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Discrete & Continuous Dynamical Systems - A

2009 , Volume 25 , Issue 1

Special Issue Dedicated to Professor Masayasu Mimura
on the occasion of his 65th birthday

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Preface
Danielle Hilhorst and  Hiroshi Matano
2009, 25(1): i-ii doi: 10.3934/dcds.2009.25.1i +[Abstract](20) +[PDF](34.6KB)
Abstract:
Professor Masayasu Mimura, known to his friends as ''Mayan", was born on Shikoku Island, Japan, on October 11th, 1941. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. Thanks to his advisor's acquaintance with a large circle of scientists, Mayan could keep close relations with a group of biologists and biophysicists - Ei Teramoto and Nanako Shigesada, among others - since his very early days.

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On the retention of the interfaces in some elliptic and parabolic nonlinear problems
Luis Alvarez and  Jesús Ildefonso Díaz
2009, 25(1): 1-17 doi: 10.3934/dcds.2009.25.1 +[Abstract](42) +[PDF](245.5KB)
Abstract:
We study some retention phenomena on the free boundaries associated to some elliptic and parabolic problems of reaction-diffusion type. This is the case, for instance, of the waiting time phenomenon for solutions of suitable parabolic equations. We find sufficient conditions in order to have a discrete version of the waiting time property (the so called nondiffusion of the support) for solutions of the associated family of elliptic equations and prove how to pass to the limit in order to get this property for the solutions of the parabolic equation.
Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains
Henri Berestycki and  Luca Rossi
2009, 25(1): 19-61 doi: 10.3934/dcds.2009.25.19 +[Abstract](40) +[PDF](509.3KB)
Abstract:
This work is the continuation of our previous paper [6]. There, we dealt with the reaction-diffusion equation

$\partial_t u=\Delta u+f(x-cte,u),\qquad t>0,\quad x\in\R^N,$

where $e\in S^{N-1}$ and $c>0$ are given and $f(x,s)$ satisfies some usual assumptions in population dynamics, together with $f_s(x,0)<0$ for $|x|$ large. The interest for such equation comes from an ecological model introduced in [1] describing the effects of global warming on biological species. In [6],we proved that existence and uniqueness of travelling wave solutions of the type $u(x,t)=U(x-cte)$ and the large time behaviour of solutions with arbitrary nonnegative bounded initial datum depend on the sign of the generalized principal in $\R^N$ of an associated linear operator. Here, we establish analogous results for the Neumann problem in domains which are asymptotically cylindrical, as well as for the problem in the whole space with $f$ periodic in some space variables, orthogonal to the direction of the shift $e$.
   The $L^1$ convergence of solution $u(t,x)$ as $t\to\infty$ is established next. In this paper, we also show that a bifurcation from the zero solution takes place as the principal crosses $0$. We are able to describe the shape of solutions close to extinction thus answering a question raised by M.~Mimura. These two results are new even in the framework considered in [6].
   Another type of problem is obtained by adding to the previous one a term $g(x-c'te,u)$ periodic in $x$ in the direction $e$. Such a model arises when considering environmental change on two different scales. Lastly, we also solve the case of an equation

$\partial_t u=\Delta u+f(t,x-cte,u),$

when $f(t,x,s)$ is periodic in $t$. This for instance represents the seasonal dependence of $f$. In both cases, we obtain a necessary and sufficient condition for the existence, uniqueness and stability of pulsating travelling waves, which are solutions with a profile which is periodic in time.

Long time convergence for a class of variational phase-field models
Pierluigi Colli , Danielle Hilhorst , Françoise Issard-Roch and  Giulio Schimperna
2009, 25(1): 63-81 doi: 10.3934/dcds.2009.25.63 +[Abstract](35) +[PDF](315.8KB)
Abstract:
In this paper we analyze a class of phase field models for the dynamics of phase transitions which extend the well-known Caginalp and Penrose-Fife phase field models. We prove the existence and uniqueness of the solution of a corresponding initial boundary value problem and deduce further regularity of the solution by exploiting the so-called regularizing effect. Finally we study the long time behavior of the solution and show that it converges algebraically fast to a stationary solution as $t$ tends to infinity.
Some remarks on half space problems
E. N. Dancer
2009, 25(1): 83-88 doi: 10.3934/dcds.2009.25.83 +[Abstract](54) +[PDF](134.6KB)
Abstract:
We study moving plane results on half spaces when the nonlinearity is negative at zero and give some simple applications.
Traveling wave speeds in rapidly oscillating media
Fathi Dkhil and  Angela Stevens
2009, 25(1): 89-108 doi: 10.3934/dcds.2009.25.89 +[Abstract](41) +[PDF](209.3KB)
Abstract:
In this paper we study the effects of periodically varying heterogeneous media on the speed of traveling waves in reaction-diffusion equations. Under suitable conditions the traveling wave speed of the non-homogenized problem can be calculated in terms of the speed of the homogenized problem. We discuss a variety of examples and focus especially on the influence of the symmetric and antisymmetric part of the diffusion matrix on the wave speed.
The two-dimensional Keller-Segel model after blow-up
Jean Dolbeault and  Christian Schmeiser
2009, 25(1): 109-121 doi: 10.3934/dcds.2009.25.109 +[Abstract](38) +[PDF](192.3KB)
Abstract:
In the two-dimensional Keller-Segel model for chemotaxis of biological cells, blow-up of solutions in finite time occurs if the total mass is above a critical value. Blow-up is a concentration event, where point aggregates are created. In this work global existence of generalized solutions is proven, allowing for measure valued densities. This extends the solution concept after blow-up. The existence result is an application of a theory developed by Poupaud, where the cell distribution is characterized by an additional defect measure, which vanishes for smooth cell densities. The global solutions are constructed as limits of solutions of a regularized problem.
   A strong formulation is derived under the assumption that the generalized solution consists of a smooth part and a number of smoothly varying point aggregates. Comparison with earlier formal asymptotic results shows that the choice of a solution concept after blow-up is not unique and depends on the type of regularization.
   This work is also concerned with local density profiles close to point aggregates. An equation for these profiles is derived by passing to the limit in a rescaled version of the regularized model. Solvability of the profile equation can also be obtained by minimizing a free energy functional.
On the long-time limit of positive solutions to the degenerate logistic equation
Yihong Du and  Yoshio Yamada
2009, 25(1): 123-132 doi: 10.3934/dcds.2009.25.123 +[Abstract](36) +[PDF](164.1KB)
Abstract:
We study the long-time behavior of positive solutions to the problem

$u_t-\Delta u=a u-b(x)u^p \mbox{ in } (0,\infty)\times \Omega, Bu=0 \mbox{ on } (0,\infty)\times \partial \Omega, $

where $a$ is a real parameter, $b\geq 0$ is in $C^\mu(\bar{\Omega})$ and $p>1$ is a constant, $\Omega$ is a $C^{2+\mu}$ bounded domain in $R^N$ ($N\geq 2$), the boundary operator $B$ is of the standard Dirichlet, Neumann or Robyn type. Under the assumption that $\overline\Omega_0$:=$\{x\in\Omega: b(x)=0\}$ has non-empty interior, is connected, has smooth boundary and is contained in $\Omega$, it is shown in [8] that when $a\geq \lambda_1^D(\Omega_0)$, for any fixed $x\in \overline{\Omega}_0$, $\overline{\lim}_{t\to\infty}u(t,x)$=$\infty$, and for any fixed $x\in \overline{\Omega}\setminus \overline{\Omega}_0$,

$\overline{\lim}_{t\to\infty}u(t,x)\leq \overline{U}_a(x), \underline{\lim}_{t\to\infty}u(t,x)\geq \underline{U}_a(x),

where $\underline{U}_a$ and $\overline{U}_a$ denote respectively the minimal and maximal positive solutions of the boundary blow-up problem

$-\Delta u=au-b(x)u^p \mbox{ in} \ \Omega\setminus\overline{\Omega}_0,\ Bu=0 \mbox{ on}\ \partial \Omega,\ \ u=\infty \mbox{ on}\ \partial \Omega_0.$

The main purpose of this paper is to show that, under the above assumptions,

$\lim_{t\to\infty} u(t,x)=\underline U_a(x),\forall x\in \overline\Omega\setminus \overline\Omega_0.$

This proves a conjecture stated in [8]. Some extensions of this result are also discussed.

On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry
J. Földes and  Peter Poláčik
2009, 25(1): 133-157 doi: 10.3934/dcds.2009.25.133 +[Abstract](39) +[PDF](340.9KB)
Abstract:
We consider fully nonlinear weakly coupled systems of parabolic equations on a bounded reflectionally symmetric domain. Assuming the system is cooperative we prove the asymptotic symmetry of positive bounded solutions. To facilitate an application of the method of moving hyperplanes, we derive Harnack type estimates for linear cooperative parabolic systems.
Energy estimates for electro-reaction-diffusion systems with partly fast kinetics
Annegret Glitzky
2009, 25(1): 159-174 doi: 10.3934/dcds.2009.25.159 +[Abstract](33) +[PDF](230.4KB)
Abstract:
We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro-reaction-diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.
A mathematical model of blood coagulation induced by activation sources
Georgy Th. Guria , Miguel A. Herrero and  Ksenia E. Zlobina
2009, 25(1): 175-194 doi: 10.3934/dcds.2009.25.175 +[Abstract](45) +[PDF](741.8KB)
Abstract:
In this work a mathematical model for blood coagulation induced by an activator source is presented. Blood coagulation is viewed as a process resulting in fibrin polymerization, which is considered as the first step towards thrombi formation. We derive and study a system for the first moments of the polymer concentrations and the activating variables. Analysis of this last model allows us to identify parameter regions which could lead to thrombi formation, both in homeostatic and pathological situations.
Singular travelling waves in a model for tumour encapsulation
John R. King , Judith Pérez-Velázquez and  H.M. Byrne
2009, 25(1): 195-230 doi: 10.3934/dcds.2009.25.195 +[Abstract](34) +[PDF](615.2KB)
Abstract:
In this paper we study an existing mathematical model of tumour encapsulation comprising two reaction-convection-diffusion equations for tumour-cell and connective-tissue densities. The existence of travelling-wave solutions has previously been shown in certain parameter regimes, corresponding to a connective tissue wave which moves in concert with an advancing front of the tumour cells. We extend these results by constructing novel classes of travelling waves for parameter regimes not previously treated asymptotically; we term these singular because they do not correspond to regular trajectories of the corresponding ODE system. Associated with this singularity is a number of further (inner) asymptotic regions in which the dynamics is not governed by the travelling-wave formulation, but which we also characterise.
Periodic traveling waves of a mean curvature flow in heterogeneous media
Bendong Lou
2009, 25(1): 231-249 doi: 10.3934/dcds.2009.25.231 +[Abstract](40) +[PDF](342.0KB)
Abstract:
We consider a curvature flow in heterogeneous media in the plane: $ V= a(x,y) \kappa + b$, where for a plane curve, $V$ denotes its normal velocity, $\kappa$ denotes its curvature, $b$ is a constant and $a(x,y)$ is a positive function, periodic in $y$. We study periodic traveling waves which travel in $y$-direction with given average speed $c \geq 0$. Four different types of traveling waves are given, whose profiles are straight lines, ''V"-like curves, cup-like curves and cap-like curves, respectively. We also show that, as $(b,c)\rightarrow (0,0)$, the profiles of the traveling waves converge to straight lines. These results are connected with spatially heterogeneous version of Bernshteĭn's Problem and De Giorgi's Conjecture, which are proposed at last.
Upper bounds for coarsening for the degenerate Cahn-Hilliard equation
Amy Novick-Cohen and  Andrey Shishkov
2009, 25(1): 251-272 doi: 10.3934/dcds.2009.25.251 +[Abstract](47) +[PDF](301.5KB)
Abstract:
The long time behavior for the degenerate Cahn-Hilliard equation [4, 5, 9],

$u_t=\nabla \cdot (1-u^2) \nabla \[ \frac{\Theta}{2} \{ \ln(1+u)-\ln(1-u)\} - \alpha u -$ Δu$],$

is characterized by the growth of domains in which $u(x,t) \approx u_{\pm},$ where $u_\pm$ denote the ''equilibrium phases;" this process is known as coarsening. The degree of coarsening can be quantified in terms of a characteristic length scale, $l(t)$, where $l(t)$ is prescribed via a Liapunov functional and a $W^{1, \infty}$ predual norm of $u(x,t).$ In this paper, we prove upper bounds on $l(t)$ for all temperatures $\Theta \in (0, \Theta_c),$ where $\Theta_c$ denotes the ''critical temperature," and for arbitrary mean concentrations, $\bar{u}\in (u_{-}, u_{+}).$ Our results generalize the upper bounds obtained by Kohn & Otto [14]. In particular, we demonstrate that transitions may take place in the nature of the coarsening bounds during the coarsening process.

Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions
Toshiyuki Ogawa and  Takashi Okuda
2009, 25(1): 273-297 doi: 10.3934/dcds.2009.25.273 +[Abstract](39) +[PDF](1822.0KB)
Abstract:
Bifurcation structure of the stationary solutions to the Swift-Hohenberg equation with a symmetry breaking boundary condition is studied. Namely, a SO(2) breaking perturbation is added to the Neumann or Dirichlet boundary conditions. As a result, half of the secondary bifurcation points change their characters by the imperfection of pitchfork bifurcations.
On a constrained reaction-diffusion system related to multiphase problems
José-Francisco Rodrigues and  Lisa Santos
2009, 25(1): 299-319 doi: 10.3934/dcds.2009.25.299 +[Abstract](44) +[PDF](291.6KB)
Abstract:
We solve and characterize the Lagrange multipliers of a reaction-diffusion system in the Gibbs simplex of $\R^{N+1}$ by considering strong solutions of a system of parabolic variational inequalities in $\R^N$. Exploring properties of the two obstacles evolution problem, we obtain and approximate a $N$-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear reaction terms. We also show continuous dependence results and we establish sufficient conditions of non-degeneracy for the stability of those phase subregions.
Homogenization and influence of fragmentation in a biological invasion model
Mohammad El Smaily , François Hamel and  Lionel Roques
2009, 25(1): 321-342 doi: 10.3934/dcds.2009.25.321 +[Abstract](32) +[PDF](281.7KB)
Abstract:
In this paper, some properties of the minimal speeds of pulsating Fisher-KPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model.
Mathematical analysis of a model of morphogenesis
José Ignacio Tello
2009, 25(1): 343-361 doi: 10.3934/dcds.2009.25.343 +[Abstract](30) +[PDF](216.6KB)
Abstract:
We consider a simple mathematical model of distribution of morphogens (signaling molecules responsible for the differentiation of cells and the creation of tissue patterns) proposed by Lander, Nie and Wan in 2002. The model consists of a system of two equations: a PDE of parabolic type modeling the distribution of free morphogens with a dynamic boundary condition and an ODE describing the evolution of bound receptors. Three biological processes are taken into account: diffusion, degradation and reversible binding. We prove existence and uniqueness of solutions and its asymptotic behavior.
On the Gierer-Meinhardt system with precursors
Juncheng Wei and  Matthias Winter
2009, 25(1): 363-398 doi: 10.3934/dcds.2009.25.363 +[Abstract](38) +[PDF](432.0KB)
Abstract:
We consider the following Gierer-Meinhardt system with a precursor $ \mu (x)$ for the activator $A$ in $\mathbb{R}^1$:

$A_t=$ε2$A^{''}- \mu (x) A+\frac{A^2}{H} \mbox{ in } (-1, 1),$
$\tau H_t=D H^{''}-H+ A^2 \mbox{ in } (-1, 1),$
$ A' (-1)= A' (1)= H' (-1) = H' (1) =0.$

Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system.
   We establish the existence and stability of $N-$peaked steady-states in terms of the precursor $\mu(x)$ and the diffusion coefficient $D$. It is shown that $\mu (x)$ plays an essential role for both existence and stability of spiky patterns. In particular, we show that precursors can give rise to instability. This is a new effect which is not present in the homogeneous case.

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