In this paper, we consider dynamics defined by the Navier–Stokes equations in the Oberbeck–Boussinesq approximation in a two dimensional domain. This model of fluid dynamics involves fundamental physical effects: convection, and diffusion. The main result is as follows: local semiflows, induced by this problem, can generate all possible structurally stable dynamics defined by \(C^1\) smooth vector

This paper concerns the global dynamics and asymptotic spreading speeds for a partially degenerate epidemic model with time delay and free boundaries. Given a suitable compatible condition for initial values, we establish the global well-posedness of solutions and provide some sufficient conditions for spreading and vanishing. When spreading occurs, we prove that the asymptotic spreading speed is uniquely

This paper is concerned with the spatio-temporal dynamics of nonnegative bounded entire solutions of some reaction–diffusion equations in \(\mathbb {R}^N\) in any space dimension N. The solutions are assumed to be localized in the past. Under certain conditions on the reaction term, the solutions are then proved to be time-independent or heteroclinic connections between different steady states. Furthermore

We consider the equation $$\begin{aligned} \Delta _x u+u_{yy}+f(u)=0,\quad x=(x_1,\dots ,x_N)\in {{\mathbb {R}}}^N,\ y\in {{\mathbb {R}}}, \end{aligned}$$(1) where \(N\ge 2\) and f is a sufficiently smooth function satisfying \(f(0)=0\), \(f'(0)<0\), and some natural additional conditions. We prove that equation (1) possesses uncountably many positive solutions (disregarding translations) which are

The time-dependent restricted \((n+1)\)-body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by n primary bodies following a periodic solution of the n-body problem. We prove that the satellite has periodic solutions close to the large-amplitude circular orbits of the Kepler problem (comet solutions), and in the case that the primaries

In this paper we present local Sternberg conjugation theorems near attracting fixed points for lattice systems. The interactions are spatially decaying and are not restricted to finite distance. The conjugations obtained retain the same spatial decay. In the presence of resonances the conjugations are to a polynomial normal form that also has decaying properties.

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carathéodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained

In Treschev (Proc Steklov Math Inst 310:262–290, 2020) we introduce the concept of a \(\mu \)-norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from Treschev (2020) and present further results on the \(\mu \)-norm. More precisely, we specify three classes of unitary operators

We start to discuss some aspects of the scattering theory for the Sturm–Liouville operator \(L:\dfrac{1}{y}\left[ -D^2+q\right] \). In particular, we pose and solve the problem of reconstructing the function q when y is fixed and when a set \({\mathcal {S}}\) of scattering data is given. In the meanwhile, several relations concerning the spectral properties of L and the solutions of the related eigenvalue

We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and unbounded weight: $$\begin{aligned} \partial _tu=\partial _{xx}(u^m) + |x|^{\sigma }u^p, \end{aligned}$$ posed for \(x\in {\mathbb {R}}\), \(t\ge 0\), where \(m>1\), \(02\) completing the analysis performed in a recent work where this very interesting critical case

An asymptotic expansion of the solution of a nonhomogeneous matrix difference equation of general form is obtained. The case when there is no bound on the differences of the arguments is considered. The effect of the roots of the characteristic equation is taken into account. The asymptotic behavior of the remainder is established depending on the asymptotics of the free term of the equation.

We extend to the infinite dimensional context the link between two completely different topics recently highlighted by the authors: the classical eigenvalue problem for real square matrices and the Brouwer degree for maps between oriented finite dimensional real manifolds. Thanks to this extension, we solve a conjecture regarding global continuation in nonlinear spectral theory that we have formulated

In this paper, we consider the following one-dimensional, quasi-periodically forced, linear KdV equations $$\begin{aligned} u_t+(1+ a_{1}(\omega t)) u_{xxx}+ a_{2}(\omega t,x) u_{xx}+ a_{3}(\omega t,x)u_{x} +a_{4}(\omega t,x)u=0 \end{aligned}$$ under the periodic boundary condition \(u(t,x+2\pi )=u(t,x)\), where \(\omega \)’s are frequency vectors lying in a bounded closed region \(\Pi _*\subset {\mathbb

In this paper we study the local and global well posedness of a fractional dissipative Klein–Gordon–Schrödinger type system in dimension 1 and establish the existence of a global attractor.

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial

Let \(\text {Homeo}_{+}(\mathbb {S}^1)\) denote the group of orientation preserving homeomorphisms of the circle \(\mathbb {S}^1\). A subgroup G of \(\text {Homeo}_{+}(\mathbb {S}^1)\) is tightly transitive if it is topologically transitive and no subgroup H of G with \([G: H]=\infty \) has this property; is almost minimal if it has at most countably many nontransitive points. In the paper, we determine

Fifth-order dispersive equations arise in the context of higher-order models for phenomena such as water waves. For fifth-order variable-coefficient linear dispersive equations, we provide conditions under which the intitial value problem is either well-posed or ill-posed. For well-posedness, a balance must be struck between the leading-order dispersion and possible backwards diffusion from the fourth-derivative

For a continuous self-map f on a compact metric space X, we provide two simple examples: the first confirms that shadowing of (X, f) is not inherited by \(({\mathcal {M}}(X),{\hat{f}})\) in general, and the other satisfies that both (X, f) and \(({\mathcal {M}}(X),{\hat{f}})\) have no Li–Yorke pair, where \({\mathcal {M}}(X)\) be the space of all Borel probability measures on X. Then we prove that

In this article we study some statistical aspects of surface diffeomorphisms. We first show that for a \(C^1\) generic diffeomorphism, a Dirac invariant measure whose statistical basin of attraction is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. We then construct an example of a \(C^1\)-diffeomorphism having a Dirac invariant measure, supported

Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times \(t\in (-\infty ,\infty )\)) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the

The presence of slow–fast Hopf (or singular Hopf) points in slow–fast systems in the plane is often deduced from the shape of a vector field brought into normal form. It can however be quite cumbersome to put a system in normal form. In De Maesschalck et al. (Canards from birth to transition, 2020), Wechselberger (Geometric singular perturbation theory beyond the standard form, Springer, New York,

In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig–Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on \(\mathbb {T}^\infty

The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space and time dependent logistic source, $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u -\chi \nabla \cdot (u\nabla v)+u(a(x,t)-b(x,t)u),&{}\quad x\in {\mathbb R}^N,\\ 0=\Delta v-\lambda v+\mu u ,&{}\quad x\in {\mathbb

Consider the Kirchhoff equation $$\begin{aligned} \partial _{tt} u - \Delta u \Big ( 1 + \int _{\mathbb {T}^d} |\nabla u|^2 \Big ) = 0 \end{aligned}$$ on the d-dimensional torus \(\mathbb {T}^d\). In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads

In this paper, we consider the normal form problem of a commutative family of germs of diffeomorphisms at a fixed point, say the origin, of \(\mathbb {K}^n (\mathbb {K}={\mathbb {C}}\text {~or~} {\mathbb {R}})\). We define a notion of integrability of such a family. We give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic (resp.

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space \(\mathbb {R}^n\), where \(n \ge 2\), assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as \(|x| \rightarrow \infty \). Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We

In this paper, we propose a mild condition, named Condition \((**)\), for collections of sequence of integers and show that for any measure preserving system the Pinsker \(\sigma \)-algebra is a characteristic \(\sigma \)-algebra for the averages along a collection satisfying Condition \((**)\). We introduce the notion of \(\Delta \)-weakly mixing subsets along a collection of sequences of integers

We study existence and multiplicity of positive ground states for the scalar curvature equation $$\begin{aligned} \varDelta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n\,, \quad n>2, \end{aligned}$$ when the function \(K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+\) is bounded above and below by two positive constants, i.e. \(0<\underline{K} \le K(r) \le \overline{K}\) for every

We consider a periodic function \(p:{\mathbb {R}}\rightarrow {\mathbb {R}}\) of minimal period 4 which satisfies a family of delay differential equations $$\begin{aligned} x'(t)=g(x(t-d_{\Delta }(x_t))),\quad \Delta \in {\mathbb {R}}, \end{aligned}$$(0.1) with a continuously differentiable function \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and delay functionals $$\begin{aligned} d_{\Delta }:C([-2

We study the asymptotic dynamics of stochastic Young differential delay equations under the regular assumptions on Lipschitz continuity of the coefficient functions. Our main results show that, if there is a linear part in the drift term which has no delay factor and has eigenvalues of negative real parts, then the generated random dynamical system possesses a random pullback attractor provided that

Let \(h:V\subset {\mathbb {R}}^{2}\longrightarrow {\mathbb {R}}^{2}\) be an embedding. The aim of this paper is to analyze the dynamical behavior of h depending on the number of fixed points and 2-cycles, their local behaviors and the features of V. Our approach allows us to extend some celebrated results of the theory of monotone flows, namely the order interval trichotomy, for non-monotone maps.

We generalize the concepts of D-stability and additive D-stability of matrices. For this, we consider a family of unbounded regions defined in terms of Linear Matrix Inequalities (so-called LMI regions). We study the problem when the localization of a matrix spectrum in an unbounded LMI region is preserved under specific multiplicative and additive perturbations of the initial matrix. The most well-known

Multistationarity in molecular systems underlies switch-like responses in cellular decision making. Determining whether and when a system displays multistationarity is in general a difficult problem. In this work we completely determine the set of kinetic parameters that enable multistationarity in a ubiquitous motif involved in cell signaling, namely a dual phosphorylation cycle. In addition we show

We study a Lotka–Volterra competition–diffusion model that describes the growth, spread and competition of two species in a shifting habitat. Some results have been obtained previously for some cases for the diffusion rates and competitions rates, and in this paper we continue to explore the remaining complementary case for the spatial dynamics of the system. Our main result in this paper reveals an

The Cauchy problem in \({\mathbb {R}}^n\), \(n\ge 1\), for the parabolic equation $$\begin{aligned} u_t=u^p \Delta u \qquad \qquad (\star ) \end{aligned}$$ is considered in the strongly degenerate regime \(p\ge 1\). The focus is firstly on the case of positive continuous and bounded initial data, in which it is known that a minimal positive classical solution exists, and that this solution satisfies

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture their long-term behaviour comprehensively, both pullback

We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first situation, the effect of the delays is dominated by non-delayed diagonal negative feedback terms, and sufficient conditions for both the asymptotic and the exponential

This paper establishes the global stability for the 2D Boussinesq system with partial dissipation and horizontal thermal diffusion. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. We extend the \(H^1\)-stability in [7] to \(H^2\)-stability which we care about and obtain the large time behavior of the linearized system.

For each given \(n \ge 2\), we study the variational existence of a new spatial periodic orbit in the 2n-body problem. Besides excluding possible collision singularities, the main challenges left are to show that the orbit is not a relative equilibrium and it is spatial. By introducing a new estimate in the collinear central configuration, we can prove that it is not a relative equilibrium in the 2n-body

The mass-in-mass (MiM) lattice consists of an infinite chain of identical beads that are both nonlinearly coupled to their nearest neighbors and linearly coupled to a distinct resonator particle; it serves as a prototypical model of wave propagation in granular crystals and metamaterials. We study traveling waves in an MiM lattice whose bead interaction is governed by the cubic Fermi–Pasta–Ulam–Tsingou

We first provide a classification of the pure rotational motion of 2 particles on a sphere interacting via a repelling potential. This is achieved by providing a simple geometric equivalence between repelling particles and attracting particles, and relying on previous work on the similar classification for attracting particles. The second theme of the paper is to study the 2-body problem on a surface

This paper is devoted to studying wave breaking phenomena for the Fornberg–Whitham equation. Making use of the structure and the conservation quantity of the Fornberg–Whitham equation, we give a sufficient condition on the initial data to ensure wave breaking.

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different

Existence of solutions connecting a singularity of a perturbed implicit differential equations is studied. It is assumed that the unperturbed differential equation has a solution of the same kind. By a suitable, nonlinear, change of coordinates these kind of solutions are associated to homoclinic solutions to a fixed point of an ordinary differential equation with a one-dimensional centre manifold

We study a quasi-one-dimensional classical Poisson–Nernst–Planck model for ionic flow through a membrane channel with two positively charged ion species (cations) and one negatively charged, and with zero permanent charges. We treat the model problem as a boundary value problem of a singularly perturbed differential system. Under the framework of the geometric singular perturbation theory, together

This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equations. Based on the regular linear systems and feedback

A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate \(\frac{kIS}{1+\beta I+\alpha I^2}\) (\(\beta >-2 \sqrt{\alpha }\) such that \(1+\beta I+\alpha I^{2}>0\) for all \(I\ge 0\)) is considered in this paper. It is shown that the basic reproduction number \(R_0\) does not act as a threshold value for the disease spread anymore, and there exists a sub-threshold

Most epidemiological models for mosquito-borne disease assume exponentially distributed residence times in disease stages in order to simplify the model formulation and analysis. However, these models do not allow for accurate description of the interaction between drug concentration and pathogen load within hosts. To improve this, we formulate a model by considering arbitrarily distributed sojourn

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index \(c>0\) and homoclinic tangencies of arbitrary codimension \(c>0\). These type of sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite, we also construct robust homoclinic tangencies

Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured

We consider a scalar delay differential equation \(\dot{x}(t)=-dx(t)+f((1-\alpha )\rho x(t-\tau )+\alpha \rho x(t-2\tau ))\) with an instant mortality rate \(d>0\), the nonlinear Rick reproductive function f, a survival rate during all development stages \(\rho \), and a proportion constant \(\alpha \in [0, 1]\) with which population undergoes a diapause development. We consider global continuation

In this paper we focus on a class of symmetric vector fields in the context of singularly perturbed fast-slow dynamical systems. Our main question is to know how symmetry properties of a dynamical system are affected by singular perturbations. In addition, our approach uses tools in geometric singular perturbation theory [8], which address the persistence of normally hyperbolic compact manifolds. We

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite

In this paper, the speed sign of the traveling wave to the bistable Lotka–Volterra competitive lattice system is investigated via the upper–lower solution method as well as the comparison principle. We provide an interval estimation for the bistable speed firstly. Two comparison principles are further established to obtain new conditions to the determinacy of the sign of the bistable speed. To our

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