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DCDS, series A includes peerreviewed original papers and invited expository papers on the theory and methods of analysis, differential equations and dynamical systems. This journal is committed to recording important new results in its field and maintains the highest standards of innovation and quality. To be published in this journal, an original paper must be correct, new, nontrivial and of interest to a substantial number of readers.
DCDS is published monthly in 2017 and is a publication of the American Institute of Mathematical Sciences. All rights reserved.
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TOP 10 Most Read Articles in DCDSA, February 2017
1 
Stability analysis of reactiondiffusion models on evolving domains: The effects of crossdiffusion
Volume 36, Number 4, Pages: 2133  2170, 2015
Anotida Madzvamuse,
Hussaini Ndakwo
and Raquel Barreira
Abstract
References
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This article presents stability analytical results of a two component reactiondiffusion system with linear crossdiffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with timedependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove crossdiffusiondriven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusiondriven instability conditions on stationary domains, in the presence of crossdiffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activatorinhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reactioncrossdiffusion models with equal diffusion coefficients in the principal components as well as those of the shortrange inhibition, longrange activation and activatoractivator form can generate patterns only in the presence of crossdiffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a shortrange inhibition, longrange activation reactiondiffusion model with linear crossdiffusion in the presence of domain evolution.

2 
$H^1$ Solutions of a class of fourth order nonlinear equations for image processing
Volume 10, Number 1/2, Pages: 349  366, 2003
John B. Greer
and Andrea L. Bertozzi
Abstract
Full Text
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Recently fourth order equations of the form
$u_t = \nabla\cdot((\mathcal G(J_\sigma u)) \nabla \Delta u)$ have been proposed
for noise reduction and simplification of two dimensional images.
The operator $\mathcal G$ is a nonlinear functional involving
the gradient or Hessian of its argument, with decay in the far field.
The operator $J_\sigma$ is a standard mollifier.
Using ODE methods on Sobolev spaces,
we prove existence and uniqueness of solutions of this problem
for $H^1$ initial data.

3 
Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations
Volume 36, Number 4, Pages: 1789  1811, 2015
Santosh Bhattarai
Abstract
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In this paper we establish existence and stability results concerning fully
nontrivial solitarywave solutions to 3coupled nonlinear Schrödinger system
\begin{equation*}
i\partial_t u_{j}+
\partial_{xx}u_{j}+ \left(\sum_{k=1}^{3} a_{kj} u_k^{p}\right)u_j^{p2}u_j = 0, \ j=1,2,3,
\end{equation*}
where $u_j$ are complexvalued functions of $(x,t)\in \mathbb{R}^{2}$ and $a_{kj}$ are
positive constants satisfying $a_{kj}=a_{jk}$ (symmetric attractive case).
Our approach improves many of the previously known results. In all variational methods used previously
to study the stability of solitary waves,
which we are aware of, the constraint functionals were not independently chosen.
Here we study a problem of minimizing the energy functional subject to
three independent $L^2$ mass constraints and
establish existence and stability results for
a true threeparameter family of solitary waves.

4 
Stability of variational eigenvalues
for the fractional $p$Laplacian
Volume 36, Number 4, Pages: 1813  1845, 2015
Lorenzo Brasco,
Enea Parini
and Marco Squassina
Abstract
References
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By virtue of $\Gamma$convergence arguments, we investigate the stability
of variational eigenvalues associated with a given topological index for the fractional $p$Laplacian operator, in the
singular limit as the nonlocal operator converges to the $p$Laplacian. We also obtain the convergence of the corresponding
normalized eigenfunctions in a suitable fractional norm.

5 
Time periodic solutions to the threedimensional equations of compressible magnetohydrodynamic flows
Volume 36, Number 4, Pages: 1847  1868, 2015
Hong Cai
and Zhong Tan
Abstract
References
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In this paper, the compressible magnetohydrodynamic system with some smallness and symmetry assumptions on the time periodic external force is considered in $\mathbb{R}^3$. Based on the uniform estimates and the topological degree theory, we prove the existence of a time periodic solution in a bounded domain. Then by a limiting process, the result in the whole space $\mathbb{R}^3$ is obtained.

6 
Flows of vector fields with point singularities and the vortexwave system
Volume 36, Number 5, Pages: 2405  2417, 2015
Gianluca Crippa,
Milton C. Lopes Filho,
Evelyne Miot
and Helena J. Nussenzveig Lopes
Abstract
References
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The vortexwave system is a version of the vorticity equation governing the motion of 2D incompressible fluids in which vorticity is split into a finite sum of Diracs, evolved
through an ODE, plus an $L^p$ part, evolved through an active scalar transport equation. Existence of a weak solution for this system
was recently proved by Lopes Filho, Miot and Nussenzveig Lopes, for $p>2$, but their result left open the existence and basic properties
of the underlying Lagrangian flow. In this article we study existence, uniqueness and the qualitative properties of the (Lagrangian flow for the)
linear transport problem associated to the vortexwave system. To this end, we study the flow associated to
a twodimensional vector field which is singular at a moving point. We first observe that existence and uniqueness of the regular Lagrangian flow are ensured by combining previous results by Ambrosio and by Lacave and Miot. In addition we prove that, generically, the Lagrangian trajectories do not collide with the point singularity. In the second part we present an approximation scheme for the flow, with explicit error estimates obtained by adapting results by Crippa and De Lellis for Sobolev vector fields.

7 
Sharp estimates for fully bubbling solutions of $B_2$ Toda system
Volume 36, Number 4, Pages: 1759  1788, 2015
Weiwei Ao
Abstract
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In this paper, we obtain sharp estimates of fully bubbling solutions of the $B_2$ Toda system in a compact Riemann surface. Our main goal in this paper are (i) to obtain sharp convergence rate, (ii) to completely determine the location of bubbles, (iii) to derive the $\partial_z^2$ condition.

8 
Recent progresses in boundary layer theory
Volume 36, Number 5, Pages: 2521  2583, 2015
GungMin Gie,
ChangYeol Jung
and Roger Temam
Abstract
References
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In this article,
we review recent progresses in boundary layer analysis of some
singular perturbation problems.
Using the techniques of differential geometry,
an asymptotic expansion of reactiondiffusion or heat equations in a domain with curved boundary
is constructed and validated in some suitable functional spaces.
In addition, we investigate the effect of curvature
as well as that
of an illprepared initial data.
Concerning convectiondiffusion equations, the asymptotic behavior of their solutions
is difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, the boundary layer analysis is performed on relatively simpler domains,
typically intervals, rectangles, or circles.
We consider also the interior transition layers at the turning point characteristics in an interval domain and classical (ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangular domain
using the technique of correctors and the tools of functional analysis.
The validity of our asymptotic expansions is also established in suitable spaces.

9 
A biharmonicmodified forward time stepping
method for fourth order nonlinear diffusion equations
Volume 29, Number 4, Pages: 1367  1391, 2010
Andrea L. Bertozzi,
Ning Ju
and HsiangWei Lu
Abstract
References
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We consider a class of splitting schemes for fourth order nonlinear
diffusion equations. Standard backwardtime differencing requires
the solution of a higher order elliptic problem, which can be both
computationally expensive and workintensive to code, in higher space
dimensions.
Recent papers in the literature provide computational evidence that
a biharmonicmodified, forward timestepping method, can provide good
results for these problems.
We provide a theoretical explanation of the results.
For a basic nonlinear 'thin film' type equation we prove $H^1$
stability of the method given very simple boundedness constraints
of the numerical solution. For a more general class of longwave
unstable problems, we prove stability and convergence, using
only constraints on the smooth solution.
Computational examples include both the model of 'thin film' type
problems and a quantitative model for electrowetting in a HeleShaw
cell (Lu et al J. Fluid Mech. 2007).
The methods considered here are related to 'convexity splitting'
methods for gradient flows with nonconvex energies.

10 
Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems
Volume 36, Number 5, Pages: 2803  2825, 2015
Lijun Wei
and Xiang Zhang
Abstract
References
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This paper deals with the maximum number of limit cycles, which can be bifurcated from periodic orbits of planar piecewise smooth Hamiltonian systems, which are located in a neighborhood of a generalized homoclinic loop with a nilpotent saddle on a switch line. First we present asymptotic expressions of the Melnikov functions near the loop. Then using these expressions we study the number of limit cycles which are bifurcated from the periodic orbits near the homoclinic loop under small perturbations. Finally we provide two concrete examples showing applications of our main results.

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