ISSN:

1531-3492

eISSN:

1553-524X

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## Discrete & Continuous Dynamical Systems - B

July 2019 , Volume 24 , Issue 7

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*+*[Abstract](1520)

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**Abstract:**

This paper studies the parabolic-elliptic Keller-Segel system with supercritical sensitivity:

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**Abstract:**

Two properties of stochastic heat equations driven by impulsive noises, which are also called Lévy space-time white noises, are mainly investigated in this paper. We first study the comparison theorem for two stochastic heat equations driven by same noises under some sufficient condition, which is proved via the application of Itô's formula. In particular, we obtain the non-negativity of solutions with non-negative initial data. Then, we investigate the positive correlation of the solutions as the application of the comparison theorem. We prove that the total masses of two solutions relative to two different stochastic heat equations with same noise become nearly uncorrelated after a long time.

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**Abstract:**

In this paper, we study a tumor growth equation along with various models for the nutrient component, including a *in vitro* model and a *in vivo* model. At the cell density level, the spatial availability of the tumor density

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**Abstract:**

We consider a compound Poisson risk process perturbed by a Brownian motion through using a potential measure where the claim sizes depend on inter-claim times via the Farlie-Gumbel-Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for the distribution of the maximum surplus before ruin. This distribution can be calculated through the probability that the surplus process attains a given level from the initial surplus without first falling below zero. The explicit expressions for this distribution are derived when the claim amounts are exponentially distributed.

*+*[Abstract](1349)

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**Abstract:**

Hysteresis is an important issue in modeling piezoelectric materials, for example, in applications to energy harvesting, where hysteresis losses may influence the efficiency of the process.The main problem in numerical simulations is the inversion of the underlying hysteresis operator.Moreover, hysteresis dissipation is accompanied with heat production, which in turn increases thetemperature of the device and may change its physical characteristics. More accurate models thereforehave to take the temperature dependence into account for a correct energy balance.We prove here that the classical Preisach operator with a fairly general parameter-dependenceadmits a Lipschitz continuous inverse in the space of right-continuous regulated functions, propose a time-discrete and memory-discrete inversion algorithm, and show that higher regularity of the inputs leads to a higher regularity of the output of the inverse.

*+*[Abstract](1532)

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**Abstract:**

In this paper, we will prove the uniqueness of traveling front solutions with critical and noncritical speeds, connecting the origin and the positive equilibrium, for the classical competitive Lotka-Volterra system with diffusion in the weak competition, which partially answers the open problem presented by Tang and Fife in [

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**Abstract:**

In this paper, the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system is investigated. By using the Lyapunov-Schmidt method, combining with the implicit function theorem, we prove that this system bifurcates from the trivial solution to the nontrivial solution branch as parameter crosses certain critical value. The expression of bifurcated solution is also obtained.

*+*[Abstract](1315)

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**Abstract:**

In this paper, we apply a Lyapunov functional approach to Lotka-Volterra systems with infinite delays and feedback controls and establish that the feedback controls have no influence on the attractivity properties of a saturated equilibrium. This improves previous results by the authors and others, where, while feedback controls were used mostly to change the position of a unique saturated equilibrium, additional conditions involving the controls had to be assumed in order to preserve its global attractivity. The situation of partial extinction is further analysed, for which the original system is reduced to a lower dimensional one which maintains its global dynamics features.

*+*[Abstract](1526)

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**Abstract:**

We consider a stochastic nonlinear evolution equation where the domain is given by a fractal set. The linear part of the equation is given by a Laplacian defined on the fractal. This equation generates a random dynamical system. The long time behavior is given by an attractor which has a finite Hausdorff dimension. We would like to reveal the connections between upper and lower estimates of this Hausdorff dimension and the geometry of the fractal. In particular, the parameter which determines these bounds is the spectral exponent of the fractal. Especially for the lower estimate we construct a local unstable random Lipschitz manifold.

*+*[Abstract](1613)

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**Abstract:**

Positive and negative feedback loops in biological regulatory networks appear often in a multi-node manner since regulatory processes are in general multi-step. Although it is well known that interlocked positive and negative feedback loops (iPNFLs) can generate sustained oscillations, how the number of nodes in each loop affects the oscillations remains elusive. By analyzing a model of iPNFLs with multiple nodes, we find that the node number of the negative loop mainly plays a role of amplifying oscillation amplitudes whereas that of the positive loop mainly plays a role of reducing oscillatory regions, both depending on the (competitive or noncompetitive) way of interaction between the two loops. We also find that given an iPNFL network of the same structure, the noncompetitive model is more likely to produce large-amplitude oscillations than the competitive model. These results not only indicate that multi-node iPNFLs are an effective mechanism of promoting oscillations but also are helpful for the design of synthetic oscillators.

*+*[Abstract](1447)

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**Abstract:**

The previous works focus on the uniqueness for the initial-value problems of stochastic primitive equations. Uniqueness for the initial-value problems means that if the two initial conditions are the same, then the two solutions coincide with each other. However there is no work to answer what will happen to the solutions if the two initial conditions are different. This problem for the stochastic three dimensional primitive equations is addressed by the backward uniqueness established in this article. The backward uniqueness means that if two solutions intersect at time $t>0, $ then they are equal everywhere on the interval $(0, t).$ In other words, given two different initial-value conditions, the corresponding two solutions will never cross in the future. Hence this article can be viewed as a further study of the dependence of the solutions on the initial data.

*+*[Abstract](1472)

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**Abstract:**

The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.

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**Abstract:**

We derive some regularity estimates of the solution to a time fractional diffusion equation by using the Galerkin method. The regularity estimates partially unravel the singularity structure of the solution with respect to the time variable. We show that the regularity of the weak solution can be improved by subtracting some particular forms of singular functions.

*+*[Abstract](1318)

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**Abstract:**

The present paper is concerned with the problem of determining the rate of convergence of global attractors of the family of dissipative semilinear thermoelastic systems with variable coefficients

where

*+*[Abstract](1511)

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**Abstract:**

In this paper, the groundwater flow equation within an unconfined aquifer is modified using the concept of new derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. Some properties and applications are given regarding the Caputo-Fabrizio fractional order derivative. The existence and the uniqueness of the solution of the modified groundwater flow equation within an unconfined aquifer is presented, the proof of the existence use the definition of Caputo-Fabrizio integral and the powerful fixed-point Theorem. A detailed analysis on the uniqueness is included. We perform on the numerical analysis on which the Crank-Nicolson scheme is used for discretisation. Then we present in particular the proof of the stability of the method, the proof combine the Fourier and Von Neumann stability analysis. A detailed analysis on the convergence is also achieved.

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**Abstract:**

We are concerned with the breakdown of strong solutions to the three-dimensional compressible magnetohydrodynamic equations with density-dependent viscosity. It is shown that for the initial density away from vacuum, the strong solution exists globally if the gradient of the velocity satisfies

*+*[Abstract](1299)

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**Abstract:**

In this paper, the asymptotic wave behavior of the solution for the nonlinear damped wave equation in

*+*[Abstract](1291)

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**Abstract:**

This paper deals with new results on existence, uniqueness and stability for a class of nonlinear beams arising in connection with nonlocal dissipative models for flight structures with *energy damping* first proposed by Balakrishnan-Taylor [

where

*+*[Abstract](1447)

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**Abstract:**

In this paper, by the use of martingale property and spectral decomposition theory, we investigate the stochastic invariance for neutral stochastic functional differential equations (NSFDEs) and provide necessary and sufficient conditions for the invariance of closed sets of

*+*[Abstract](1291)

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**Abstract:**

We investigate the collective behavior of synchrony for the Kuramoto and Winfree models. We first prove the *global convergence* of frequency synchronization for the non-identical Kuramoto system of three oscillators. It is shown that the uniform boundedness of the diameter of the phase functions implies complete frequency synchronization. In light of this, we show, under a suitable condition on the coupling strength and deviation of the intrinsic frequencies, that the diameter function of the phases is uniformly bounded. In a similar spirit, we also prove the *global convergence* of phase-locked synchronization for the Winfree model of

*+*[Abstract](1474)

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**Abstract:**

In this article, we study the initial-value problem for inhomogeneous fractional nonlinear Schrödinger equation

where

*+*[Abstract](1393)

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**Abstract:**

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source

in a smooth bounded domain

the considered problem possesses a global classical solution which is bounded, where

*+*[Abstract](1290)

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**Abstract:**

This paper investigates the issue of weighted exponentially input to state stability (EISS, in short) of stochastic coupled systems on networks with time-varying delay driven by

*+*[Abstract](1253)

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**Abstract:**

In this article, a notion of *bi-spatial continuous random dynamical system* is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bi-spatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the non-autonomous stochastic fractional power dissipative equation on

*+*[Abstract](1209)

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**Abstract:**

An inhibitory uptake function is incorporated into the discrete, size-structured nonlinear chemostat model developed by Arino et al. (Journal of Mathematical Biology, 45(2002)). Different from the model with a monotonically increasing uptake function, we show that the inhibitory kinetics can induce very complex dynamics including stable equilibria, cycles and chaos (via the period-doubling cascade). In particular, when the nutrient concentration in the input feed to the chemostat

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