arXiv subjects

Xiaoyan Wang

Publications and source records attributed to Xiaoyan Wang.

41 records · Page 3Linked to original sources

arXiv2015

Oscillation and Instability in Chemical Reactions

We prove that the famous diffusive Brusselator model can support more complicated spatial-temporal wave structure than the usual temporal-oscillation from a standard Hopf bifurcation. In our investigation, we discover that the diffusion term in the model is neither a usual parabolic stabilizer nor a destabilizer as in the Turing instability of uniform state, but rather plays the role of maintaining an equivariant Hopf bifurcation spectral mechanism. At the same time, we show that such a mechanism can occur around any nonzero wave number and this finding is also different from the former works where oscillations caused by diffusion can cause the growth of wave structure only at a particular wavelength. Our analysis also demonstrates that the complicated spatial-temporal oscillation is not solely driven by the inhomogeneity of the reactants.

math.AP↗

arXiv2015

Dynamical behavior of a system modeling wave bifurcations with higher order viscosity

We rigorously show that a class of systems of partial differential equations modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics on the center manifold(s). A direct consequence of our result is that the oscillations of the dynamics are \textit{not} due to rotation waves though the system exhibits Euclidean symmetries. The main difficulties of carrying out the program are: 1) the system under study contains multi bifurcation parameters and we do not know \textit{a priori} how they come into play in the bifurcation dynamics. 2) the representation of the linear operator on the center space is a $2\times 2$ zero matrix, which makes the characteristic condition in the well-known normal form theorem trivial. We overcome the first difficulty by using projection method. We managed to overcome the second subtle difficulty by using a conjugate pair coordinate for the center space and applying duality and projection arguments. Due to the specific complex pair parametrization, we could naturally get a form of the center manifold reduction function, which makes the study of the current dynamics on the center manifold possible. The symmetry of the system plays an essential role in excluding the possibility of bifurcating rotation waves.

math.AP↗

arXiv2012

High Energy Solutions to $p(x)-$Laplacian Equations of Schrödinger type

In this paper, we investigate nonlinear Schr$\ddot{o}$dinger type equations in $R^N$ under the framework of variable exponent spaces. We propose new assumptions on the nonlinear term to yield bounded Palais-Smale sequences and then prove the special sequences we find converge to critical points respectively. The main arguments are based on the geometry supplied by Fountain Theorem. Consequently, we show that the equation has a sequence of solutions with high energies.

math.AP↗

arXiv2012

Solutions for Neumann boundary value problems involving $\big(p_{1}(x), p_{2}(x)\big)$-Laplace operators

In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=λf(x,u) in a bounded smooth domain $Ω\subset\mathbb{R}^{N}$ with Neumann boundary condition given by |\nabla u|^{p_{1}(x)-2}\frac{\partial u}{\partialν}+|\nabla u|^{p_{2}(x)-2}\frac{\partial u}{\partialν}=μg(x,u) on $\partialΩ$. Under appropriate conditions on the source and boundary nonlinearities, we obtain a number of results on existence and multiplicity of solutions by variational methods in the framework of variable exponent Lebesgue and Sobolev spaces.

math.AP↗

arXiv2012

On $\big(p_{1}(x), p_{2}(x)\big)$-Laplace Equations

In this paper, we investigate the $(p_{1}(x), p_{2}(x))$-Laplace operator, the properties of the corresponding integral functional and weak solutions to the related differential equations. We show that the integral functional admits a derivative of type $(S_+)$ which induces a homeomorphism between duality space pairs. As applications of the above results, we gave some existence results of the $(p_{1}(x), p_{2}(x))$-Laplace equation -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)=f(x,u) in a bounded smooth domain $Ω\subset\mathbb{R}^{N}$ with Dirichlet boundary condition.

math.FA↗