On the decay of solutions in nonsimple elastic solids with memory

The decay of solutions in nonsimple elasticity with memory is addressed, analyzing how the decay rate is influenced by the different dissipation mechanisms appearing in the equations. In particular, a first order dissipation is shown to guarantee the asymptotic stability of the related solution semigroup, but is not strong enough to entail exponential stability. The latter occurs for a dissipation mechanism of the second order, that is, the same order as the one of the leading operator.     

Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain

In this paper, we study the asymptotic behavior of solutions for the plate equation with a localized damping and a critical exponent. We prove the existence, regularity and finite dimensionality of a global attractor in .     

Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and an integral equation . We characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.     

具有强阻尼的非自治吊桥方程一致吸引子的存在性

In this paper we show the existence of the uniform attractors for the family of processes corresponding to the suspension bridge equations in H02 × L2 by a new concept of Condition (C*) and the enegy estimats methods.     

The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form.     

Local dependence functions for some families of bivariate distributions and total positivity

The purpose of this paper is to investigate a very useful application of a certain local dependence function γ_f(x,y), which was considered recently by Holland and Wang [20]. An interesting property of γ_f(x,y) is that the underlying joint density f(x,y) is TP₂ (that is, totally positive of order 2) if and only if . This gives an elegant way to investigate the TP₂ property of any bivariate distribution. For the Saramanov family, the Ali-Mikhail-Haq family of bivariate distributions and the family of bivariate elliptical distributions, we derive the local dependence function and obtain conditions for f(x,y) to be TP₂. These families are quite rich and include many other large classes of bivariate distributions as their special cases. Similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with Pareto conditionals.     

Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci

In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equilibrium structure of the chaotic system and can be applied to degenerate Hopf bifurcation at the desired location with preferred stability.     

Long Time Behavior of the Cahn-Hilliard Equation with Irregular Potentials and Dynamic Boundary Conditions

The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered.The existence of the global attractor is proved and the long time behavior of the trajectories,namely,the convergence to steady states,is studied.     

Well-posedness and dynamics for the fractional Ginzburg-Landau equation

This article studies the global well-posedness and long-time dynamics for the nonlinear complex Ginzburg–Landau equation involving fractional Laplacian. The global existence and some uniqueness criterion of weak solutions are given with compactness method. To study the strong solutions with the semigroup method, we generalize some pointwise estimates for the fractional Laplacian to the complex background and study carefully the linear evolution of the equation. Finally, the existence of global attractors is studied.