In this paper, we consider a class of self-similar sets, denoted by $ \mathcal{A} $, and investigate the set of points in the self-similar sets having unique codings. We call such set the univoque set and denote it by $ U_1 $. We analyze the isomorphism and bi-Lipschitz equivalence between the univoque sets. The main result of this paper, in terms of the dimension of $ U_1 $, is to give several equivalent

A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernel and distribution functions. The model describes the dynamics of particle growth when binary collisions occur to form either a single particle via coalescence or two/more particles via breakup with possible transfer of mass. Each

We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results.

The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

We study equivariant Gromov-Hausdorff distances for general actions which are not necessarily isometric as Fukaya introduced. We prove that if an action is expansive and has the pseudo-orbit tracing property then it is stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using Lott and Villani's ideas of optimal transport, we investigate equivariant Gromov-Hausdorff convergence

In this paper, we prove the existence of local stable and unstable invariant manifolds for a class of random differential equations driven by nonlinear colored noise defined in a fractional power of a separable Banach space. In the case of linear noise, we show the pathwise convergence of these random invariant manifolds as well as invariant foliations as the correlation time of the colored noise approaches

We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In the general case in dimension $ d = 1 $, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and

We obtain some new Morawetz estimates for the Klein-Gordon flow of the form $ \begin{equation*} \big\| |\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^{2}_{x, t}(|(x, t)|^{-\alpha})} \lesssim \| f \|_{H^s} \end{equation*} $

In this paper we study the initial boundary value problem for the system $ -\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right] = s(x), \ \ \mathbf{m}_t-\alpha^2\Delta\mathbf{m}+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m} = \beta^2(\mathbf{m}\cdot\nabla p)\nabla p $ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical

As a natural counterpart to Nakada's $ \alpha $-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension

Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions

We study the dynamics and energy exchanges between a linear oscillator and a nonlinear interaction state for the one dimensional, quintic nonlinear Schrödinger equation. Grébert and Thomann [9] proved that there exist solutions with initial data built on four Fourier modes, that confirm the conservative exchange of wave energy. Captured multi resonance in multiple Fourier modes, we simulate a similar

In this paper, we consider the chemotaxis–shallow water system in a bounded domain $ \Omega\subset\mathbb{R}^2 $. By energy method, we establish the global existence of strong solution with small initial perturbation and obtain the exponential decaying rate of the solution. We divide the bounded domain into interior domain and the domain up to the boundary. In the interior domain, the problem is treated

We are concerned with the inverse problem of identifying magnetic anomalies with varying parameters beneath the Earth using geomagnetic monitoring. Observations of the change in Earth's magnetic field–the secular variation–provide information about the anomalies as well as their variations. In this paper, we rigorously establish the unique recovery results for this magnetic anomaly detection problem

We consider the Cauchy problem for focusing nonlinear Schrödinger equation $ i\partial_t u + \Delta u = - |u|^\alpha u, \quad (t, x) \in \mathbb R \times \mathbb R^N, $

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in [19,20]. Roughly speaking, we show essentially that if the

Studied herein is the local well-posedness of solutions with the low regularity in the periodic setting for a class of one-dimensional generalized Rotation-Camassa-Holm equation, which is considered as an asymptotic model to describe the propagation of shallow-water waves in the equatorial region with the weak Coriolis effect due to the Earth's rotation. With the aid of the semigroup approach and a

We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

Previously in [14], we considered a diffusive logistic equation with two parameters, $ r(x) $ – intrinsic growth rate and $ K(x) $ – carrying capacity. We investigated and compared two special cases of the way in which $ r(x) $ and $ K(x) $ are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra

This paper studies the very weak solution to the steady MHD type system in a bounded domain. We prove the existence of very weak solutions to the MHD type system for arbitrary large external forces $ ({\bf f},{\bf{g}}) $ in $ L^r({\Omega})\times [X_{\theta',q'}({\Omega})]' $ and suitable boundary data $ ({\mathcal B}^0,{\mathcal U}^0) $ in $ W^{-1/p,p}({\partial}{\Omega})\times W^{-1/q,q}({\partial}{\Omega})

This article is a review of Lévy processes, stochastic integration and existence results for stochastic differential equations and stochastic partial differential equations driven by Lévy noise. An abstract PDE of the typical type encountered in fluid mechanics is considered in a stochastic setting driven by a general Lévy noise. Existence and uniqueness of a local pathwise solution is established

The goal of this paper is to study the two-dimensional inviscid Boussinesq equations with temperature-dependent thermal diffusivity. Firstly we establish the global existence theory and regularity estimates for this system with Yudovich's type initial data. Then we investigate the vortex patch problem, and proving that the patch remains in Hölder class $ C^{1+s}\; (0

We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting $ N $-particle system, and the $ N $ times tensorized product of the solution to the corresponding limit nonlinear conservation law. It then enables to recover classical propagation of chaos results [20] in the case of Lipschitz coefficients, uniform in time propagation of chaos in [17] in

We provide the simple proof of the Adams type inequalities in whole space $ {\mathbb R}^{2m} $. The main tools are the Fourier rearrangement technique introduced by Lenzmann and Sok [16], a Hardy–Rellich type inequality for radial functions, and the sharp Moser–Trudinger type inequalities in $ {\mathbb R}^2 $.

A boolean network is a map $ F:\{0,1\}^n \to \{0,1\}^n $ that defines a discrete dynamical system by the subsequent iterations of $ F $. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested

By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain $ C^{2} $ $ \mathbb{N}^2 $-actions

The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates

We prove the existence of at least one positive solution for a Schrödinger equation in $ \mathbb{R}^N $

In this paper, we propose and investigate a memory-based reaction-diffusion equation with nonlocal maturation delay and homogeneous Dirichlet boundary condition. We first study the existence of the spatially inhomogeneous steady state. By analyzing the associated characteristic equation, we obtain sufficient conditions for local stability and Hopf bifurcation of this inhomogeneous steady state, respectively

The centralizer (automorphism group) and normalizer (extended symmetry group) of the shift action inside the group of self-homeomorphisms are studied, in the context of certain $ \mathbb{Z}^d $ subshifts with a hierarchical supertile structure, such as bijective substitutive subshifts and the Robinson tiling. Restrictions on these groups via geometrical considerations are used to characterize explicitly

We study KMS states for gauge actions with potential functions on Cuntz–Krieger algebras whose underlying one-sided topological Markov shifts are continuously orbit equivalent. As a result, we have a certain relationship between topological entropy of continuously orbit equivalent one-sided topological Markov shifts.

In this paper, an infinite dimensional KAM theorem with double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with simple normal frequencies in [3] by Bambusi, Berti and Magistrelli. As applications, for nonlinear wave equation and nonlinear Schr$ \ddot{\mbox{o}} $dinger equation with periodic boundary conditions, quasi-periodic

This paper is mainly concerned with the classification of the general two-component $ \mu $-Camassa-Holm systems with quadratic nonlinearities. As a conclusion of such classification, a two-component $ \mu $-Camassa-Holm system admitting multi-peaked solutions and $ H^1 $-norm conservation law is found, which is a $ \mu $-version of the two-component modified Camassa-Holm system and can be derived

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations $ \partial_t u-|Du|^\gamma\Delta_p^N u = f, $

We study the influence of Szegő projector on the $ L^2- $

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: $ u_t = u_{xx}+x^m|u_x|^p, \ \ t>0, \ \ 0

We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling

We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.

We study the regularity of exceptional actions of groups by $ C^{1, \alpha} $ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $ \alpha $. Let $ G $ be a finitely generated group admitting a $ C^{1, \alpha} $ action $ \rho $ with a free orbit on the circle, and such that the

Some neurons in the nervous system do not show repetitive firing for steady currents. For time-varying inputs, they fire once if the input rise is fast enough. This property of phasic firing is known as Type III excitability. Type III excitability has been observed in neurons in the auditory brainstem (MSO), which show strong phase-locking and accurate coincidence detection. In this paper, we consider

A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A rigorous classification approach is to characterize the bifurcations found in the

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.