ISSN:

1078-0947

eISSN:

1553-5231

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## Discrete & Continuous Dynamical Systems

October 2020 , Volume 40 , Issue 10

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**Abstract:**

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

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**Abstract:**

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 as a demonstration of general techniques in the area.

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**Abstract:**

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

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We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting

*+*[Abstract](1276)

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We provide the simple proof of the Adams type inequalities in whole space

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**Abstract:**

A boolean network is a map *firing memory* and it effects in the dynamics of symmetric (non-directed) *conjunctive networks*. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is **PSPACE**-complete.

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**Abstract:**

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

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**Abstract:**

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. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, *Calc. Var. Partial Differential Equations*, **55** (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion.

*+*[Abstract](1010)

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**Abstract:**

We prove the existence of at least one positive solution for a Schrödinger equation in

with a vanishing potential at infinity and subcritical nonlinearity

*+*[Abstract](1053)

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**Abstract:**

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. For the Hopf bifurcation analysis, a geometric method and prior estimation techniques are combined to find all bifurcation values because the characteristic equation includes a non-self-adjoint operator and two time delays. In addition, we provide an explicit formula to determine the crossing direction of the purely imaginary eigenvalues. The bifurcation analysis reveals that the diffusion with memory effect could induce spatiotemporal patterns which were never possessed by an equation without memory-based diffusion. Furthermore, these patterns are different from the ones of a spatial memory equation with Neumann boundary condition.

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**Abstract:**

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

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**Abstract:**

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.

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**Abstract:**

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 [

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**Abstract:**

This paper is mainly concerned with the classification of the general two-component

*+*[Abstract](868)

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In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations

where

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**Abstract:**

We study the influence of Szegő projector on the

It has no Galilean invariance but the momentum

for some

*+*[Abstract](1061)

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**Abstract:**

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term:

where

*+*[Abstract](833)

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**Abstract:**

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 offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.

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**Abstract:**

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.

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**Abstract:**

We study the regularity of exceptional actions of groups by

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