DCDS
Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations
Yaping Wu Xiuxia Xing
Discrete & Continuous Dynamical Systems - A 2008, 20(4): 1123-1139 doi: 10.3934/dcds.2008.20.1123
This paper is concerned with the stability of the traveling front solutions with critical speeds for a class of $p$-degree Fisher-type equations. By detailed spectral analysis and sub-supper solution method, we first show that the traveling front solutions with critical speeds are globally exponentially stable in some exponentially weighted spaces. Furthermore by Evans's function method, appropriate space decomposition and detailed semigroup decaying estimates, we can prove that the waves with critical speeds are locally asymptotically stable in some polynomially weighted spaces, which verifies some asymptotic phenomena obtained by numerical simulations.
keywords: sub-supper solutions traveling fronts Evans's function. asymptotic stability spectral analysis
DCDS-B
Stability of traveling waves for autocatalytic reaction systems with strong decay
Yaping Wu Niannian Yan
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1601-1633 doi: 10.3934/dcdsb.2017033

This paper is concerned with the spatial decay and stability of travelling wave solutions for a reaction diffusion system $u_{t}=d u_{xx}-uv$, $v_{t}=v_{xx}+uv-Kv^{q}$ with $q>1$. By applying Centre Manifold Theorem and detailed asymptotic analysis, we get the precise spatial decaying rate of the travelling waves with noncritical speeds. Further by applying spectral analysis, Evans function method and some numerical simulation, we proved the spectral stability and the linear exponential stability of the waves with noncritical speeds in some weighted spaces.

keywords: Stability traveling waves spectral analysis Evans function numerical computation
CPAA
Existence of traveling waves with transition layers for some degenerate cross-diffusion systems
Yanxia Wu Yaping Wu
Communications on Pure & Applied Analysis 2012, 11(3): 911-934 doi: 10.3934/cpaa.2012.11.911
This paper is concerned with the existence of traveling waves for a class of degenerate reaction-diffusion systems with cross-diffusion. By applying the analytic singular perturbation method and the theorem of center manifold approximation, we prove the existence of traveling waves with transition layers for the more general degenerate systems with cross-diffusion. Especially for the degenerate S-K-T competition system with cross-diffusion we prove that some new wave patterns exhibiting competition exclusion are induced by the cross-diffusion. We also extend some of the existence results in [5] for the non-cross diffusion systems to the more general degenerate biological systems with cross-diffusion, however the detailed fast-slow structure of the waves for the systems with cross-diffusion is a little different from those for the systems without cross-diffusion.
keywords: degenerate reaction-diffusion system. traveling wave solutions cross-diffusion transition layers Existence
DCDS
The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion
Yaping Wu Qian Xu
Discrete & Continuous Dynamical Systems - A 2011, 29(1): 367-385 doi: 10.3934/dcds.2011.29.367
This paper is concerned with the existence of large positive spiky steady states for S-K-T competition systems with cross-diffusion. Firstly by detailed integral and perturbation estimates, the existence and detailed fast-slow structure of a class of spiky steady states are obtained for the corresponding shadow system, which also verify and extend some existence results on spiky steady states obtained in [10] by different method of proof. Further by applying special perturbation method, we prove the existence of large positive spiky steady states for the original competition systems with large cross-diffusion rate.
keywords: shadow system singular perturbation. existence spiky steady state cross diffusion
KRM
On a chemotaxis model with saturated chemotactic flux
Alina Chertock Alexander Kurganov Xuefeng Wang Yaping Wu
Kinetic & Related Models 2012, 5(1): 51-95 doi: 10.3934/krm.2012.5.51
We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS) system. Our modification is based on a fundamental physical property of the chemotactic flux function---its boundedness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient only when the latter is small, while when the chemoattractant gradient tends to infinity the cell velocity saturates. Unlike the original PKS system, the solutions of the modified model do not blow up in either finite or infinite time in any number of spatial dimensions, thus making it possible to use bounded spiky steady states to model cell aggregation. After obtaining local and global existence results, we use the local and global bifurcation theories to show the existence of one-dimensional spiky steady states; we also study the stability of bifurcating steady states. Finally, we numerically verify these analytical results, and then demonstrate that solutions of the two-dimensional model with nonlinear saturated chemotactic flux function typically develop very complicated spiky structures.
keywords: spiky steady states local and global existence Chemotaxis bifurcation saturation of flux second-order positivity preserving upwind scheme. hybrid finite-volume-finite-difference method
DCDS-B
Stability of travelling waves with noncritical speeds for double degenerate Fisher-Type equations
Yi Li Yaping Wu
Discrete & Continuous Dynamical Systems - B 2008, 10(1): 149-170 doi: 10.3934/dcdsb.2008.10.149
This paper is concerned with the asymptotic stability of travel- ling wave solutions for double degenerate Fisher-type equations. By spectral analysis, each travelling front solution with non-critical speed is proved to be linearly exponentially stable in some exponentially weighted spaces. Further by Evans function method and detailed semigroup estimates each travelling front solution with non-critical speed is proved to be locally algebraically stable to perturbations in some polynomially weighted spaces, and it is also locally exponentially stable to perturbations in some polynomially and exponentially weighted spaces.
keywords: asymptotic stability semigroup estimates travelling waves algebraic decay Evans function. spectral analysis
DCDS
Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations
Yaping Wu Xiuxia Xing Qixiao Ye
Discrete & Continuous Dynamical Systems - A 2006, 16(1): 47-66 doi: 10.3934/dcds.2006.16.47
This paper is concerned with the asymptotic stability of travelling wave front solutions with algebraic decay for $n$-degree Fisher-type equations. By detailed spectral analysis, each travelling wave front solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted $L^{\infty}$ spaces. Further by Evans function method and detailed semigroup estimates, the travelling wave fronts with non-critical speed are proved to be locally algebraically stable to perturbations in some polynomially weighted $L^{\infty}$ spaces. It's remarked that due to the slow algebraic decay rate of the wave at $+\infty,$ the Evans function constructed in this paper is an extension of the definitions in [1, 3, 7, 11, 21] to some extent, and the Evans function can be extended analytically in the neighborhood of the origin.
keywords: travelling wave fronts Asymptotic stability algebraic decay rate spectral analysis semigroup estimates.
DCDS
On the global existence of a cross-diffusion system
Yuan Lou Wei-Ming Ni Yaping Wu
Discrete & Continuous Dynamical Systems - A 1998, 4(2): 193-203 doi: 10.3934/dcds.1998.4.193
We consider a strongly-coupled nonlinear parabolic system which arises from population dynamics. The global existence of classical solutions is established when the space dimension is two and one of the cross-diffusion pressures is zero.
keywords: a priori estimates. Cross-diffusion global existence
DCDS
The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion
Wei-Ming Ni Yaping Wu Qian Xu
Discrete & Continuous Dynamical Systems - A 2014, 34(12): 5271-5298 doi: 10.3934/dcds.2014.34.5271
This paper concerns with the existence and stability properties of non-constant positive steady states in one dimensional space for the following competition system with cross diffusion $$\left\{ \begin{array}{ll} u_t=[(d_{1}+\rho_{12}v)u]_{xx}+u(a_{1}-b_{1}u-c_{1}v),&x\in(0,1), t>0, \\ v_t= d_{2}v_{xx}+v(a_{2}-b_{2}u-c_{2}v),& x\in(0,1),t>0,                    (1) \\ u_{x}=v_{x}=0, &x=0,1, t>0. \end{array}\right. $$ First, by Lyapunov-Schmidt method, we obtain the existence and the detailed structure of a type of small nontrivial positive steady states to the shadow system of (1) as $\rho_{12}\to \infty$ and when $d_2$ is near $a_2/\pi^2$, which also verifies some related existence results obtained earlier in [11] by a different method. Then, based on the detailed structure of the steady states, we further establish the stability of the small nontrivial positive steady states for the shadow system by spectral analysis. Finally, we prove the existence and stability of the corresponding nontrivial positive steady states for the original cross diffusion system (1) when $\rho_{12}$ is large enough and $d_2$ is near $a_2/\pi^2$.
keywords: steady states cross diffusion spectral analysis shadow system. Existence stability

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