We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization

For $ n\ge 3 $

We study the stochastic stability in the zero-noise limit from a quantitative point of view.We consider smooth expanding maps of the circle perturbed by additive noise. We show that in this case the zero-noise limit has a quadratic speed of convergence, as suggested by numerical experiments and heuristics published by Lin, in 2005 (see [25]). This is obtained by providing an explicit formula for the

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a $ N $-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least $ N $ orthogonal Finsler geodesic chords that are geometrically distinct. If the reversibility

In this paper we study a $ 2\times2 $ semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $ L^\infty $ in the space-time domain $ (0,1)\times [0,+\infty) $. Then we address the problem of the time-asymptotic stability of the

Inspired by the generalized Christoffel problem, we consider a class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $. Under some sufficient conditions, we prove the a priori estimates for solutions to the Monge-Ampère type equation $ \det(\kappa-\mathbf{1}) = f(X, \nu(X)) $. Moreover, we obtain an existence result for the compact horo-convex hypersurface

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation $ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $ and $ \alpha>0. $ Only partial results are known for the local existence in the subcritical case $ \alpha<(4-2b)/(N-2s) $ and much more less in the critical case $ \alpha = (4-2b)/(N-2s). $

This note treats several problems for the fractional perimeter or $ s $-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $ s \nearrow 1 $ is proven. It is shown that their limit as $ s \searrow -\infty $ can be expressed

Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves

In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with $ n $ degrees of freedom we provide a criterion to obtain a type of formal stability, called Lie stability. Our result generalises previous approaches, as exponential stability in the sense of Nekhoroshev (excepting a few situations) and other classical results on formal stability of equilibria. In case of Lie

We consider the following nonlocal critical equation $\begin{equation} -\Delta u = (I_{\mu_1}\ast|u|^{2_{\mu_1}^\ast})|u|^{2_{\mu_1}^\ast-2}u +(I_{\mu_2}\ast|u|^{2_{\mu_2}^\ast})|u|^{2_{\mu_2}^\ast-2}u,\; x\in\mathbb{R}^N, \;\;\;\;\;\;\;(1) \end{equation}$

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node $ A $ to node $ B $ if, using arbitrary small perturbations, a trajectory starting from any point of $ A $ can be steered to any point of $ B $. In this article we describe the graph of the logistic map. Our main result is

We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global

We give a proof by foliation that the cones over $ \mathbb{S}^k \times \mathbb{S}^l $ minimize parametric elliptic functionals for each $ k, \, l \geq 1 $. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of

In this work we obtain positive bounded solutions of various perturbations of $ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u - \gamma \sum_{i, j = 1}^N \frac{x_i x_j}{|x|^2} u_{x_i x_j} & = & u^p \qquad \mbox{ in } B_1, \\ \hfill u & = & 0 \hfill\qquad\ \mbox{ on } \partial B_1, \end{array}\right. \end{equation} \ \ \ \ \ \ \ \ \ \ \ (1) $

A restricted permutation of a locally finite directed graph $ G = (V, E) $ is a vertex permutation $ \pi: V\to V $ for which $ (v, \pi(v))\in E $, for any vertex $ v\in V $. The set of such permutations, denoted by $ \Omega(G) $, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser [18]

This paper showcases the use of PDE-based graph methods for modern machine learning applications. We consider a case study of body-worn video classification because of the large volume of data and the lack of training data due to sensitivity of the information. Many modern artificial intelligence methods are turning to deep learning which typically requires a lot of training data to be effective. They

In this paper we study the weak two-sided limit shadowing for flows on a compact metric space which is different with the usual shadowing, two-sided limit shadowing and L-shadowing, and characterize the weak two-sided limit shadowing flows from the pointwise and measurable viewpoints. Moreover, we prove that if a flow $ \phi $ has the weak two-sided limit shadowing property on its chain recurrent $

We study the partial regularity problem for a three dimensional simplified Ericksen–Leslie system, which consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier–Stokes equations, we first prove some new local

The first aim of this paper is to show well-posedness of Dirichlet and transmission problems in bounded and exterior Lipschitz domains in $ {\mathbb R}^n $, $ n\geq 3 $, for the anisotropic Stokes system with $ L_{\infty } $ viscosity tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices with zero matrix trace, with data from standard and weighted Sobolev spaces. To

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations $ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $ is established via Wolff type potentials. It is worthwhile to note that the model case of $ \mathbb{A} $ here is the non-degenerate $ p $-Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc

In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu [10] who recently classified the centralizer for perturbations of time-$ 1 $ maps of geodesic flows in negative curvature. We strongly rely on recent classification

We consider a wide family of non-uniformly expanding maps and hyperbolic Hölder continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in

We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the critical exponents. At this boundary there is a phase transition in the geometry of the system. Differently from the previous approaches, this is achieved by studying

In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely

On a closed $ 3 $

In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [10] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain:

We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish $ L^q $ time-decay for the linear homogeneous

This paper shows the unique solvability of elliptic problems associated with two-phase incompressible flows, which are governed by the two-phase Navier-Stokes equations with a sharp moving interface, in unbounded domains such as the whole space separated by a compact interface and the whole space separated by a non-compact interface. As a by-product, we obtain the Helmholtz-Weyl decomposition for two-phase

In this paper we discuss some thermoelastic and thermoviscoelastic models obtained from the Gurtin theory, based on the invariance of the entropy under time reversal. We derive differential systems where the temperature and the velocity are ruled by generalized versions of the Moore-Gibson-Thompson equation. In the one-dimensional case, we provide a complete analysis of the evolution, establishing

We consider a map $ F $ of class $ C^r $ with a fixed point of parabolic type whose differential is not diagonalizable, and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of $ F $, there exist invariant curves of class $ C^r $ away from the fixed

The present paper deals with a class of Schrödinger-poisson system. Under some suitable assumptions on the decay rate of the coefficients, we derive the existence of infinitely many positive solutions to the problem by using purely variational methods. Comparing to the previous works, we encounter some new challenges because of nonlocal term. By doing some delicate estimates for the nonlocal term we

In this paper we construct a unique global in time weak nonnegative solution to the corrected Derrida-Lebowitz-Speer-Spohn equation, which statistically describes the interface fluctuations between two phases in a certain spin system. The construction of the weak solution is based on the dissipation of a Lyapunov functional which equals to the square of the Hellinger distance between the solution and

We introduce a weighted $ L_{p} $

We investigate the problem of determining the planar curves that describe ramps where a particle of mass $ m $ moves with constant-speed when is subject to the action of the friction force and a force whose magnitude $ F(r) $ depends only on the distance $ r $ from the origin. In this paper we describe all the constant-speed ramps for the case $ F(r) = -m/r $. We show the circles and the logarithmic

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when at least one of the components of the gradient vanishes. We extend here the results in [16], [10], [24].

The present work aims at the mathematical derivation of the equations for the isentropic flow from those for the non-isentropic flow for perfect gases in the whole space. Suppose that the following things hold for the entropy equation: (1). both conduction of heat and its generation by dissipation of mechanical energy are sufficiently weak(with the order of $ \varepsilon $

We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to $ r $-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary

We are concerned with blow-up mechanisms in a semilinear heat equation: $ u_t = \Delta u + |x|^{2a} u^p , \quad x \in \textbf{R}^N , \, t>0, $

In this paper we consider the problem $ (P_{\lambda})\ \ \ \ \ \ \left\{ \begin{array}{rcl} -\Delta u+V_{\lambda}(x)u = (I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right. $

In this paper, we consider the compressible Euler equations with time-dependent damping $ \frac{{\alpha}}{(1+t)^\lambda}u $ in one space dimension. By constructing "decoupled" Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation

In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a $ C^1 $-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain

We investigate spreading properties of solutions for a spatially distributed system of equations modelling the evolutionary epidemiology of plant-pathogen interactions. In this work the mutation process is described using a non-local convolution operator in the phenotype space. Initially equipped with a localized amount of infection, we prove that spreading occurs with a definite spreading speed that

We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result

In this paper, we use a variational approach to study traveling wave solutions of a gradient system in an infinite strip. As the even-symmetric potential of the system has three local minima, we prove the existence of a traveling wave that propagates from one phase to the other two phases, where these phases corresponds to the three local minima of the potential. To control the asymptotic behavior

For the coupled Ginzburg-Landau system in $ {\mathbb R}^2 $

If $ (M,g) $ is a smooth compact rank $ 1 $ Riemannian manifold without focal points, it is shown that the measure $ \mu_{\max} $ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $ \mu_{\max} $ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $ SM $ with respect to $ \mu_{\max} $

We prove the existence of infinitely many sign changing radial solutions for a $ p $-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the

In this paper, we present two constructions of forward self-similar solutions to the $ 3 $D incompressible Navier-Stokes system, as the singular limit of forward self-similar solutions to certain parabolic systems.

For a discrete system $ (X,T) $, the flip system $ (X,T,F) $ can be regarded as the action of infinite dihedral group $ D_\infty $ on the space $ X $. Under this action, $ X $ is partitioned into a set of orbits. We are interested in counting the finite orbits in this partition via the prime orbit counting function. In this paper, we prove the asymptotic behaviour of this counting function for the

In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential $ u(\omega t) $. We show that if the frequency vector $ \omega $ is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy $ E $. In contrast with previous results, the conditions of small potential $ u $ or large energy

We consider the wave equation with a cubic convolution $ \partial_{t}^2 u-\Delta u = (|x|^{- \gamma}*u^2)u $

We show that weak solutions to parabolic equations in divergence form with conormal boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain integrability conditions.

Recently, Lindenstrauss and Tsukamoto established a double variational principle between mean dimension theory and rate distortion theory. The main purpose of this paper is to develop some new variational relations for the metric mean dimension and the rate distortion dimension. Inspired by the dimension theory of topological entropy, we introduce and explore the Bowen metric mean dimension of subsets

We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $ M^{2,p}_{s}( \mathbb{R}) $ for $ s \ge \frac14 $ and $ 2\leq p < \infty $. For $ s < \frac 14 $, we show that the solution map for mKdV is not locally uniformly continuous in $ M^{2,p}_{s}( \mathbb{R}) $. By combining this local well-posedness

In this paper we study the local and global existence of solutions for a class of heat equation in whole $ \mathbb{R}^{N} $ where the nonlinearity has a critical growth for $ N \geq 2 $. In order to prove the global existence, we will use the potential well theory combined with the Nehari manifold, and also with the Pohozaev manifold that is a novelty for this type of problem. Moreover, the blow-up

This work investigates the dynamics of competitive Kolmogorov systems formulated in a semi-Markov regime-switching framework. The conditional holding time of each environmental regime is allowed to follow arbitrary probability distribution on the nonnegative half-line in the sense of approximations. Sharp sufficient conditions of the coexistence and competitive exclusion of species are established

We study the quasilinear elliptic equation $ \begin{equation*} -Qu = e^u ~~~~\mbox{in}~~~~ \Omega\subset \mathbb{R}^{N}, \end{equation*} $

Considering the Cauchy problem of the derivative NLS $ \begin{align} {\rm i} u_{t} + \partial_{xx} u = {\rm i} \mu \partial_x (|u|^2u),\quad u(0,x) = u_0(x), \end{align} $

In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers $ {\lambda}_0 $ and $ {\lambda}_1 $ for the control parameter $ {\lambda} $ in the equation. Motivated by [9], we assume that $ {\lambda}_0< {\lambda}_1 $ and the linearized operator at the trivial solution has multiple critical eigenvalues $ \beta_N^+