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Discrete & Continuous Dynamical Systems - A
June 2016 , Volume 36 , Issue 6
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2016, 36(6): 2931-2944
doi: 10.3934/dcds.2016.36.2931
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Abstract:
Let G ↷ X be a topological action of a topological semigroup $G$ on a compact metric space $X$. We show in this paper that for any given point $x$ in $X$, the following two properties that both approximate to periodicity are equivalent to each other:
$\bullet$ For any neighborhood $U$ of $x$, the return times set $\{g\in G ： gx\in U\}$ is syndetic of Furstenburg in $G$.
$\bullet$ Given any $\varepsilon>0$, there exists a finite subset $K$ of $G$ such that for each $g$ in $G$, the $\varepsilon$-neighborhood of the orbit-arc $K[gx]$ contains the entire orbit $G[x]$.
This is a generalization of a classical theorem of Birkhoff for the case where $G=\mathbb{R}$ or $\mathbb{Z}$. In addition, a counterexample is constructed to this statement, while $X$ is merely a complete but not locally compact metric space.
Let G ↷ X be a topological action of a topological semigroup $G$ on a compact metric space $X$. We show in this paper that for any given point $x$ in $X$, the following two properties that both approximate to periodicity are equivalent to each other:
$\bullet$ For any neighborhood $U$ of $x$, the return times set $\{g\in G ： gx\in U\}$ is syndetic of Furstenburg in $G$.
$\bullet$ Given any $\varepsilon>0$, there exists a finite subset $K$ of $G$ such that for each $g$ in $G$, the $\varepsilon$-neighborhood of the orbit-arc $K[gx]$ contains the entire orbit $G[x]$.
This is a generalization of a classical theorem of Birkhoff for the case where $G=\mathbb{R}$ or $\mathbb{Z}$. In addition, a counterexample is constructed to this statement, while $X$ is merely a complete but not locally compact metric space.
2016, 36(6): 2945-2967
doi: 10.3934/dcds.2016.36.2945
+[Abstract](1459)
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Abstract:
In this paper, we show the 3D nonhomogeneous incompressible MHD equations have a global solution provided that the initial data in critical Besov spaces $\dot{B}_{q,1}^{{3}/{q}}(\mathbb{R}^{3})\times\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3}) \times\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3})$ satisfy a nonlinear smallness condition for all $1< q\leq p<6$, ${1}/{q}-{1}/{p}<{1}/{3}$ if the initial density is near a positive constant. Moreover, this solution is unique under the restriction condition ${1}/{p}+{1}/{q}\geq{2}/{3}$. Motivated by Chemin and Gallagher [7], we also provide an example of initial data satisfying that nonlinear smallness condition, but the norms of $u_{0},b_{0}$ (even all their components) can be arbitrarily large in $\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3})$. In particular, when $b$ identically equals 0, our results improve that of Paicu and Zhang [28].
In this paper, we show the 3D nonhomogeneous incompressible MHD equations have a global solution provided that the initial data in critical Besov spaces $\dot{B}_{q,1}^{{3}/{q}}(\mathbb{R}^{3})\times\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3}) \times\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3})$ satisfy a nonlinear smallness condition for all $1< q\leq p<6$, ${1}/{q}-{1}/{p}<{1}/{3}$ if the initial density is near a positive constant. Moreover, this solution is unique under the restriction condition ${1}/{p}+{1}/{q}\geq{2}/{3}$. Motivated by Chemin and Gallagher [7], we also provide an example of initial data satisfying that nonlinear smallness condition, but the norms of $u_{0},b_{0}$ (even all their components) can be arbitrarily large in $\dot{B}_{p,1}^{-1+{3}/{p}}(\mathbb{R}^{3})$. In particular, when $b$ identically equals 0, our results improve that of Paicu and Zhang [28].
2016, 36(6): 2969-2979
doi: 10.3934/dcds.2016.36.2969
+[Abstract](1325)
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Let $(X,T,\mathcal{B}, \mu)$ be a measure-theoretical dynamical system with a compatible metric $d.$ Following Boshernitzan, call a point $x\in X$ is $\{n_{k}\}$-moving recurrent if $$\inf_{k\geq1} d\big(T^{n_{k}}x, \ T^{n_k+{k}}x\big)=0,$$ where $\{n_{k}\}_{k\in \mathbb{N}}$ is a given sequence of integers. It was asked whether the set of $\{n_{k}\}$-moving recurrent points is of full $\mu$-measure. In this paper, we restrict our attention to the doubling map and quantify the size of the set of $\{n_{k}\}$-moving recurrent points in the sense of measure (a class of $2$-fold mixing measures) and Hausdorff dimension.
Let $(X,T,\mathcal{B}, \mu)$ be a measure-theoretical dynamical system with a compatible metric $d.$ Following Boshernitzan, call a point $x\in X$ is $\{n_{k}\}$-moving recurrent if $$\inf_{k\geq1} d\big(T^{n_{k}}x, \ T^{n_k+{k}}x\big)=0,$$ where $\{n_{k}\}_{k\in \mathbb{N}}$ is a given sequence of integers. It was asked whether the set of $\{n_{k}\}$-moving recurrent points is of full $\mu$-measure. In this paper, we restrict our attention to the doubling map and quantify the size of the set of $\{n_{k}\}$-moving recurrent points in the sense of measure (a class of $2$-fold mixing measures) and Hausdorff dimension.
2016, 36(6): 2981-2990
doi: 10.3934/dcds.2016.36.2981
+[Abstract](1295)
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Abstract:
We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.
We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.
2016, 36(6): 2991-3009
doi: 10.3934/dcds.2016.36.2991
+[Abstract](1323)
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Abstract:
This work deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.
This work deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.
2016, 36(6): 3011-3034
doi: 10.3934/dcds.2016.36.3011
+[Abstract](953)
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Abstract:
This paper is concerned with a system of differential equations related to a circuit model for microwave heating, complemented by suitable initial and boundary conditions. A RLC circuit with a thermistor is representing the microwave heating process with temperature-induced modulations on the electric field. The unknowns of the PDE system are the absolute temperature in the body, the voltage across the capacitor and the electrostatic potential. Using techniques based on monotonicity arguments and sharp estimates, we can prove the existence of a weak solution to the initial-boundary value problem.
This paper is concerned with a system of differential equations related to a circuit model for microwave heating, complemented by suitable initial and boundary conditions. A RLC circuit with a thermistor is representing the microwave heating process with temperature-induced modulations on the electric field. The unknowns of the PDE system are the absolute temperature in the body, the voltage across the capacitor and the electrostatic potential. Using techniques based on monotonicity arguments and sharp estimates, we can prove the existence of a weak solution to the initial-boundary value problem.
2016, 36(6): 3035-3076
doi: 10.3934/dcds.2016.36.3035
+[Abstract](1459)
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Abstract:
In this paper we consider the following problem \begin{eqnarray} \label{abstract} \quad \left\{ \begin{array}{ll}-\Delta u +u= u^{{n-k+2\over n-k-2} \pm\epsilon} & \mbox{ in } \Omega \\ u>0& \mbox{ in }\Omega (0.1)\\ {\partial u\over\partial\nu}=0 & \mbox{ on } \partial\Omega \end{array} \right. \end{eqnarray} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\ge 7$, $k$ is an integer with $k\ge 1$, and $\epsilon >0$ is a small parameter. Assume there exists a $k$-dimensional closed, embedded, non degenerate minimal submanifold $K$ in $\partial \Omega$. Under a sign condition on a certain weighted avarage of sectional curvatures of $\partial \Omega$ along $K$, we prove the existence of a sequence $\epsilon = \epsilon_j \to 0$ and of solutions $u_\epsilon$ to (0.1) such that $$ |\nabla u_\epsilon |^2 \, \rightharpoonup \, S \delta_K , \quad {\mbox {as}} \quad \epsilon \to 0 $$ in the sense of measure, where $\delta_K$ denotes a Dirac delta along $K$ and $S$ is a universal positive constant.
In this paper we consider the following problem \begin{eqnarray} \label{abstract} \quad \left\{ \begin{array}{ll}-\Delta u +u= u^{{n-k+2\over n-k-2} \pm\epsilon} & \mbox{ in } \Omega \\ u>0& \mbox{ in }\Omega (0.1)\\ {\partial u\over\partial\nu}=0 & \mbox{ on } \partial\Omega \end{array} \right. \end{eqnarray} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\ge 7$, $k$ is an integer with $k\ge 1$, and $\epsilon >0$ is a small parameter. Assume there exists a $k$-dimensional closed, embedded, non degenerate minimal submanifold $K$ in $\partial \Omega$. Under a sign condition on a certain weighted avarage of sectional curvatures of $\partial \Omega$ along $K$, we prove the existence of a sequence $\epsilon = \epsilon_j \to 0$ and of solutions $u_\epsilon$ to (0.1) such that $$ |\nabla u_\epsilon |^2 \, \rightharpoonup \, S \delta_K , \quad {\mbox {as}} \quad \epsilon \to 0 $$ in the sense of measure, where $\delta_K$ denotes a Dirac delta along $K$ and $S$ is a universal positive constant.
2016, 36(6): 3077-3106
doi: 10.3934/dcds.2016.36.3077
+[Abstract](1485)
+[PDF](557.4KB)
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In this paper, we are concerned with global existence and optimal decay rates of solutions for the compressible Hall-MHD equations in dimension three. First, we prove the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Second, optimal decay rates of strong solutions in $L^2$-norm are obtained if the initial data belong to $L^1$ additionally. Finally, we apply Fourier splitting method by Schonbek [Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for higher order spatial derivatives of classical solutions in $H^3$-framework, which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].
In this paper, we are concerned with global existence and optimal decay rates of solutions for the compressible Hall-MHD equations in dimension three. First, we prove the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Second, optimal decay rates of strong solutions in $L^2$-norm are obtained if the initial data belong to $L^1$ additionally. Finally, we apply Fourier splitting method by Schonbek [Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for higher order spatial derivatives of classical solutions in $H^3$-framework, which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].
2016, 36(6): 3107-3123
doi: 10.3934/dcds.2016.36.3107
+[Abstract](1180)
+[PDF](461.5KB)
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We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to $\epsilon^{-1/2}$ for $\epsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations --$\rho_\epsilon$ converge to the unique solution to the porous medium equation [14,13]. For viscosity coefficient $\mu(\rho_\epsilon)=\rho_\epsilon^\alpha$ with $\alpha>1$, we obtain a rate of convergence of $\rho_\epsilon$ in $L^\infty(0,T; H^{-1}(\mathbb{R}))$; for $1<\alpha\leq\frac{3}{2}$ the solution $\rho_\epsilon$ converges in $L^\infty(0,T;L^2(\mathbb{R}))$. For compactly supported initial data, we prove that most of the mass corresponding to solution $\rho_\epsilon$ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of $\epsilon$.
We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to $\epsilon^{-1/2}$ for $\epsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, the densities solving the compressible Navier-Stokes equations --$\rho_\epsilon$ converge to the unique solution to the porous medium equation [14,13]. For viscosity coefficient $\mu(\rho_\epsilon)=\rho_\epsilon^\alpha$ with $\alpha>1$, we obtain a rate of convergence of $\rho_\epsilon$ in $L^\infty(0,T; H^{-1}(\mathbb{R}))$; for $1<\alpha\leq\frac{3}{2}$ the solution $\rho_\epsilon$ converges in $L^\infty(0,T;L^2(\mathbb{R}))$. For compactly supported initial data, we prove that most of the mass corresponding to solution $\rho_\epsilon$ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of $\epsilon$.
2016, 36(6): 3125-3152
doi: 10.3934/dcds.2016.36.3125
+[Abstract](1175)
+[PDF](562.1KB)
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We consider analytic cocycles on $\mathbb{T}^d\times U(n)$. We prove that, if a cocycle $(\alpha,A)$ with Diophantine $\alpha$ in an analytic class of radius $h$ can be conjugated to a constant cocycle $(\alpha,C)$ via some measurable conjugacy, then for almost all $C$, for any $h_*$ smaller than $h$, it can be conjugated to $(\alpha,C)$ in the analytic class of radius $h_*$, provided that $A$ is sufficiently close to some constant (the closeness depend only on $h-h_*$ and the Diophantine condition of $\alpha$).
We consider analytic cocycles on $\mathbb{T}^d\times U(n)$. We prove that, if a cocycle $(\alpha,A)$ with Diophantine $\alpha$ in an analytic class of radius $h$ can be conjugated to a constant cocycle $(\alpha,C)$ via some measurable conjugacy, then for almost all $C$, for any $h_*$ smaller than $h$, it can be conjugated to $(\alpha,C)$ in the analytic class of radius $h_*$, provided that $A$ is sufficiently close to some constant (the closeness depend only on $h-h_*$ and the Diophantine condition of $\alpha$).
2016, 36(6): 3153-3225
doi: 10.3934/dcds.2016.36.3153
+[Abstract](1010)
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In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2i\omega$ resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we show here the existence of homoclinic connections with several loops for every periodic orbit close to the origin, except the origin itself. The same problem was studied before for reversible non Hamiltonian vector fields, and the splitting of the homoclinic orbits lead to exponentially small terms which prevent the existence of homoclinic connections with one loop to exponentially small periodic orbits. The same phenomenon occurs here but we get round this difficulty thanks to geometric arguments specific to Hamiltonian systems and by studying homoclinic orbits with many loops.
In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2i\omega$ resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we show here the existence of homoclinic connections with several loops for every periodic orbit close to the origin, except the origin itself. The same problem was studied before for reversible non Hamiltonian vector fields, and the splitting of the homoclinic orbits lead to exponentially small terms which prevent the existence of homoclinic connections with one loop to exponentially small periodic orbits. The same phenomenon occurs here but we get round this difficulty thanks to geometric arguments specific to Hamiltonian systems and by studying homoclinic orbits with many loops.
2016, 36(6): 3227-3250
doi: 10.3934/dcds.2016.36.3227
+[Abstract](1181)
+[PDF](531.8KB)
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We are interested in the following differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(u(t))$ where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and determine the Conley index of the set $K_\infty$, that is the union of the bounded orbits of this equation.
We are interested in the following differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(u(t))$ where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and determine the Conley index of the set $K_\infty$, that is the union of the bounded orbits of this equation.
2016, 36(6): 3251-3276
doi: 10.3934/dcds.2016.36.3251
+[Abstract](984)
+[PDF](575.2KB)
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Concerning the influence of the dielectric constant on the electrostatic potential in the bulk of electrolyte solutions, we investigate a charge conserving Poisson-Boltzmann (CCPB) equation [31,32] with a variable dielectric coefficient and a small parameter $\epsilon$ (related to the Debye screening length) in a bounded connected domain with smooth boundary. Under the Robin boundary condition with a given applied potential, the limiting behavior (as $\epsilon\downarrow0$) of the solution (the electrostatic potential) has been rigorously studied. In particular, under the charge neutrality constraint, our result exactly shows the effects of the dielectric coefficient and the applied potential on the limiting value of the solution in the interior domain. The main approach is the Pohozaev's identity of this model. On the other hand, under the charge non-neutrality constraint, we show that the maximum difference between the boundary and interior values of the solution has a lower bound $\log\frac{1}{\epsilon}$ as $\epsilon$ goes to zero. Such an asymptotic blow-up behavior describes an unstable phenomenon which is totally different from the behavior of the solution under the charge neutrality constraint.
Concerning the influence of the dielectric constant on the electrostatic potential in the bulk of electrolyte solutions, we investigate a charge conserving Poisson-Boltzmann (CCPB) equation [31,32] with a variable dielectric coefficient and a small parameter $\epsilon$ (related to the Debye screening length) in a bounded connected domain with smooth boundary. Under the Robin boundary condition with a given applied potential, the limiting behavior (as $\epsilon\downarrow0$) of the solution (the electrostatic potential) has been rigorously studied. In particular, under the charge neutrality constraint, our result exactly shows the effects of the dielectric coefficient and the applied potential on the limiting value of the solution in the interior domain. The main approach is the Pohozaev's identity of this model. On the other hand, under the charge non-neutrality constraint, we show that the maximum difference between the boundary and interior values of the solution has a lower bound $\log\frac{1}{\epsilon}$ as $\epsilon$ goes to zero. Such an asymptotic blow-up behavior describes an unstable phenomenon which is totally different from the behavior of the solution under the charge neutrality constraint.
2016, 36(6): 3277-3315
doi: 10.3934/dcds.2016.36.3277
+[Abstract](1125)
+[PDF](628.4KB)
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In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type. These nonexistence results, known as Liouville theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type of integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.
In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type. These nonexistence results, known as Liouville theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type of integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.
2016, 36(6): 3317-3338
doi: 10.3934/dcds.2016.36.3317
+[Abstract](1327)
+[PDF](461.0KB)
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In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In this first part, we show that a spreading-vanishing dichotomy occurs for such free boundary problems, that is, the species either successfully spreads to all the new environment and stabilizes at a time almost periodic positive solution, or it fails to establish and dies out eventually. The results of this part extend the existing results on spreading-vanishing dichotomy for time and space independent, or time periodic and space independent, or time independent and space periodic diffusive KPP equations with free boundaries. The extension is nontrivial and is ever done for the first time.
In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In this first part, we show that a spreading-vanishing dichotomy occurs for such free boundary problems, that is, the species either successfully spreads to all the new environment and stabilizes at a time almost periodic positive solution, or it fails to establish and dies out eventually. The results of this part extend the existing results on spreading-vanishing dichotomy for time and space independent, or time periodic and space independent, or time independent and space periodic diffusive KPP equations with free boundaries. The extension is nontrivial and is ever done for the first time.
2016, 36(6): 3339-3356
doi: 10.3934/dcds.2016.36.3339
+[Abstract](1005)
+[PDF](464.0KB)
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The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms $||u||_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order.
The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms $||u||_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order.
2016, 36(6): 3357-3373
doi: 10.3934/dcds.2016.36.3357
+[Abstract](1376)
+[PDF](444.9KB)
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We study the following elliptic system with Sobolev critical exponent \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=|u|^{2^*-2}u + \frac{\lambda\alpha}{2^*}|u|^{\alpha-2}|v|^{\beta}u,\, &x\in \mathbb{R}^N, \\ -\Delta v=|v|^{2^*-2}v + \frac{\lambda\beta}{2^*}|u|^{\alpha}|v|^{\beta-2}v,\, &x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $\lambda>0$ is a parameter, $N\geq 3$, $\alpha, \beta>1,$ $\alpha+\beta=2^*:=\frac{2N}{N-2}$, the critical Sobolev exponent. We obtain a uniqueness result on the least energy solutions and show that a manifold of a type of positive solutions is non-degenerate in some ranges of $\lambda,\alpha,\beta,N$ for the above system.
We study the following elliptic system with Sobolev critical exponent \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=|u|^{2^*-2}u + \frac{\lambda\alpha}{2^*}|u|^{\alpha-2}|v|^{\beta}u,\, &x\in \mathbb{R}^N, \\ -\Delta v=|v|^{2^*-2}v + \frac{\lambda\beta}{2^*}|u|^{\alpha}|v|^{\beta-2}v,\, &x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $\lambda>0$ is a parameter, $N\geq 3$, $\alpha, \beta>1,$ $\alpha+\beta=2^*:=\frac{2N}{N-2}$, the critical Sobolev exponent. We obtain a uniqueness result on the least energy solutions and show that a manifold of a type of positive solutions is non-degenerate in some ranges of $\lambda,\alpha,\beta,N$ for the above system.
2016, 36(6): 3375-3416
doi: 10.3934/dcds.2016.36.3375
+[Abstract](1149)
+[PDF](1840.9KB)
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We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.
We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.
2016, 36(6): 3417-3433
doi: 10.3934/dcds.2016.36.3417
+[Abstract](930)
+[PDF](418.4KB)
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The aim of this work is to study a kind of refinement of the entropy conjecture, in the context of partially hyperbolic diffeomorphism with one dimensional central direction, of $d$-dimensional torus. We start by establishing a connection between the unstable index of hyperbolic sets and the index at algebraic level. Two examples are given which might shed light on which are the good questions in the higher dimensional center case.
The aim of this work is to study a kind of refinement of the entropy conjecture, in the context of partially hyperbolic diffeomorphism with one dimensional central direction, of $d$-dimensional torus. We start by establishing a connection between the unstable index of hyperbolic sets and the index at algebraic level. Two examples are given which might shed light on which are the good questions in the higher dimensional center case.
2016, 36(6): 3435-3443
doi: 10.3934/dcds.2016.36.3435
+[Abstract](1337)
+[PDF](386.7KB)
Abstract:
We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. It is well-known that $f$ has positive topological entropy iff $\{ f_n\}_{n\ge 1}$ has. On the other hand, for systems with zero topological entropy, $\{ f_n\}_{n\ge 1}$ with very complex dynamics can converge even to the identity map. We study the following question: Which properties of the limit function $f$ are inherited by nonautonomous system $\{ f_n\}_{n\ge 1}$? We show that Li-Yorke chaos, distributional chaos DC1 and, for zero entropy maps, infinite $\omega$-limit sets are inherited by nonautonomous systems and, for zero entropy maps, we give a criterion on $f$ under which $\{ f_n\}_{n\ge 1}$ is DC1. More precisely, our main results are: (i) If $f$ is Li-Yorke chaotic then $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and the analogous implication is true for distributional chaos DC1; (ii) If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $\omega$-limit sets; (iii) We introduce new notion of a quasi horseshoe, a generalization of horseshoe. It turns out that $\{f_n\}_{n\ge 1}$ exhibits distributional chaos DC1 if $f$ has a quasi horseshoe. The last result is true for maps defined on arbitrary compact metric spaces.
We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. It is well-known that $f$ has positive topological entropy iff $\{ f_n\}_{n\ge 1}$ has. On the other hand, for systems with zero topological entropy, $\{ f_n\}_{n\ge 1}$ with very complex dynamics can converge even to the identity map. We study the following question: Which properties of the limit function $f$ are inherited by nonautonomous system $\{ f_n\}_{n\ge 1}$? We show that Li-Yorke chaos, distributional chaos DC1 and, for zero entropy maps, infinite $\omega$-limit sets are inherited by nonautonomous systems and, for zero entropy maps, we give a criterion on $f$ under which $\{ f_n\}_{n\ge 1}$ is DC1. More precisely, our main results are: (i) If $f$ is Li-Yorke chaotic then $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and the analogous implication is true for distributional chaos DC1; (ii) If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $\omega$-limit sets; (iii) We introduce new notion of a quasi horseshoe, a generalization of horseshoe. It turns out that $\{f_n\}_{n\ge 1}$ exhibits distributional chaos DC1 if $f$ has a quasi horseshoe. The last result is true for maps defined on arbitrary compact metric spaces.
2016, 36(6): 3445-3461
doi: 10.3934/dcds.2016.36.3445
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Abstract:
In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.
In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.
2016, 36(6): 3463-3481
doi: 10.3934/dcds.2016.36.3463
+[Abstract](1064)
+[PDF](465.4KB)
Abstract:
Let $f: M \to M$ be a $C^{1+\theta}$-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by $f$ and played on any unstable manifold. Utilizing it we generalize some results of [25] as follows. Consider a set of points with non-dense forward orbit: $$E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$ for some $y \in M$ and $$E_{x}(f, y) := E(f, y) \cap W^u(x)$$ for any $x\in M$. We show that $E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimension equal to $\dim W^u(x)$. Then for any nonempty open set $V \subset M$ we show that $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by using a technique of constructing measures supported on $E(f, y)$ with lower pointwise dimension approximating $\dim M$.
Let $f: M \to M$ be a $C^{1+\theta}$-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by $f$ and played on any unstable manifold. Utilizing it we generalize some results of [25] as follows. Consider a set of points with non-dense forward orbit: $$E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$ for some $y \in M$ and $$E_{x}(f, y) := E(f, y) \cap W^u(x)$$ for any $x\in M$. We show that $E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimension equal to $\dim W^u(x)$. Then for any nonempty open set $V \subset M$ we show that $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by using a technique of constructing measures supported on $E(f, y)$ with lower pointwise dimension approximating $\dim M$.
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