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### Open Access Journals

DCDS

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.

DCDS-B

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.

CPAA

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.

DCDS

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.

KRM

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.

DCDS-B

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.

DCDS

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.

DCDS

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

DCDS

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.

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