For the curved n-body problem in \(\mathbb {S}^3\), we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium if and only if n is odd and the masses are equal. The equilibrium is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on \(\mathbb {S}^1\) embedded in \(\mathbb {S}^2\). We then study the stability

In the current series of two papers, we study the long time behavior of nonnegative solutions to the following random Fisher–KPP equation, $$\begin{aligned} u_t =u_{xx}+a(\theta _t\omega )u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$(1) where \(\omega \in \Omega \), \((\Omega , {\mathcal {F}},{\mathbb {P}})\) is a given probability space, \(\theta _t\) is an ergodic metric dynamical system on

We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE $$\begin{aligned} u_t = u_{xx} + f(x,u,u_x) \end{aligned}$$(*) on the unit interval \(0< x<1\) with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria \(v\in {\mathcal {E}}\). The global attractor \({\mathcal {A}}\) of (*), also called

We consider stability problems for the compressible viscous and diffusive magnetohydrodynamic (MHD) equations arising in the modeling of magnetic field confinement nuclear fusion. In the first part, we investigate the Cauchy problem to the barotropic MHD system. With the help of the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy

We study a free boundary problem for Fisher-KPP equation: \(u_t=u_{xx}+f(u)\) (\(g(t)< x < h(t)\)) with free boundary conditions \(h'(t)=-u_x(t,h(t))-\beta \) and \(g'(t)=-u_x(t,g(t))-\alpha \) for \(\alpha >0\) and \(\beta \in \mathbb {R}\). Such a free boundary problem can model the spreading of a biological or chemical species affected by the boundary environment. \(\beta >0\) means that there is

A compact space X is said to be minimal if there exists a map \(f:X\rightarrow X\) such that the forward orbit of any point is dense in X. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha and Tywoniuk (J Dyn Differ Equ, 29:243–257, 2017) on spaces with cyclic group of homeomorphisms generated by a minimal homeomorphism, and results of the first author, Clark and Oprocha

In this paper, we will define the index pair \((i_A(B),\nu _A(B))\) by the dual variational method, and show the relationship between the indices defined by different methods. As applications, we apply the index \((i_A(B),\nu _A(B))\) to study the existence and multiplicity of homoclinic orbits of nonlinear Hamiltonian systems and solutions of nonlinear Dirac equations.

In the study of aperiodic order via dynamical methods, topological entropy is an important concept. In this paper, parts of the theory, like Bowen’s formula for fibre wise entropy or the independence of the definition from the choice of a Van Hove sequence, are extended to actions of several non-discrete groups. To establish these results, we will show that the Ornstein–Weiss lemma is valid for all

We show that for any uniformly bounded in time \(H^1\cap L^1\) solution of the dispersion generalized Benjamin–Ono equation, the limit infimum, as time t goes to infinity, converges to zero locally in an increasing-in-time region of space of order \(t/\log t\). This result is in accordance with the one established by Muñoz and Ponce (Proc Am Math Soc 147(12):5303–5312, 2019) for solutions of the Benjamin–Ono

We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected media with Dirichlet conditions outside the domain. After showing the existence and uniqueness of a non-trivial bounded stationary state \(n_+\), we prove that it invades the unstable state zero exponentially fast in time.

We improve the Gagliardo–Nirenberg inequality $$\begin{aligned} \Vert \varphi \Vert _{L^q({\mathbb {R}}^n)} \le C \Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)} {\mathcal {L}}^{-(\frac{1}{q} - \frac{n-2}{2n})} (\Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)}), \end{aligned}$$ \(r=2\), \(0

We are interested in the global-in-time existence results for quasilinear wave equations satisfying null condition in one dimension. The main difficulty is that waves do not decay in one dimension space, and we rely on the new type of weighted energy estimates introduced in Luli et al. (Adv Math 329:174–188, 2018), where the authors proved that small data lead to global-in-time solutions for semilinear

We consider the nonlinear Schrödinger equation on \({\mathbb R}^N \), \(N\ge 1\), $$\begin{aligned} \partial _t u = i \varDelta u + \lambda | u |^\alpha u \end{aligned}$$ with \(\lambda \in {\mathbb C}\) and \(\mathfrak {R}\lambda >0\), for \(H^1\)-subcritical nonlinearities, i.e. \(\alpha >0\) and \((N-2) \alpha < 4\). Given a compact set \(K \subset {\mathbb {R}}^N \), we construct \(H^1\) solutions

A dynamical system with a plastic self-organising velocity vector field was introduced in Janson and Marsden (Sci Rep 7:17007, 2017) as a mathematical prototype of new explainable intelligent systems. Although inspired by the brain plasticity, it does not model or explain any specific brain mechanisms or processes, but instead expresses a hypothesised principle possibly implemented by the brain. The

The original version of this article unfortunately contained a mistake in Eq. 1.4. The correct equation is given below.

The vanishing shear viscosity limit to an initial-boundary value problem of planar MHD equations for compressible viscous heat-conducting fluids is justified and the convergence rates are obtained. More important, to capture the behavior of the solutions at small shear viscosity, both the boundary-layer thickness and the boundary-layer corrector are discussed.

We establish the robustness of the notion of an exponential dichotomy for a nonautonomous linear delay equation. We consider the general cases of equations satisfying the Carathéodory conditions and of noninvertible evolution families defined by the linear equations. The stable and unstable spaces of the perturbed equations are obtained as images of linear operators that in their turn are fixed points

We prove for a large class of n-body problems including a subclass of quasihomogeneous n-body problems, the classical n-body problem, the n-body problem in spaces of negative constant Gaussian curvature and a restricted case of the n-body problem in spaces of positive constant curvature for the case that all masses are equal and not necessarily constant that any solution for which the point masses

This article determines all possible (proper as well as pseudo) equilibrium arrangements under a repulsive power law force of three point particles on the unit circle. These are the critical points of the sum over the three (standardized) Riesz pair interaction terms, each given by \(V_s(r)= s^{-1}\left( r^{-s}-1 \right) \) when the real parameter \(s \ne 0\), and by \(V_0(r) := \lim _{s\rightarrow

In this paper we prove that the following delay differential equation $$\begin{aligned} \frac{d}{dt}x(t)=rx(t)\left( 1-\int _{0}^{1}x(t-s)ds\right) , \end{aligned}$$ has a periodic solution of period two for \(r>\frac{\pi ^{2}}{2}\) (when the steady state, \(x=1\), is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea

The Hopf bifurcations for the classical Gray–Scott system and the Schnakenberg system in an one-dimensional interval are considered. For each system, the existence of time-periodic solutions near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of limit cycles are determined and it is shown that the

New elementary, self-contained proofs are presented for the topological and the smooth classification theorems of linear flows on finite-dimensional normed spaces. The arguments, and the examples that accompany them, highlight the fundamental roles of linearity and smoothness more clearly than does the existing literature.

In this paper we define the index at infinity of an asymptotically linear autonomous Hamiltonian system. We use this index to prove the existence and bifurcation from infinity of periodic solutions of the system. We apply the degree for G-invariant strongly indefinite functionals defined by Gołȩbiewska and Rybicki in (Nonlinear Anal 74:1823-1834, 2011).

In this paper we prove that under mild conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, the construction of greedy sequence of times, and Gronwall-type lemma with the help of Shauder theorem of fixed points.

The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as t→±∞ to stationary states. The attraction is caused by nonlinear energy radiation.

We develop the Bohl spectrum for nonautonomous linear differential equations on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker-Sell spectra. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker-Sell spectrum in general even for bounded

A predator–prey model with general Holling type functional response is considered. The aim of this paper is to investigate the global stability of the interior equilibrium of the model when it exists. By transforming the model to an equivalent system and with the detailed analysis of the crucial behavior of a corresponding function we are able to show that the local stability and the global stability

West Nile virus (WNv) transmission shows both seasonal pattern in every single year and cyclic pattern over years. In this paper we formulate a compartmental model with bird demographics and maturation time of mosquitoes during metamorphosis to study the impact of ambient temperature on the transmission and recurrence of disease. We show that avian birds serve as a reservoir of viruses, whilst maturation

A diffusive chemostat model with two competing species and one nutrient is revisited in this paper. It is shown that at large diffusion rate, both species are washed out, while competition exclusion occurs at small diffusion rate. This implies that a stable coexistence only occurs at intermediate diffusion rate, and an explicit way of determining parameter ranges which support a stable coexistence

We consider the Modified Kuramoto–Sivashinky Equation (MKSE) in one and two space dimensions and we obtain explicit and accurate estimates of various Sobolev norms of the solutions. In particular, by using the sharp constants which appear in the functional interpolation inequalities used in the analysis of partial differential equations, we evaluate explicitly the sup-norm of the solutions of the MKSE

We prove the convergence of the solutions \(u_{m,p}\) of the equation \(u_t+(u^m)_x=-\,u^p\) in \({{\mathbb {R}}}\times (0,\infty )\), \(u(x,0)=u_0(x)\ge 0\) in \({{\mathbb {R}}}\), as \(m\rightarrow \infty \) for any \(p>1\) and \(u_0\in L^1({{\mathbb {R}}})\cap L^{\infty }({{\mathbb {R}}})\) or as \(p\rightarrow \infty \) for any \(m>1\) and \(u_0\in L^{\infty }({{\mathbb {R}}})\). We also show that

This paper is devoted to the study of propagation dynamics for a time periodic nonlocal dispersal model with stage structure. In the case where the birth rate function is monotone, we establish the existence of the spreading speed and its coincidence with the minimal wave speed for monotone periodic traveling waves by appealing to the theory developed for monotone semiflows. In the case where the birth

In this paper, we prove a blow-up criterion in terms of the magnetic field H and the mass density \(\rho \) for the strong solutions to the initial boundary value problem of the 3D isentropic compressible MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the upper bounds of \((H,\rho )\) control the possible blow-up (see Rozanova in Proceedings of symposia

Animal movements and their underlying mechanisms are an extremely important research area in biology and have been extensively studied for centuries. However, spatial memory and cognition, which is the most significant difference between animal movements and chemical movements, has been ignored in the modeling of animal movements. To incorporate cognition and memory of “clever” animals in the simplest

In this paper, we prove that for \(C^1\)-generic diffeomorphisms, if a homoclinic class contains periodic orbits of indices i and j with \(j>i+1\), and the homoclinic class has no-domination of index l for any \(l\in \{i+1,\ldots ,j-1\}\), then there exists a non-hyperbolic ergodic measure with more than one vanishing Lyapunov exponents and whose support is the whole homoclinic class. Some other results

We study the asymptotic development at infinity of an integral operator. We use this development to give sufficient conditions to upper bound the number of critical periodic orbits that bifurcate from the outer boundary of the period function of planar potential centers. We apply the main results to two different families: the power-like potential family \(\ddot{x}=x^q-x^p\), \(p,q\in \mathbb {R}\)

For the one-dimensional gas dynamics in thermal nonequilibrium which is a \(4\times 4\) nonlinear hyperbolic system with relaxation, global existence for Cauchy problem is established, and the asymptotic stability of the traveling wave solution with shock profile is proved under the condition that the shock strength and the initial disturbance are small and the integral zero of the initial disturbance

The object of study is an autonomous impulsive system proposed as a model of drugs absorption by living organisms consisting of a linear differential delay equation and an impulsive self-support condition. We get a representation of the general solution in terms of the fundamental solution of the differential delay equation. The impulsive self-support generates periodic auto-oscillations given by fixed

Existence of travelling waves is studied for a delay reaction–diffusion system of equations describing the distribution of viruses and immune cells in the tissue. The proof uses the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces.

We revisit Wu and Zou non-standard quasi-monotonicity approach for proving existence of monotone wavefronts in monostable reaction–diffusion equations with delays. This allows to solve the problem of existence of monotone wavefronts in a neutral KPP–Fisher equation. In addition, using some new ideas proposed recently by Solar et al., we establish the uniqueness (up to a translation) of these monotone

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional \(p(\cdot )\)-Laplacian with variable exponents, which is a fractional version of the nonhomogeneous \(p(\cdot )\)-Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem \(({\mathscr {P}}_{1})\) in a bounded

This work is concerned with the fast–slow dynamics for intraguild predation models with evolutionary effects. Assuming the survival pressure of the weaker predator induces evolution of it to the intraguild predator, then the system can be viewed as a singularly perturbed problem with two different time scales—predation time scale and evolution time scale. Using the geometric singular perturbation theory

We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equations with a sign-changing potential V that converges at infinity and a nonlinear term being asymptotically linear at infinity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is that, due to the convergency of V at infinity

In this paper, we investigate a delayed reaction–diffusion–advection equation, which models the population dynamics in the advective heterogeneous environment. The existence of the nonconstant positive steady state and associated Hopf bifurcation are obtained. A weighted inner product associated with the advection rate is introduced to compute the normal forms, which is the main difference between

We consider a model of multi-species competition in the chemostat that includes demographic stochasticity and discrete delays. We prove that for any given initial data, there exists a unique global positive solution for the stochastic delayed system. By employing the method of stochastic Lyapunov functionals, we determine the asymptotic behaviors of the stochastic solution and show that although the

This paper concerns the Cauchy problem of the compressible non-resistive magnetohydrodynamic equations on the whole two-dimensional space with vacuum as far field density. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, we prove that there exists a unique local strong solution provided the initial density and the initial magnetic field