Discrete & Continuous Dynamical Systems - S最新文献

For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency begin{document}$ omega $end{document} of an integrable system by a trigonometric polynomial of degree begin{document}$ N $end{document} perturbation begin{document}$ R_N $end{document} with begin{document}$ |R_N|_{C^r}. We obtain a relation among begin{document}$ N $end{document}, begin{document}$ r $end{document}, begin{document}$ epsilon $end{document} and the arithmetic property of begin{document}$ omega $end{document}, for which the area-preserving map admit no invariant circles with begin{document}$ omega $end{document}.

The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in begin{document}$ {{mathbb{R}}}^d $end{document}, begin{document}$ 1leq dleq 3 $end{document}, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case begin{document}$ d = 2 $end{document}. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.

Let begin{document}$ (X,T) $end{document} be a topological dynamical system and begin{document}$ ngeq 2 $end{document}. We say that begin{document}$ (X,T) $end{document} is begin{document}$ n $end{document}-tuplewise IP-sensitive (resp. begin{document}$ n $end{document}-tuplewise thickly sensitive) if there exists a constant begin{document}$ delta>0 $end{document} with the property that for each non-empty open subset begin{document}$ U $end{document} of begin{document}$ X $end{document}, there exist begin{document}$ x_1,x_2,dotsc,x_nin U $end{document} such that

begin{document}$ Bigl{kin mathbb{N}colon minlimits_{1le ideltaBigr} $end{document}

is an IP-set (resp. a thick set).

We obtain several sufficient and necessary conditions of a dynamical system to be begin{document}$ n $end{document}-tuplewise IP-sensitive or begin{document}$ n $end{document}-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is begin{document}$ n $end{document}-tuplewise IP-sensitive for all begin{document}$ ngeq 2 $end{document}, while it is begin{document}$ n $end{document}-tuplewise thickly sensitive if and only if it has at least begin{document}$ n $end{document} minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IPbegin{document}$ ^* $end{document}-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IPbegin{document}$ ^* $end{document}-equicontinuous. We show that every minimal system adm

The aim of this paper is to give global nonexistence and blow–up results for the problem

begin{document}$ begin{cases} u_{tt}-Delta u+P(x,u_t) = f(x,u) qquad &text{in $(0,infty)timesOmega$,} u = 0 &text{on $(0,infty)times Gamma_0$,} u_{tt}+partial_nu u-Delta_Gamma u+Q(x,u_t) = g(x,u)qquad &text{on $(0,infty)times Gamma_1$,} u(0,x) = u_0(x),quad u_t(0,x) = u_1(x) & text{in $overline{Omega}$,} end{cases} $end{document}

where begin{document}$ Omega $end{document} is a bounded open begin{document}$ C^1 $end{document} subset of begin{document}$ {mathbb R}^N $end{document}, begin{document}$ Nge 2 $end{document}, begin{document}$ Gamma = partialOmega $end{document}, begin{document}$ (Gamma_0,Gamma_1) $end{document} is a partition of begin{document}$ Gamma $end{document}, begin{document}$ Gamma_1not = emptyset $end{document} being relatively open in begin{document}$ Gamma $end{document}, begin{document}$ Delta_Gamma $end{document} denotes the Laplace–Beltrami operator on begin{document}$ Gamma $end{document}, begin{document}$ nu $end{document} is the outward normal to begin{document}$ Omega $end{document}, and the terms begin{document}$ P $end{document} and begin{document}$ Q $end{document} represent nonlinear damping terms, while begin{document}$ f $end{document} and begin{document}$ g $end{document} are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems begin{document}$ H(x,u,p) $end{document} with certain dependence on the contact variable begin{document}$ u $end{document}. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set begin{document}$ tilde{mathcal{S}}_s $end{document} consists of strongly static orbits, which coincides with the Aubry set begin{document}$ tilde{mathcal{A}} $end{document} in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show begin{document}$ tilde{mathcal{S}}_ssubsetneqqtilde{mathcal{A}} $end{document} in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of begin{document}$ H $end{document} on begin{document}$ u $end{document} fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.