Discrete & Continuous Dynamical Systems - S最新文献

In this paper we completely solve the problem of when a Cantor dynamical system begin{document}$ (X, f) $end{document} can be embedded in begin{document}$ mathbb{R} $end{document} with vanishing derivative. For this purpose we construct a refining sequence of marked clopen partitions of begin{document}$ X $end{document} which is adapted to a dynamical system of this kind. It turns out that there is a huge class of such systems.

By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as begin{document}$ det D^2 u = |u|^{-n-2-k} (xcdot Du -u)^{-k} $end{document} with zero boundary data, have unexpected degenerate nature.

We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in begin{document}$ {{bf R}}^{n} $end{document}, and study the asymptotic profile and optimal decay rates of solutions as begin{document}$ t to infty $end{document} in begin{document}$ L^{2} $end{document}-sense. The operator begin{document}$ L $end{document} considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [7]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.

In this article we consider the following weighted nonlinear eigenvalue problem for the begin{document}$ g- $end{document}Laplacian

begin{document}$ -{text{ div}}left( g(|nabla u|)frac{nabla u}{|nabla u|}right) = lambda w(x) h(|u|)frac{u}{|u|} quad text{ in }Omegasubset mathbb R^n, ngeq 1 $end{document}

with Dirichlet boundary conditions. Here begin{document}$ w $end{document} is a suitable weight and begin{document}$ g = G' $end{document} and begin{document}$ h = H' $end{document} are appropriated Young functions satisfying the so called begin{document}$ Delta' $end{document} condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of begin{document}$ G $end{document}, begin{document}$ H $end{document}, begin{document}$ w $end{document} and the normalization begin{document}$ mu $end{document} of the corresponding eigenfunctions.

We introduce some new strategies to obtain results that generalize several inequalities from the literature of begin{document}$ p- $end{document}Laplacian type eigenvalues.

Any begin{document}$ C^d $end{document} conservative map begin{document}$ f $end{document} of the begin{document}$ d $end{document}-dimensional unit ball begin{document}$ {mathbb B}^d $end{document}, begin{document}$ dgeq 2 $end{document}, can be realized by renormalized iteration of a begin{document}$ C^d $end{document} perturbation of identity: there exists a conservative diffeomorphism of begin{document}$ {mathbb B}^d $end{document}, arbitrarily close to identity in the begin{document}$ C^d $end{document} topology, that has a periodic disc on which the return dynamics after a begin{document}$ C^d $end{document} change of coordinates is exactly begin{document}$ f $end{document}.

A set begin{document}$ E subset mathbb{N} $end{document} is an interpolation set for nilsequences if every bounded function on begin{document}$ E $end{document} can be extended to a nilsequence on begin{document}$ mathbb{N} $end{document}. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here begin{document}$ {r_n: n in mathbb{N}} subset mathbb{N} $end{document} with begin{document}$ r_1 < r_2 < ldots $end{document} is sublacunary if begin{document}$ lim_{n to infty} (log r_n)/n = 0 $end{document}. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for begin{document}$ 2 $end{document}-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.