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Discrete & Continuous Dynamical Systems - A

2013 , Volume 33 , Issue 5

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Persistence of Hölder continuity for non-local integro-differential equations
Kyudong Choi
2013, 33(5): 1741-1771 doi: 10.3934/dcds.2013.33.1741 +[Abstract](455) +[PDF](567.4KB)
Abstract:
In this paper, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in $C^\beta$ for all time if its initial data lies in $C^\beta$. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption $b\in L^\infty C^{1-\alpha}$ on the divergent-free drift velocity. The proof is in the spirit of [23] where Kiselev and Nazarov established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation.
Formal Poincaré-Dulac renormalization for holomorphic germs
Marco Abate and Jasmin Raissy
2013, 33(5): 1773-1807 doi: 10.3934/dcds.2013.33.1773 +[Abstract](240) +[PDF](228.1KB)
Abstract:
We shall describe an alternative approach to a general renormalization procedure for formal self-maps, originally suggested by Chen-Della Dora and Wang-Zheng-Peng, giving formal normal forms simpler than the classical Poincaré-Dulac normal form. As example of application we shall compute a complete list of normal forms for bi-dimensional superattracting germs with non-vanishing quadratic term; in most cases, our normal forms will be the simplest possible ones (in the sense of Wang-Zheng-Peng). We shall also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena.
Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation
Frédéric Abergel and Jean-Michel Rakotoson
2013, 33(5): 1809-1818 doi: 10.3934/dcds.2013.33.1809 +[Abstract](231) +[PDF](379.2KB)
Abstract:
It is known that the very weak solution of $-∫_Ω u\Deltaφ dx=∫_Ω fφ dx$, $∀φ∈ C^2(\overline{Ω}),$ $φ=0$ on $∂Ω$, $u\in L^1(Ω)$ has its gradient in $Ł^1(Ω)$ whenever $f∈ L^1(Ω;δ(1+|Lnδ|))$, $δ(x)$ being the distance of $x∈Ω$ to the boundary. In this paper, we show that if $f≥0$ is not in this weighted space $L^1(Ω;δ(1+|Lnδ|))$, then its gradient blows up in $L(\log L)$ at least. Moreover, we show that there exist a domain $Ω$ of class $C^\infty$ and a function $f∈ L^1_+(Ω,δ)$ such that the associated very weak solution has its gradient being non integrable on $Ω$.
Characterizations of $\omega$-limit sets in topologically hyperbolic systems
Andrew D. Barwell, Chris Good, Piotr Oprocha and Brian E. Raines
2013, 33(5): 1819-1833 doi: 10.3934/dcds.2013.33.1819 +[Abstract](273) +[PDF](250.4KB)
Abstract:
It is well known that $\omega$-limit sets are internally chain transitive and have weak incompressibility; the converse is not generally true, in either case. However, it has been shown that a set is weakly incompressible if and only if it is an abstract $\omega$-limit set, and separately that in shifts of finite type, a set is internally chain transitive if and only if it is a (regular) $\omega$-limit set. In this paper we generalise these and other results, proving that the characterization for shifts of finite type holds in a variety of topologically hyperbolic systems (defined in terms of expansive and shadowing properties), and also show that the notions of internal chain transitivity and weak incompressibility coincide in compact metric spaces.
Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance
Anna Capietto, Walter Dambrosio, Tiantian Ma and Zaihong Wang
2013, 33(5): 1835-1856 doi: 10.3934/dcds.2013.33.1835 +[Abstract](257) +[PDF](413.0KB)
Abstract:
In this paper we deal with the existence of unbounded orbits of the map $$ \left\{\begin{array}{l} θ_1= θ+\frac{1}{ρ} [u(θ)-l_1(ρ)]+h_1(ρ, θ), ρ_1=ρ-u'(θ)+l_2(ρ)+h_2(ρ, θ), \end{array} \right. $$ where $\mu$ is continuous and $2\pi$-periodic, $l_1$, $l_2$ are continuous and bounded, $h_1(\rho, \theta)=o(\rho^{-1})$, $h_2(\rho, \theta)=o(1)$, for $\rho\to+\infty$. We prove that every orbit of the map tends to infinity in the future or in the past for $\rho$ large enough provided that $$[\liminf_{\rho\to+\infty}l_1(\rho), \limsup_{\rho\to+\infty}l_1(\rho)]\cap Range(\mu)=\emptyset$$ and other conditions hold. On the basis of this conclusion, we prove that the system $ Jz'=\nabla H(z)+f(z)+p(t)$ has unbounded solutions when $H$ is positively homogeneous of degree 2 and positive. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this system.
Almost periodic and almost automorphic solutions of linear differential equations
Tomás Caraballo and David Cheban
2013, 33(5): 1857-1882 doi: 10.3934/dcds.2013.33.1857 +[Abstract](233) +[PDF](495.8KB)
Abstract:
We analyze the existence of almost periodic (respectively, almost automorphic, recurrent) solutions of a linear non-homogeneous differential (or difference) equation in a Banach space, with almost periodic (respectively, almost automorphic, recurrent) coefficients. Under some conditions we prove that one of the following alternatives is fulfilled:
  (i) There exists a complete trajectory of the corresponding homogeneous equation with constant positive norm;
  (ii) The trivial solution of the homogeneous equation is uniformly asymptotically stable.
If the second alternative holds, then the non-homogeneous equation with almost periodic (respectively, almost automorphic, recurrent) coefficients possesses a unique almost periodic (respectively, almost automorphic, recurrent) solution. We investigate this problem within the framework of general linear nonautonomous dynamical systems. We apply our general results also to the cases of functional-differential equations and difference equations.
No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$
Tao Chen, Yunping Jiang and Gaofei Zhang
2013, 33(5): 1883-1890 doi: 10.3934/dcds.2013.33.1883 +[Abstract](303) +[PDF](329.5KB)
Abstract:
Consider the family $f_{\lambda, \gamma}(z) = \lambda e^{iz}+\gamma e^{-iz}$ where $\lambda$ and $\gamma$ are non-zero complex numbers. It contains the sine family $\lambda \sin z$ and is a natural extension of the sine family. We give a direct proof of that the escaping set $I_{\lambda, \gamma}$ of $f_{\lambda, \gamma}$ supports no $f_{\lambda,\gamma}$-invariant line fields.
Continuous limit and the moments system for the globally coupled phase oscillators
Hayato Chiba
2013, 33(5): 1891-1903 doi: 10.3934/dcds.2013.33.1891 +[Abstract](221) +[PDF](359.4KB)
Abstract:
The Kuramoto model, which describes synchronization phenomena, is a system of ordinary differential equations on $N$-torus defined as coupled harmonic oscillators. The order parameter is often used to measure the degree of synchronization. In this paper, the moments systems are introduced for both of the Kuramoto model and its continuous model. It is shown that the moments systems for both systems take the same form. This fact allows one to prove that the order parameter of the $N$-dimensional Kuramoto model converges to that of the continuous model as $N\to \infty$.
Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition
Benjamin Dodson
2013, 33(5): 1905-1926 doi: 10.3934/dcds.2013.33.1905 +[Abstract](358) +[PDF](469.0KB)
Abstract:
In this paper, we prove global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ and $u_{0} \in H^{s}(\mathbf{R}^{3})$, $s > 5/7$. To this end, we utilize a linear-nonlinear decomposition, similar to the decomposition used in [20] for the wave equation.
From log Sobolev to Talagrand: A quick proof
Nicola Gigli and Michel Ledoux
2013, 33(5): 1927-1935 doi: 10.3934/dcds.2013.33.1927 +[Abstract](196) +[PDF](367.2KB)
Abstract:
We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent development [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincaré inequality.
Stochastic perturbations and Ulam's method for W-shaped maps
Paweł Góra and Abraham Boyarsky
2013, 33(5): 1937-1944 doi: 10.3934/dcds.2013.33.1937 +[Abstract](266) +[PDF](349.0KB)
Abstract:
For a discrete dynamical system given by a map $\tau :I\rightarrow I$, the long term behavior is described by the probability density function (pdf) of an absolutely continuous invariant measure. This pdf is the fixed point of the Frobenius-Perron operator on $L^{1}(I)$ induced by $\tau$. Ulam suggested a numerical procedure for approximating a pdf by using matrix approximations to the Frobenius-Perron operator. In [12] Li proved the convergence for maps which are piecewise $C^{2}$ and satisfy $| \tau'| >2.$ In this paper we will consider a larger class of maps with weaker smoothness conditions and a harmonic slope condition which permits slopes equal to $\pm $2. Using a generalized Lasota-Yorke inequality [4], we establish convergence for the Ulam approximation method for this larger class of maps. Ulam's method is a special case of small stochastic perturbations. We obtain stability of the pdf under such perturbations. Although our conditions apply to many maps, there are important examples which do not satisfy these conditions, for example the $W$-map [7]. The $W$-map is highly unstable in the sense that it is possible to construct perturbations $W_a$ with absolutely continuous invariant measures (acim) $\mu_a$ such that $\mu_a$ converge to a singular measure although $W_a$ converge to $W$. We prove the convergence of Ulam's method for the $W$-map by direct calculations.
Actions of Baumslag-Solitar groups on surfaces
Nancy Guelman and Isabelle Liousse
2013, 33(5): 1945-1964 doi: 10.3934/dcds.2013.33.1945 +[Abstract](274) +[PDF](456.8KB)
Abstract:
Let $BS(1, n) =< a, b \ | \ aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that $BS(1, n)$ is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $.
    This paper deals with the dynamics of actions of $BS(1, n)$ on closed orientable surfaces. We exhibit a smooth $BS(1,n)$-action without finite orbits on $\mathbb{T} ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid.
    We develop a general dynamical study for faithful topological $BS(1,n)$-actions on closed surfaces $S$. We prove that such actions $ < f, h \ | \ h o f o h^{-1} = f^n >$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty.
    When $S= \mathbb{T}^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of $BS(1,n)$ on $\mathbb{T}^2$.
    When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ and isotopic to identity then $fix(f)$ contains any minimal set.
Phase transitions in one-dimensional subshifts
Nicolai T. A. Haydn
2013, 33(5): 1965-1973 doi: 10.3934/dcds.2013.33.1965 +[Abstract](228) +[PDF](347.5KB)
Abstract:
In this note we give simple examples of one-dimensional mixing subshift with positive topological entropy which have two distinct measures of maximal entropy. We also give examples of subshifts which have two mutually singular equilibrium states for Hölder continuous functions. We also indicate how the construction can be extended to yield examples with any number of equilibrium states.
Two problems related to prescribed curvature measures
Yong Huang and Lu Xu
2013, 33(5): 1975-1986 doi: 10.3934/dcds.2013.33.1975 +[Abstract](178) +[PDF](378.8KB)
Abstract:
Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. Moreover, we promote Ivochkina's $C^2$ estimates for prescribed curvature equation in [12,13].
Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality
Yutian Lei and Zhongxue Lü
2013, 33(5): 1987-2005 doi: 10.3934/dcds.2013.33.1987 +[Abstract](296) +[PDF](445.7KB)
Abstract:
This paper is concerned with the symmetry results for the $2k$-order singular Lane-Emden type partial differential system $$ \left\{\begin{array}{ll} (-\Delta)^k(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^k(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x), \end{array} \right. $$ and the weighted Hardy-Littlewood-Sobolev type integral system $$ \left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy. \end{array} \right. $$ Here $x \in R^n \setminus \{0\}$. We first establish the equivalence of this integral system and an fractional order partial differential system, which includes the $2k$-order PDE system above. For the integral system, we prove that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. In addition, we also show that the integrable solutions are locally bounded. Thus, the equivalence implies the positive solutions of the PDE system, particularly including the higher integer-order PDE system, also have the corresponding properties.
The diffusive logistic model with a free boundary and seasonal succession
Rui Peng and Xiao-Qiang Zhao
2013, 33(5): 2007-2031 doi: 10.3934/dcds.2013.33.2007 +[Abstract](465) +[PDF](461.3KB)
Abstract:
This paper concerns a diffusive logistic equation with a free boundary and seasonal succession, which is formulated to investigate the spreading of a new or invasive species, where the free boundary represents the expanding front and the time periodicity accounts for the effect of the bad and good seasons. The condition to determine whether the species spatially spreads to infinity or vanishes at a finite space interval is derived, and when the spreading happens, the asymptotic spreading speed of the species is also given. The obtained results reveal the effect of seasonal succession on the dynamical behavior of the spreading of the single species.
Initial trace of positive solutions of a class of degenerate heat equation with absorption
Tai Nguyen Phuoc and Laurent Véron
2013, 33(5): 2033-2063 doi: 10.3934/dcds.2013.33.2033 +[Abstract](237) +[PDF](598.3KB)
Abstract:
We study the initial value problem with unbounded nonnegative functions or measures for the equation $ ∂_t u-Δ_p u+f(u)=0$ in $\mathbb{R}^ × (0,\infty)$ where $p>1$, $Δ_p u = div(|∇ u|^{p-2} ∇ u )$ and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data $kΔ_0$ and we study their limit when $k → ∞$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)= ∫_0^s f(σ)dσ$. We also give new results dealing with uniqueness and non uniqueness for the initial value problem with unbounded initial data. If $p>2$, we prove that, for a large class of nonlinearities $f$, any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case $f(u)=u^α ln^β (u+1)$, where $α>0$ and $β ≥ 0$.
An $H^1$ model for inextensible strings
Stephen C. Preston and Ralph Saxton
2013, 33(5): 2065-2083 doi: 10.3934/dcds.2013.33.2065 +[Abstract](331) +[PDF](667.7KB)
Abstract:
We study geodesics of the $H^1$ Riemannian metric $$ « u,v » = ∫_0^1 ‹ u(s), v(s)› + α^2 ‹ u'(s), v'(s)› ds$$ on the space of inextensible curves $\gamma\colon [0,1]\to\mathbb{R}^2$ with $| γ'|≡ 1$. This metric is a regularization of the usual $L^2$ metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The $H^1$ geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is $C^{\infty}$ in the Banach topology $C^1([0,1], \mathbb{R}^2)$, and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one endpoint of the curves fixed, we have global-in-time solutions. We conclude with some surprising features in the periodic case, along with an analogy to the equations of incompressible fluid mechanics.
Cohomology of $GL(2,\mathbb{R})$-valued cocycles over hyperbolic systems
Victoria Sadovskaya
2013, 33(5): 2085-2104 doi: 10.3934/dcds.2013.33.2085 +[Abstract](227) +[PDF](489.7KB)
Abstract:
We consider Hölder continuous $GL(2,\mathbb{R})$-valued cocycles over a transitive Anosov diffeomorphism. We give a complete classification up to Hölder cohomology of cocycles with one Lyapunov exponent and of cocycles that preserve two transverse Hölder continuous sub-bundles. We prove that a measurable cohomology between two such cocycles is Hölder continuous. We also show that conjugacy of periodic data for two such cocycles does not always imply cohomology, but a slightly stronger assumption does. We describe examples that indicate that our main results do not extend to general $GL(2,\mathbb{R})$-valued cocycles.
Variational methods for non-local operators of elliptic type
Raffaella Servadei and Enrico Valdinoci
2013, 33(5): 2105-2137 doi: 10.3934/dcds.2013.33.2105 +[Abstract](474) +[PDF](418.8KB)
Abstract:
In this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem $$ \left\{ \begin{array}{ll} \mathcal L_K u+\lambda u+f(x,u)=0        in   Ω \\ u=0                                 in   \mathbb{R}^n \backslash Ω , \end{array} \right. $$ where $\lambda$ is a real parameter and the nonlinear term $f$ satisfies superlinear and subcritical growth conditions at zero and at infinity. This equation has a variational nature, and so its solutions can be found as critical points of the energy functional $\mathcal J_\lambda$ associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq \lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the operator $-\mathcal L_K$. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u)        in   Ω \\ u=0                                in   \mathbb{R}^n \backslash Ω. \end{array} \right. $$ Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.
Resonance problems for Kirchhoff type equations
Jijiang Sun and Chun-Lei Tang
2013, 33(5): 2139-2154 doi: 10.3934/dcds.2013.33.2139 +[Abstract](210) +[PDF](417.7KB)
Abstract:
The existence of weak solutions is obtained for some Kirchhoff type equations with Dirichlet boundary conditions which are resonant at an arbitrary eigenvalue under a Landesman-Lazer type condition by the minimax methods.
Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations
Pedro J. Torres, Zhibo Cheng and Jingli Ren
2013, 33(5): 2155-2168 doi: 10.3934/dcds.2013.33.2155 +[Abstract](242) +[PDF](414.0KB)
Abstract:
We analyze the non-degeneracy of the linear $2n$-order differential equation $u^{(2n)}+\sum\limits_{m=1}^{2n-1}a_{m}u^{(m)}=q(t)u$ with potential $q(t)\in L^p(\mathbb{R}/T\mathbb{Z})$, by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.
Pushed traveling fronts in monostable equations with monotone delayed reaction
Elena Trofimchuk, Manuel Pinto and Sergei Trofimchuk
2013, 33(5): 2169-2187 doi: 10.3934/dcds.2013.33.2169 +[Abstract](272) +[PDF](493.0KB)
Abstract:
We study the wavefront solutions of the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone reaction term $g: \mathbb{R}_{+} → \mathbb{R}_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. To this end, we describe the asymptotics of all wavefronts (including critical and non-critical fronts) at $-\infty$. We also prove the uniqueness of wavefronts (up to a translation). In addition, a new uniqueness result for a class of nonlocal lattice equations is presented.
Application of the subharmonic Melnikov method to piecewise-smooth systems
Kazuyuki Yagasaki
2013, 33(5): 2189-2209 doi: 10.3934/dcds.2013.33.2189 +[Abstract](467) +[PDF](311.7KB)
Abstract:
We extend a refined version of the subharmonic Melnikov method to piecewise-smooth systems and demonstrate the theory for bi- and trilinear oscillators. Fundamental results for approximating solutions of piecewise-smooth systems by those of smooth systems are given and used to obtain the main result. Special attention is paid to degenerate resonance behavior, and analytical results are illustrated by numerical ones.
Note on the blowup criterion of smooth solution to the incompressible viscoelastic flow
Baoquan Yuan
2013, 33(5): 2211-2219 doi: 10.3934/dcds.2013.33.2211 +[Abstract](154) +[PDF](345.8KB)
Abstract:
We study the blowup criterion of smooth solution to the Oldroyd model. Let $(u(t,x), F(t,x)$ be a smooth solution in $[0,T)$, it is shown that the solution $(u(t,x), F(t,x)$ does not appear breakdown until $t=T$ provided $∇ u(t,x)∈ L^1([0,T]; L^∞(\mathbb{R}^n))$ for $n=2,3$.
Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow
Shu-Yi Zhang
2013, 33(5): 2221-2239 doi: 10.3934/dcds.2013.33.2221 +[Abstract](253) +[PDF](457.0KB)
Abstract:
This paper is concerned with the existence of multi-dimensional non-isothermal subsonic phase transitions in a steady supersonic flow with the van der Waals type state function. Due to the subsonic property, the Lax entropy inequality [15] is no longer valid for subsonic phase transitions. Hence, physical admissible planar waves are chosen by the viscosity capillarity criterion [24]. Based on the uniform stability result in [28], we perform the iteration scheme [20] and establish the existence.

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