# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2009 , Volume 23 , Issue 3

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2009, 23(3): 617-638 doi: 10.3934/dcds.2009.23.617 +[Abstract](263) +[PDF](298.8KB)
Abstract:
We consider the impact of noise on the stability and propagation of fronts in an excitable media with a piece-wise smooth, discontinuous ignition process. In a neighborhood of the ignition threshold the system interacts strongly with noise, the front can loose monotonicity, resulting in multiple crossings of the ignition threshold. We adapt the renormalization group methods developed for coherent structure interaction, a key step being to determine pairs of function spaces for which the the ignition function is Frechet differentiable, but for which the associated semi-group, $S(t)$, is integrable at $t=0$. We parameterize a neighborhood of the front solution through a dynamic front position and a co-dimension one remainder. The front evolution and the asymptotic decay of the remainder are on the same time scale, the RG approach shows that the remainder becomes asymptotically small, in terms of the noise strength and regularity, and the front propagation is driven by a competition between the ignition process and the noise.
2009, 23(3): 639-654 doi: 10.3934/dcds.2009.23.639 +[Abstract](178) +[PDF](220.9KB)
Abstract:
We introduce a class of non-symmetric bilinear forms on the d-dimen\-sional canonical simplex, related with Fleming-Viot type operators.
Strong continuity, closedness and results in the spirit of Beurling-Deny criteria are established. Moreover, under suitable assumptions, we prove that the forms satisfy the Log-Sobolev inequality. As a consequence, regularity results for semigroups generated by a class of Fleming-Viot type operators are given.
2009, 23(3): 655-683 doi: 10.3934/dcds.2009.23.655 +[Abstract](228) +[PDF](345.5KB)
Abstract:
We consider the existence and multiplicity of riemannian metrics of prescribed mean curvature and zero boundary mean curvature on the three dimensional half sphere $(S^3_+,g_c)$ endowed with its standard metric $g_c$. Due to Kazdan-Warner type obstructions, conditions on the function to be realized as a scalar curvature have to be given. Moreover the existence of critical point at infinity for the associated Euler Lagrange functional makes the existence results harder to be proved. However it turns out that such noncompact orbits of the gradient can be treated as a usual critical point once a Morse Lemma at infinity is performed. In particular their topological contribution to the level sets of the functional can be computed. In this paper we prove that, under generic conditions on $K$, this topology at infinity is a lower bound for the number of metrics in the conformal class of $g_c$ having prescribed scalar curvature and zero boundary mean curvature.
2009, 23(3): 685-703 doi: 10.3934/dcds.2009.23.685 +[Abstract](310) +[PDF](243.3KB)
Abstract:
In the space of $C^k$ piecewise expanding unimodal maps, $k\geq 1$, we characterize the $C^1$ smooth families of maps where the topological dynamics does not change (the "smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of $C^{k-1+Lip}$ deformations tangent to every given $C^k$ horizontal direction, for $k\ge 2$.
2009, 23(3): 705-725 doi: 10.3934/dcds.2009.23.705 +[Abstract](275) +[PDF](735.2KB)
Abstract:
We present a topological method for the detection of normally hyperbolic type invariant sets for maps. The invariant set covers a sub-manifold without a boundary in $\mathbb{R}^k$. For the method to hold we only need to assume that the movement of the system transversal to the manifold has directions of topological expansion and contraction. The movement in the direction of the manifold can be arbitrary. The result is based on the method of covering relations and local Brouwer degree theory.
2009, 23(3): 727-732 doi: 10.3934/dcds.2009.23.727 +[Abstract](249) +[PDF](131.9KB)
Abstract:
We study the existence of solution for the one-dimensional $\phi$-laplacian equation $(\phi(u'))'=\lambda f(t,u,u')$ with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate $\lambda_0$ is given such that the problem possesses a solution for any $|\lambda|<\lambda_0$.
2009, 23(3): 733-753 doi: 10.3934/dcds.2009.23.733 +[Abstract](262) +[PDF](285.4KB)
Abstract:
This paper proves the local well posedness of differential equations in metric spaces under assumptions that allow to comprise several different applications. We consider below a system of balance laws with a dissipative non local source, the Hille-Yosida Theorem, a generalization of a recent result on nonlinear operator splitting, an extension of Trotter formula for linear semigroups and the heat equation.
2009, 23(3): 755-764 doi: 10.3934/dcds.2009.23.755 +[Abstract](200) +[PDF](159.3KB)
Abstract:
For two-dimensional Euler equation on the torus, we prove that the $L^\infty$ norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for finite time.
2009, 23(3): 765-784 doi: 10.3934/dcds.2009.23.765 +[Abstract](421) +[PDF](233.8KB)
Abstract:
This paper deals with asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows in the whole space $R^2$. Based on the spectral decomposition of linearized micropolar fluid flows, the sharp algebraic time decay estimates of the micropolar fluid flows in $L_2$ and $L_\infty$ norms are obtained.
2009, 23(3): 785-826 doi: 10.3934/dcds.2009.23.785 +[Abstract](252) +[PDF](447.2KB)
Abstract:
This paper is concerned with the homogenization of some particle systems with two-body interactions in dimension one and of dislocation dynamics in higher dimensions.
The dynamics of our particle systems are described by some ODEs. We prove that the rescaled "cumulative distribution function'' of the particles converges towards the solution of a Hamilton-Jacobi equation. In the case when the interactions between particles have a slow decay at infinity as $1/x$, we show that this Hamilton-Jacobi equation contains an extra diffusion term which is a half Laplacian. We get the same result in the particular case where the repulsive interactions are exactly $1/x$, which creates some additional difficulties at short distances.
We also study a higher dimensional generalisation of these particle systems which is particularly meaningful to describe the dynamics of dislocations lines. One main result of this paper is the discovery of a satisfactory mathematical formulation of this dynamics, namely a Slepčev formulation. We show in particular that the system of ODEs for particle systems can be naturally imbedded in this Slepčev formulation. Finally, with this formulation in hand, we get homogenization results which contain the particular case of particle systems.
2009, 23(3): 827-845 doi: 10.3934/dcds.2009.23.827 +[Abstract](286) +[PDF](268.1KB)
Abstract:
In this paper, we introduce a penalisation method in order to approximate the solutions of the initial boundary value problem for a semi-linear first order symmetric hyperbolic system, with dissipative boundary conditions. The penalization is carefully chosen in order that the convergence to the wished solution is sharp, does not generate any boundary layer, and applies to fictitious domains.
2009, 23(3): 847-865 doi: 10.3934/dcds.2009.23.847 +[Abstract](202) +[PDF](228.7KB)
Abstract:
A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which tiles can slide past one another. We investigate the topology of a certain class of tiling spaces of this type. We show that they can be written as inverse limits of CW complexes, and their Čech cohomology is related to properties of the fault lines.
2009, 23(3): 867-885 doi: 10.3934/dcds.2009.23.867 +[Abstract](249) +[PDF](852.8KB)
Abstract:
We study the early stages of the nonlinear dynamics of pattern formation for unstable generalized thin film equations. For unstable constant steady states, we obtain rigorous estimates for the short- to intermediate-time nonlinear evolution which extends the mathematical characterization for pattern formation derived from linear analysis: formation of patterns can be bounded by the finitely many dominant growing eigenmodes from the initial perturbation.
2009, 23(3): 887-918 doi: 10.3934/dcds.2009.23.887 +[Abstract](189) +[PDF](331.3KB)
Abstract:
In this paper, we study the asymptotic behavior and the convergence rates of solutions to the so-called $p$-system with nonlinear damping on quadrant $\mathbb{R^+}\times \mathbb{R^+}=(0,\infty)\times (0,\infty)$,

$v_t$-u_x=0, $u_t$+p(v)_x=-αu-g(u)

with the Dirichlet boundary condition $u|_{x=0}=0$ or the Neumann boundary condition $u_x|_{x=0}=0$. The initial data $(v_0,u_0)(x)$ has the constant states $(v_+,u_+)$ at $x=\infty$. In the case of null-Dirichlet boundary condition on $u$, we show that the corresponding problem admits a unique global solution $(v(x,t), u(x,t))$ and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave $(\bar{v}(x,t), \bar{u}(x,t))$ governed by the classical Darcy's law provided that the corresponding prescribed initial error function $(w_0(x), z_0(x))$ lies in $(H^3\times H^2)(\mathbb{R}^+)$ and $||v_0(x)-v_+||_{L^1}+||w_0||_3+||z_0||_2+||V_0||_5+||Z_0||_4$ is sufficiently small. Its optimal $L^\infty$ convergence rate is also obtained by using the Green function of the diffusion equation. In the case of null-Neumann boundary condition on $u$, the global existence of smooth solution with small initial data is obtained in both of the case of $v_0(0)= v_+$ and $v_0(0)\neq v_+$. The solution $(v(x,t), u(x,t))$ is proved to tend to $(\bar v(x,t), 0)$ as $t$ tends to infinity, and we also get the optimal $L^\infty$ convergence rate in the case of $v_0(0)= v_+$.

2009, 23(3): 919-935 doi: 10.3934/dcds.2009.23.919 +[Abstract](278) +[PDF](281.2KB)
Abstract:
The paper is concerned with the equation $-\Delta_{h}u=f(u)$ on $S^d$ where $\Delta_{h}$ denotes the Laplace-Beltrami operator on the standard unit sphere $(S^d,h)$, while the continuous nonlinearity $f:\mathbb R\to \mathbb R$ oscillates either at zero or at infinity having an asymptotically critical growth in the Sobolev sense. In both cases, by using a group-theoretical argument and an appropriate variational approach, we establish the existence of $[{d}/{2}]+(-1)^{d+1}-1$ sequences of sign-changing weak solutions in $H_1^2(S^d)$ whose elements in different sequences are mutually symmetrically distinct whenever $f$ has certain symmetry and $d\geq 5$. Although we are dealing with a smooth problem, we are forced to use a non-smooth version of the principle of symmetric criticality (see Kobayashi-Ôtani, J. Funct. Anal. 214 (2004), 428-449). The $L^\infty$-- and $H_1^2$--asymptotic behaviour of the sequences of solutions are also fully characterized.
2009, 23(3): 937-955 doi: 10.3934/dcds.2009.23.937 +[Abstract](296) +[PDF](242.2KB)
Abstract:
An IFS ( iterated function system), $([0,1], \tau_{i})$, on the interval $[0,1]$, is a family of continuous functions $\tau_{0},\tau_{1}, ..., \tau_{d-1} : [0,1] \to [0,1]$. Associated to a IFS one can consider a continuous map $\hat{\sigma} : [0,1]\times \Sigma \to [0,1]\times \Sigma$, defined by $\hat{\sigma}(x,w)=(\tau_{X_{1}(w)}(x), \sigma(w))$ where $\Sigma=\{0,1, ..., d-1\}^{\mathbb{N}}$, $\sigma: \Sigma \to \Sigma$ is given by $\sigma(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : \Sigma \to \{0,1, ..., n-1\}$ is the projection on the coordinate $k$. A $\rho$-weighted system, $\rho \geq 0$, is a weighted system $([0,1], \tau_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] \to \mathbb{R}$ and a probability $\nu$ on $[0,1]$ satisfying $P_{u}(h)=\rho h, \quad P_{u}^{*}(\nu)=\rho \nu$.
A probability $\hat{\nu}$ on $[0,1]\times \Sigma$ is called holonomic for $\hat{\sigma}$, if, $\int\, g \circ \hat{\sigma}\, d\hat{\nu}= \int \,g \,d\hat{\nu}, \, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by $\mathcal H$.
For a holonomic probability $\hat{\nu}$ on $[0,1]\times \Sigma$ we define the entropy $h(\hat{\nu})$=inf$_f \in \mathbb{B}^{+} \int \ln(\frac{P_{\psi}f}{\psi f}) d\hat{\nu}\geq 0$, where, $\psi \in \mathbb{B}^{+}$ is a fixed (any one) positive potential.
Finally, we analyze the problem: given $\phi \in \mathbb{B}^{+}$, find solutions of the maximization problem $p(\phi)$= sup$_\hat{\nu} \in \mathcal{H} \{ \,h(\hat{\nu}) + \int \ln(\phi) d\hat{\nu} \,\}.$ We show an example where a holonomic not-$\hat{\sigma}$-invariant probability attains the supremum value. In the last section we consider maximizing probabilities, sub-actions and duality for potentials $A(x,w)$, $(x,w)\in [0,1]\times \Sigma$, for IFS.
2009, 23(3): 957-972 doi: 10.3934/dcds.2009.23.957 +[Abstract](270) +[PDF](283.3KB)
Abstract:
The pre-image topological pressure is defined for bundle random dynamical systems. A variational principle for it has also been given.
2009, 23(3): 973-990 doi: 10.3934/dcds.2009.23.973 +[Abstract](233) +[PDF](263.0KB)
Abstract:
In this work, we analyze a 3-d dynamic optimal design problem in conductivity governed by the two-dimensional wave equation. Under this dynamic perspective, the optimal design problem consists in seeking the time-dependent optimal layout of two isotropic materials on a 2-d domain ($\Omega\subsetR^2$); this is done by minimizing a cost functional depending on the square of the gradient of the state function involving coefficients which can depend on time, space and design. The lack of classical solutions of this type of problem is well-known, so that a relaxation must be sought. We utilize a specially appropriate characterization of 3-d ($(t,x)\inR\timesR^2$) divergence free vector fields through Clebsh potentials; this lets us transform the optimal design problem into a typical non-convex vector variational problem, to which Young measure theory can be applied to compute explicitly the "constrained quasiconvexification" of the cost density. Moreover this relaxation is recovered by dynamic (time-space) first- or second-order laminates. There are two main concerns in this work: the 2-d hyperbolic state law, and the vector character of the problem. Though these two ingredients have been previously considered separately, we put them together in this work.
2009, 23(3): 991-1008 doi: 10.3934/dcds.2009.23.991 +[Abstract](168) +[PDF](202.2KB)
Abstract:
Sufficient conditions on the function $\,f\,$ are given which ensure the boundedness of the solutions of the second order linear differential equation $\,u''+(1+f(t))\,u=0\,$ as $\, t\rightarrow +\infty\,$. To do this, a suitable class of quadratic forms is introduced.
2009, 23(3): 1009-1033 doi: 10.3934/dcds.2009.23.1009 +[Abstract](332) +[PDF](261.2KB)
Abstract:
Consider the functional differential equation (FDE) $\dot{x}(t)=f(x_t)$ with $f$ defined on an open subset of the space $C^1=C^1([-h,0],\R^n)$. Under mild smoothness assumptions, which are designed for the application to differential equations with state-dependent delays, the FDE generates a semiflow on a submanifold of $C^1$ with continuously differentiable time-$t$-maps. We show that at a stationary point continuously differentiable local center-stable manifolds of the semiflow exist. The proof uses results of Chen, Hale and Tan and of Krisztin about invariant manifolds of time-$t$-maps and their invariance properties with respect to the semiflow.
2009, 23(3): 1035-1040 doi: 10.3934/dcds.2009.23.1035 +[Abstract](241) +[PDF](125.0KB)
Abstract:
We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points $A,B$ such that no finite set of points can block all billiard trajectories from $A$ to $B$.
2009, 23(3): 1041-1060 doi: 10.3934/dcds.2009.23.1041 +[Abstract](238) +[PDF](263.0KB)
Abstract:
The goal of this paper is to present an approximation scheme for a reaction-diffusion equation with finite delay, which has been used as a model to study the evolution of a population with density distribution $u$, in such a way that the resulting finite dimensional ordinary differential system contains the same asymptotic dynamics as the reaction-diffusion equation.
2009, 23(3): 1061-1072 doi: 10.3934/dcds.2009.23.1061 +[Abstract](232) +[PDF](171.4KB)
Abstract:
In this paper, we study the variable-territory prey-predator model. We first establish the global stability of the unique positive constant steady state for the ODE system and the reaction diffusion system, and then prove the existence, uniqueness and stability of positive stationary solutions for the heterogeneous environment case.
2009, 23(3): 1073-1086 doi: 10.3934/dcds.2009.23.1073 +[Abstract](212) +[PDF](208.4KB)
Abstract:
This paper is concerned with the following non-periodic diffusion system

$\partial_tu-\Delta_x u+b(t,x)\cdot\nabla_x u+V(x)u=H_v(t,x,u,v)$in $\mathbb{R}\times\mathbb{R}^N,$ $-\partial_tv-\Delta_x v-b(t,x)\cdot\nabla_x v+V(x)v=H_u(t,x,u,v)$in$\mathbb{R}\times\mathbb{R}^N,$
$u(t,x)\to 0$and$v(t,x)\to0$as$|t|+|x|\to\infty.$

Assuming the potential $V$ is bounded and has a positive bound from below, existence and multiplicity of solutions are obtained for the system with asymptotically quadratic nonlinearities via variational approach.

2009, 23(3): 1087-1098 doi: 10.3934/dcds.2009.23.1087 +[Abstract](309) +[PDF](200.7KB)
Abstract:
It is shown that a trick introduced by H. R. Thieme [6] to study a one-species integral equation model with a nonmonotone operator can be used to show that some multispecies reaction-diffusion systems which are cooperative for small population densities but not for large ones have a spreading speed. The ideas are explained by considering a model for the interaction between ungulates and grassland.

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