Discrete & Continuous Dynamical Systems - S最新文献

Motivated by radiation hydrodynamics, we analyse a begin{document}$ 2times2 $end{document} system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called sub-shock– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on Geometric Singular Perturbation Theory (GSPT) as introduced in the pioneering work of Fenichel [5] and subsequently developed by Szmolyan [21]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.

We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential begin{document}$ ell^1 $end{document}-stability and the existence of the equilibrium solution.

We prove existence and uniqueness of eternal solutions in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation

begin{document}$ partial_tu = Delta u^m+|x|^{sigma}u^p, $end{document}

posed in begin{document}$ mathbb{R}^N $end{document}, with begin{document}$ m>1 $end{document}, begin{document}$ 0 and the critical value for the weight

begin{document}$ sigma = frac{2(1-p)}{m-1}. $end{document}

Existence and uniqueness of some specific solution holds true when begin{document}$ m+pgeq2 $end{document}. On the contrary, no eternal solution exists if begin{document}$ m+p<2 $end{document}. We also classify exponential self-similar solutions with a different interface behavior when begin{document}$ m+p>2 $end{document}. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.

Consider begin{document}$ beta > 1 $end{document} and begin{document}$ lfloor beta rfloor $end{document} its integer part. It is widely known that any real number begin{document}$ alpha in Bigl[0, frac{lfloor beta rfloor}{beta - 1}Bigr] $end{document} can be represented in base begin{document}$ beta $end{document} using a development in series of the form begin{document}$ alpha = sum_{n = 1}^infty x_nbeta^{-n} $end{document}, where begin{document}$ x = (x_n)_{n geq 1} $end{document} is a sequence taking values into the alphabet begin{document}$ {0,; ...; ,; lfloor beta rfloor} $end{document}. The so called begin{document}$ beta $end{document}-shift, denoted by begin{document}$ Sigma_beta $end{document}, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy begin{document}$ beta $end{document}-expansion of begin{document}$ 1 $end{document}. Fixing a Hölder continuous potential begin{document}$ A $end{document}, we show an explicit expression for the main eigenfunction of the Ruelle operator begin{document}$ psi_A $end{document}, in order to obtain a natural extension to the bilateral begin{document}$ beta $end{document}-shift of its corresponding Gibbs state begin{document}$ mu_A $end{document}. Our main goal here is to prove a first level large deviations principle for the family begin{document}$ (mu_{tA})_{t>1} $end{document} with a rate function begin{document}$ I $end{document} attaining its maximum value on the union of the supports of all the maximizing measures of begin{document}$ A $end{document}. The above is proved through a technique using the representation of begin{document}$ Sigma_beta $end{document} and its bilateral extension begin{docu