arXiv subjects

Yuncheng You

Publications and source records attributed to Yuncheng You.

25 records · Page 2Linked to original sources

arXiv2017

Global attractors for the damped nonlinear wave equation in unbounded domains

The existence of a global attractor for wave equations in unbounded domains is a challenging problem due to the non-compactness of the Sobolev embeddings. To overcome this difficulty, some authors have worked with weighted Sobolev spaces which restrict the choice of the initial data. Using the "tail estimation method" introduced by B. Wang for reaction diffusion equations, we establish in this paper the existence of a global attractor for two wave equations in the traditional Hilbert spaces $\displaystyle H^1(Ω)\times L^2(Ω)$ where $Ω$ is an unbounded domain of $\R^N$. The first equation, with a mass term is studied in the whole space $\R^N$ and the second one without mass term is considered in a domain bounded in only one direction so that Poincaré inequality will hold.

math.AP↗

arXiv2014

Random Attractor for Stochastic Wave Equation with Arbitrary Exponent and Additive Noise on $\mathbb{R}^n$

Asymptotic random dynamics of weak solutions for a damped stochastic wave equation with the nonlinearity of arbitrarily large exponent and the additive noise on $\mathbb{R}^n$ is investigated. The existence of a pullback random attractor is proved in a parameter region with a breakthrough in proving the pullback asymptotic compactness of the cocycle with the quasi-trajectories defined on the integrable function space of arbitrary exponent and on the unbounded domain of arbitrary dimension.

math.AP↗

arXiv2011

Longtime Dynamics of The Oregonator System

In this work the existence and properties of a global attractor for the solution semiflow of the Oregonator system are proved. The Oregonator system is the mathematical model of the famous Belousov-Zhabotinskii reaction. A rescaling and grouping estimation method is developed to show the absorbing property and the asymptotic compactness of the solution trajectories of this three-variable reaction-diffusion system with quadratic nonlinearity from the autocatalytic kinetics. It is proved that the fractal dimension of the global attractor is finite. The existence of an exponential attractor for this Oregonator semiflow is also shown.

math.DS↗

arXiv2011

2D Backward Stochastic Navier-Stokes Equations with Nonlinear Forcing

The paper is concerned with the existence and uniqueness of a strong solution to a two-dimensional backward stochastic Navier-Stokes equation with nonlinear forcing, driven by a Brownian motion. We use the spectral approximation and the truncation and variational techniques. The methodology features an interactive analysis on basis of the regularity of the deterministic Navier-Stokes dynamics and the stochastic properties of the Itô-type diffusion processes.

math.PR↗

arXiv2011

Global Dissipative Dynamics of the Extended Brusselator System

The existence of a global attractor for the solution semiflow of the extended Brusselator system in the $L^2$ phase space is proved, which is a cubic-autocatalytic and partially reversible reaction-diffusion system with linear coupling between two compartments. The method of grouping and re-scaling estimation is developed to deal with the challenge in proving the absorbing property and the asymptotic compactness of this typical multi-component reaction-diffusion systems. It is also proved that the global attractor is an $(H, E)$ global attractor with the $L^\infty$ regularity and that the Hausdorff dimension and the fractal dimension of the global attractor are finite. The results and methodology can find many applications and further extensions in complex biological and biochemical dynamical systems.

math.AP↗

arXiv2009

Global Attractor of a coupled Two-Cell Brusselator Model

In this work the existence of a global attractor for the solution semiflow of the coupled two-cell Brusselator model equations is proved. A grouping estimation method and a new decomposition approach are introduced to deal with the challenge in proving the absorbing property and the asymptotic compactness of this type of four-variable reaction-diffusion systems with cubic autocatalytic nonlinearity and with linear coupling. It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite.

math.DS↗

arXiv2005

Solving the Korteweg-de Vries Equation by Its Bilinear Form: Wronskian Solutions

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

nlin.SI↗