Hopf bifurcation of interior equilibrium point. Further we also investigate the existence and non-existence of non-constant positive steady state solutions.

Density-dependent effects on Turing patterns and steady state bifurcation in a Beddington\u2013DeAngelis-type predator\u2013prey model

In this paper, Turing patterns and steady state bifurcation of a diffusive Beddington–DeAngelis-type predator–prey model with density-dependent death rate for the predator are considered. We first investigate the stability and Turing instability of the unique positive equilibrium point for the model. Then the existence/nonexistence, the local/global structure of nonconstant positive steady state solutions, and the direction of the local bifurcation are established. Our results demonstrate that a Turing instability is induced by the density-dependent death rate under appropriate conditions, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion. Moreover, some specific examples are presented to illustrate our analytical results.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

On the weakly degenerate Allen-Cahn equation

AbstractIn this paper we consider a one-dimensional Allen-Cahn equation with degeneracy in the interior of the domain and Neumann boundary conditions. We allow the diffusivity coefficient vanish at some point of the space domain and we are addressed on the existence of stable non-constant solution.