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Discrete & Continuous Dynamical Systems
March 2016 , Volume 36 , Issue 3
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2016, 36(3): 1143-1157
doi: 10.3934/dcds.2016.36.1143
+[Abstract](3195)
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Abstract:
Let $0 < s < 1$ and $1< p < 2$ be such that $ps < N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality:
Given $1 \le q < p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality （1.1）.
Let $0 < s < 1$ and $1< p < 2$ be such that $ps < N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality:
Given $1 \le q < p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality （1.1）.
2016, 36(3): 1159-1173
doi: 10.3934/dcds.2016.36.1159
+[Abstract](2238)
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Abstract:
We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the $\mathbb{R}$-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all $\beta$-substitutions for $\beta$ a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from substitutions associated with the Jacobi-Perron and Brun continued fraction algorithms.
We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the $\mathbb{R}$-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all $\beta$-substitutions for $\beta$ a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from substitutions associated with the Jacobi-Perron and Brun continued fraction algorithms.
2016, 36(3): 1175-1208
doi: 10.3934/dcds.2016.36.1175
+[Abstract](2621)
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We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact.
We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact.
2016, 36(3): 1209-1247
doi: 10.3934/dcds.2016.36.1209
+[Abstract](2715)
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Abstract:
We study the well-posedness of a class of nonlocal-interaction equations on general domains $\Omega\subset \mathbb{R}^{d}$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $\lambda$-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity. Our approach is based on the theory of gradient flows in spaces of probability measures.
We study the well-posedness of a class of nonlocal-interaction equations on general domains $\Omega\subset \mathbb{R}^{d}$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $\lambda$-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity. Our approach is based on the theory of gradient flows in spaces of probability measures.
2016, 36(3): 1249-1269
doi: 10.3934/dcds.2016.36.1249
+[Abstract](2208)
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This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_\beta(x)=\beta x$ mod 1. Let $\mathcal{J}_\beta(a,b) := \{ x \in (0,1) : T_\beta^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every periodic point of period $n$ for $T_\beta$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\beta(a,b)$. In this paper we completely describe the following sets: \begin{align*} D_0(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \neq \emptyset \}, \\ D_1(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \text{ is uncountable} \}, \\ D_2(\beta) &= \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}. \end{align*}
This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\beta$-transformation. Let $\beta \in (1,2)$ and consider the $\beta$-transformation $T_\beta(x)=\beta x$ mod 1. Let $\mathcal{J}_\beta(a,b) := \{ x \in (0,1) : T_\beta^n(x) \notin (a,b) \text{ for all } n \geq 0 \}$. An integer $n$ is bad for $(a,b)$ if every periodic point of period $n$ for $T_\beta$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\beta(a,b)$. In this paper we completely describe the following sets: \begin{align*} D_0(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \neq \emptyset \}, \\ D_1(\beta) &= \{ (a,b) \in [0,1)^2 : \mathcal{J}(a,b) \text{ is uncountable} \}, \\ D_2(\beta) &= \{ (a,b) \in [0,1)^2 : B_\beta(a,b) \text{ is finite} \}. \end{align*}
2016, 36(3): 1271-1278
doi: 10.3934/dcds.2016.36.1271
+[Abstract](2035)
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Abstract:
Let $\varphi$ be a $C^\infty$ transversely symplectic topologically mixing Anosov flow such that $dim E^{su}\geq 2$. We suppose that the weak distributions of $\varphi$ are $C^1$. If the length Hamenstädt metrics of $\varphi$ are sub-Riemannian then we prove that the weak distributions of $\varphi$ are necessarily $C^\infty$. Combined with our previous rigidity result in [5] we deduce the classification of such Anosov flows with $C^1$ weak distributions provided that the length Hamenstädt metrics are sub-Riemannian.
Let $\varphi$ be a $C^\infty$ transversely symplectic topologically mixing Anosov flow such that $dim E^{su}\geq 2$. We suppose that the weak distributions of $\varphi$ are $C^1$. If the length Hamenstädt metrics of $\varphi$ are sub-Riemannian then we prove that the weak distributions of $\varphi$ are necessarily $C^\infty$. Combined with our previous rigidity result in [5] we deduce the classification of such Anosov flows with $C^1$ weak distributions provided that the length Hamenstädt metrics are sub-Riemannian.
2016, 36(3): 1279-1319
doi: 10.3934/dcds.2016.36.1279
+[Abstract](3423)
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We investigate the long term behavior in terms of finite dimensional global attractors and (global) asymptotic stabilization to steady states, as time goes to infinity, of solutions to a non-local semilinear reaction-diffusion equation associated with the fractional Laplace operator on non-smooth domains subject to Dirichlet, fractional Neumann and Robin boundary conditions.
We investigate the long term behavior in terms of finite dimensional global attractors and (global) asymptotic stabilization to steady states, as time goes to infinity, of solutions to a non-local semilinear reaction-diffusion equation associated with the fractional Laplace operator on non-smooth domains subject to Dirichlet, fractional Neumann and Robin boundary conditions.
2016, 36(3): 1321-1329
doi: 10.3934/dcds.2016.36.1321
+[Abstract](2171)
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We prove that a Lattès map admits an always full wandering continuum if and only if it is flexible. The full wandering continuum is a line segment in a bi-infinite or one-side-infinite geodesic under the flat metric.
We prove that a Lattès map admits an always full wandering continuum if and only if it is flexible. The full wandering continuum is a line segment in a bi-infinite or one-side-infinite geodesic under the flat metric.
2016, 36(3): 1331-1353
doi: 10.3934/dcds.2016.36.1331
+[Abstract](2767)
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This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.
This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.
2016, 36(3): 1355-1382
doi: 10.3934/dcds.2016.36.1355
+[Abstract](1914)
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Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a self-consistent interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)) [18], a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developed. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developed in the spirit of James and Vauchelet (NoDEA (2013)) [26]. However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.
Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a self-consistent interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)) [18], a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developed. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developed in the spirit of James and Vauchelet (NoDEA (2013)) [26]. However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.
2016, 36(3): 1383-1413
doi: 10.3934/dcds.2016.36.1383
+[Abstract](2269)
+[PDF](522.5KB)
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We study the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton's law of cooling, which lead to nontrivial equilibrium configurations. We study the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain.
We study the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton's law of cooling, which lead to nontrivial equilibrium configurations. We study the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain.
2016, 36(3): 1415-1430
doi: 10.3934/dcds.2016.36.1415
+[Abstract](2089)
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We consider the linear heat equation on a bounded domain, which has two components with a thin coating surrounding a body (of metallic nature), subject to the Dirichlet boundary condition. The coating is composed of two layers, the pure ceramic part and the mixed part. The mixed part is considered to be functionally graded material (FGM) that is meant to make a smooth transition from being metallic to being ceramic. The diffusion tensor is isotropic on the body, and allowed to be anisotropic on the coating; and the size of diffusion tensor may differ significantly in these components. We find effective boundary conditions (EBCs) that are approximately satisfied by the solution of the heat equation on the boundary of the body. A concrete example is considered to study the effect of FGM coating. We also provide numerical simulations to verify our theoretical results.
We consider the linear heat equation on a bounded domain, which has two components with a thin coating surrounding a body (of metallic nature), subject to the Dirichlet boundary condition. The coating is composed of two layers, the pure ceramic part and the mixed part. The mixed part is considered to be functionally graded material (FGM) that is meant to make a smooth transition from being metallic to being ceramic. The diffusion tensor is isotropic on the body, and allowed to be anisotropic on the coating; and the size of diffusion tensor may differ significantly in these components. We find effective boundary conditions (EBCs) that are approximately satisfied by the solution of the heat equation on the boundary of the body. A concrete example is considered to study the effect of FGM coating. We also provide numerical simulations to verify our theoretical results.
2016, 36(3): 1431-1464
doi: 10.3934/dcds.2016.36.1431
+[Abstract](2609)
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Existence of a solution of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{aligned} & - \Delta u + V_1(x) u=\mu_1(x) u^3 + \beta(x) u v^2 \qquad\mbox{in}\ \mathbb{R}^N, \\ & - \Delta v + V_2(x) v=\beta(x) u^2 v + \mu_2(x) v^3 \qquad \mbox{in}\ \mathbb{R}^N, \\ & u>0,\ v>0,\quad u,\ v\in H^1(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $N=1,2,3$, and $V_j,\mu_j,\beta$ are continuous functions of $x\in\mathbb{R}^N$, is proved provided that either $V_j,\mu_j,\beta$ are invariant under the action of a finite subgroup of $O(N)$ or there is no such invariance assumption. In either case the result is obtained both for $\beta$ small and for $\beta$ large in terms of $V_j$ and $\mu_j$.
Existence of a solution of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{aligned} & - \Delta u + V_1(x) u=\mu_1(x) u^3 + \beta(x) u v^2 \qquad\mbox{in}\ \mathbb{R}^N, \\ & - \Delta v + V_2(x) v=\beta(x) u^2 v + \mu_2(x) v^3 \qquad \mbox{in}\ \mathbb{R}^N, \\ & u>0,\ v>0,\quad u,\ v\in H^1(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $N=1,2,3$, and $V_j,\mu_j,\beta$ are continuous functions of $x\in\mathbb{R}^N$, is proved provided that either $V_j,\mu_j,\beta$ are invariant under the action of a finite subgroup of $O(N)$ or there is no such invariance assumption. In either case the result is obtained both for $\beta$ small and for $\beta$ large in terms of $V_j$ and $\mu_j$.
2016, 36(3): 1465-1491
doi: 10.3934/dcds.2016.36.1465
+[Abstract](2616)
+[PDF](828.5KB)
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We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.
We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.
2016, 36(3): 1493-1537
doi: 10.3934/dcds.2016.36.1493
+[Abstract](2340)
+[PDF](724.1KB)
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In this paper we prove that in appropriate weighted Sobolev spaces, in the case of no bound states, the scattering map of the Korteweg-de Vries (KdV) on $\mathbb{R}$ is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV flow and the corresponding Airy flow is 1-smoothing.
In this paper we prove that in appropriate weighted Sobolev spaces, in the case of no bound states, the scattering map of the Korteweg-de Vries (KdV) on $\mathbb{R}$ is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV flow and the corresponding Airy flow is 1-smoothing.
2016, 36(3): 1539-1562
doi: 10.3934/dcds.2016.36.1539
+[Abstract](2774)
+[PDF](560.1KB)
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The paper deals with the global existence of strong solution to the equations modeling a motion of a rigid body around viscous fluid. Moreover, the estimates of second gradients of velocity and pressure are given.
The paper deals with the global existence of strong solution to the equations modeling a motion of a rigid body around viscous fluid. Moreover, the estimates of second gradients of velocity and pressure are given.
2016, 36(3): 1563-1581
doi: 10.3934/dcds.2016.36.1563
+[Abstract](2710)
+[PDF](445.5KB)
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In this paper, we study the three-dimensional axisymmetric Boussinesq equations with swirl. We establish the global existence of solutions for the three-dimensional axisymmetric Boussinesq equations for a family of anisotropic initial data.
In this paper, we study the three-dimensional axisymmetric Boussinesq equations with swirl. We establish the global existence of solutions for the three-dimensional axisymmetric Boussinesq equations for a family of anisotropic initial data.
2016, 36(3): 1583-1601
doi: 10.3934/dcds.2016.36.1583
+[Abstract](2870)
+[PDF](473.5KB)
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We study the three-dimensional full compressible Euler-Poisson system without the temperature damping. Using a general energy method, we prove the optimal decay rates of the solutions and their higher order derivatives. We show that the optimal decay rates is algebraic but not exponential since the absence of temperature damping.
We study the three-dimensional full compressible Euler-Poisson system without the temperature damping. Using a general energy method, we prove the optimal decay rates of the solutions and their higher order derivatives. We show that the optimal decay rates is algebraic but not exponential since the absence of temperature damping.
2016, 36(3): 1603-1628
doi: 10.3934/dcds.2016.36.1603
+[Abstract](2423)
+[PDF](573.4KB)
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In this paper, by an approximating argument, we obtain infinitely many solutions for the following problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = \mu \frac{|u|^{2^{*}(t)-2}u}{|y|^{t}} + \frac{|u|^{2^{*}(s)-2}u}{|y|^{s}} + a(x) u, & \hbox{$\text{in} \Omega$}, \\ u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array} \right. \end{equation*} where $\mu\geq0,2^{*}(t)=\frac{2(N-t)}{N-2},2^{*}(s) = \frac{2(N-s)}{N-2},0\leq t < s < 2,x = (y,z)\in \mathbb{R}^{k} \times \mathbb{R}^{N-k},2 \leq k < N,(0,z^*)\subset \bar{\Omega}$ and $\Omega$ is an open bounded domain in $\mathbb{R}^{N}.$ We prove that if $N > 6+t$ when $\mu>0$ and $N > 6+s$ when $\mu=0,$ $a((0,z^*)) > 0,$ $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many solutions.
In this paper, by an approximating argument, we obtain infinitely many solutions for the following problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = \mu \frac{|u|^{2^{*}(t)-2}u}{|y|^{t}} + \frac{|u|^{2^{*}(s)-2}u}{|y|^{s}} + a(x) u, & \hbox{$\text{in} \Omega$}, \\ u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array} \right. \end{equation*} where $\mu\geq0,2^{*}(t)=\frac{2(N-t)}{N-2},2^{*}(s) = \frac{2(N-s)}{N-2},0\leq t < s < 2,x = (y,z)\in \mathbb{R}^{k} \times \mathbb{R}^{N-k},2 \leq k < N,(0,z^*)\subset \bar{\Omega}$ and $\Omega$ is an open bounded domain in $\mathbb{R}^{N}.$ We prove that if $N > 6+t$ when $\mu>0$ and $N > 6+s$ when $\mu=0,$ $a((0,z^*)) > 0,$ $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many solutions.
2016, 36(3): 1629-1647
doi: 10.3934/dcds.2016.36.1629
+[Abstract](2539)
+[PDF](1933.8KB)
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In this work we develop the shape Conley index theory for local semiflows on complete metric spaces by using a weaker notion of shape index pairs. This allows us to calculate the shape index of a compact isolated invariant set $K$ by restricting the system on any closed subset that contains a local unstable manifold of $K$, and hence significantly increases the flexibility of the calculation of shape indices and Morse equations. In particular, it allows to calculate shape indices and Morse equations for an infinite dimensional system by using only the unstable manifolds of the invariant sets, without requiring the system to be two-sided on the unstable manifolds.
In this work we develop the shape Conley index theory for local semiflows on complete metric spaces by using a weaker notion of shape index pairs. This allows us to calculate the shape index of a compact isolated invariant set $K$ by restricting the system on any closed subset that contains a local unstable manifold of $K$, and hence significantly increases the flexibility of the calculation of shape indices and Morse equations. In particular, it allows to calculate shape indices and Morse equations for an infinite dimensional system by using only the unstable manifolds of the invariant sets, without requiring the system to be two-sided on the unstable manifolds.
2016, 36(3): 1649-1659
doi: 10.3934/dcds.2016.36.1649
+[Abstract](2289)
+[PDF](420.4KB)
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Let $M$ be a closed and smooth manifold and $H_\varepsilon:T^*M\to\mathbf{R}^1$ be a family of Tonelli Hamiltonians for $\varepsilon\geq0$ small. For each $\varphi\in C(M,\mathbf{R}^1)$, $T^\varepsilon_t\varphi(x)$ is the unique viscosity solution of the Cauchy problem \begin{align*} \left\{ \begin{array}{ll} d_tw+H_\varepsilon(x,d_xw)=0, & \ \mathrm{in}\ M\times(0,+\infty),\\ w|_{t=0}=\varphi, & \ \mathrm{on}\ M, \end{array} \right. \end{align*} where $T^\varepsilon_t$ is the Lax-Oleinik operator associated with $H_\varepsilon$. A result of Fathi asserts that the uniform limit, for $t\to+\infty$, of $T^\varepsilon_t\varphi+c_\varepsilon t$ exists and the limit $\bar{\varphi}_\varepsilon$ is a viscosity solution of the stationary Hamilton-Jacobi equation \begin{align*} H_\varepsilon(x,d_xu)=c_\varepsilon, \end{align*} where $c_\varepsilon$ is the unique $k$ for which the equation $H_\varepsilon(x,d_xu)=k$ admits viscosity solutions. In the present paper we discuss the continuous dependence of the viscosity solution $\bar{\varphi}_\varepsilon$ with respect to the parameter $\varepsilon$.
Let $M$ be a closed and smooth manifold and $H_\varepsilon:T^*M\to\mathbf{R}^1$ be a family of Tonelli Hamiltonians for $\varepsilon\geq0$ small. For each $\varphi\in C(M,\mathbf{R}^1)$, $T^\varepsilon_t\varphi(x)$ is the unique viscosity solution of the Cauchy problem \begin{align*} \left\{ \begin{array}{ll} d_tw+H_\varepsilon(x,d_xw)=0, & \ \mathrm{in}\ M\times(0,+\infty),\\ w|_{t=0}=\varphi, & \ \mathrm{on}\ M, \end{array} \right. \end{align*} where $T^\varepsilon_t$ is the Lax-Oleinik operator associated with $H_\varepsilon$. A result of Fathi asserts that the uniform limit, for $t\to+\infty$, of $T^\varepsilon_t\varphi+c_\varepsilon t$ exists and the limit $\bar{\varphi}_\varepsilon$ is a viscosity solution of the stationary Hamilton-Jacobi equation \begin{align*} H_\varepsilon(x,d_xu)=c_\varepsilon, \end{align*} where $c_\varepsilon$ is the unique $k$ for which the equation $H_\varepsilon(x,d_xu)=k$ admits viscosity solutions. In the present paper we discuss the continuous dependence of the viscosity solution $\bar{\varphi}_\varepsilon$ with respect to the parameter $\varepsilon$.
2016, 36(3): 1661-1675
doi: 10.3934/dcds.2016.36.1661
+[Abstract](2039)
+[PDF](474.0KB)
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We study the regularity of sonic curves from a two-dimensional Riemann problem for the nonlinear wave system of Chaplygin gas, which is an essential step for the global existence of solutions to the two-dimensional Riemann problems. As a result, we establish the global existence of uniformly smooth solutions in the semi-hyperbolic patches up to the sonic boundary, where the degeneracy of hyperbolicity occurs. Furthermore, we show the $C^1$-regularity of sonic curves.
We study the regularity of sonic curves from a two-dimensional Riemann problem for the nonlinear wave system of Chaplygin gas, which is an essential step for the global existence of solutions to the two-dimensional Riemann problems. As a result, we establish the global existence of uniformly smooth solutions in the semi-hyperbolic patches up to the sonic boundary, where the degeneracy of hyperbolicity occurs. Furthermore, we show the $C^1$-regularity of sonic curves.
2016, 36(3): 1677-1692
doi: 10.3934/dcds.2016.36.1677
+[Abstract](2387)
+[PDF](428.1KB)
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This work focuses on the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. By KAM method and the special structure of unperturbed nonlinear terms, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of [22].
This work focuses on the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. By KAM method and the special structure of unperturbed nonlinear terms, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of [22].
2016, 36(3): 1693-1707
doi: 10.3934/dcds.2016.36.1693
+[Abstract](2113)
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Abstract:
In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the classes is not innately locally maximal, it is hard to answer whether structurally stable homoclinic classes are hyperbolic. In this article, we make some progress on this question. We prove that if a homoclinic class is structurally stable, then it admits a dominated splitting. Moreover we prove that codimension one structurally stable classes are hyperbolic. Also, if the diffeomorphism is far away from homoclinic tangencies, then structurally stable homoclinic classes are hyperbolic.
In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the classes is not innately locally maximal, it is hard to answer whether structurally stable homoclinic classes are hyperbolic. In this article, we make some progress on this question. We prove that if a homoclinic class is structurally stable, then it admits a dominated splitting. Moreover we prove that codimension one structurally stable classes are hyperbolic. Also, if the diffeomorphism is far away from homoclinic tangencies, then structurally stable homoclinic classes are hyperbolic.
2016, 36(3): 1709-1719
doi: 10.3934/dcds.2016.36.1709
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Abstract:
In this paper, we consider the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge. By using the fact that when the spatial gauge potentials are in the Coulomb gauge, their $\dot{H}^1$ norms can be controlled by the energy of the corresponding system and their $L^2$ norms, and the gauge invariance of the systems, we show that finite energy solutions of these two systems exist globally in this gauge.
In this paper, we consider the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge. By using the fact that when the spatial gauge potentials are in the Coulomb gauge, their $\dot{H}^1$ norms can be controlled by the energy of the corresponding system and their $L^2$ norms, and the gauge invariance of the systems, we show that finite energy solutions of these two systems exist globally in this gauge.
2016, 36(3): 1721-1736
doi: 10.3934/dcds.2016.36.1721
+[Abstract](2967)
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Abstract:
In this paper, we consider $\alpha$-harmonic functions in the half space $\mathbb{R}^n_+$: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2} u(x)=0,~u(x)\geq0, & \qquad x\in\mathbb{R}^n_+, \\ u(x)\equiv0, & \qquad x\notin\mathbb{R}^{n}_{+}. \end{array}\right. (1) \end{equation} We prove that all solutions of (1) are either identically zero or assuming the form \begin{equation} u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+}, \end{array}\right. \label{2} \end{equation} for some positive constant $C$.
In this paper, we consider $\alpha$-harmonic functions in the half space $\mathbb{R}^n_+$: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2} u(x)=0,~u(x)\geq0, & \qquad x\in\mathbb{R}^n_+, \\ u(x)\equiv0, & \qquad x\notin\mathbb{R}^{n}_{+}. \end{array}\right. (1) \end{equation} We prove that all solutions of (1) are either identically zero or assuming the form \begin{equation} u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+}, \end{array}\right. \label{2} \end{equation} for some positive constant $C$.
2016, 36(3): 1737-1757
doi: 10.3934/dcds.2016.36.1737
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Abstract:
This paper deals with a parabolic-parabolic-ODE chemotaxis haptotaxis system with nonlinear diffusion \begin{eqnarray*}\label{1a} \left\{ \begin{split}{} &u_{t}=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &v_{t}=\Delta v-v+u, \\ &w_{t}=-vw, \end{split} \right. \end{eqnarray*} under Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{2}$, where $\chi$, $\xi$ and $\mu$ are positive parameters and $\varphi(u)$ is a nonlinear diffusion function. Firstly, under the case of non-degenerate diffusion, it is proved that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in $\Omega\times(0,\infty)$. Moreover, under the case of degenerate diffusion, we prove that the corresponding problem admits at least one nonnegative global bounded-in-time weak solution. Finally, under some additional conditions, we derive the temporal decay estimate of $w$.
This paper deals with a parabolic-parabolic-ODE chemotaxis haptotaxis system with nonlinear diffusion \begin{eqnarray*}\label{1a} \left\{ \begin{split}{} &u_{t}=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &v_{t}=\Delta v-v+u, \\ &w_{t}=-vw, \end{split} \right. \end{eqnarray*} under Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{2}$, where $\chi$, $\xi$ and $\mu$ are positive parameters and $\varphi(u)$ is a nonlinear diffusion function. Firstly, under the case of non-degenerate diffusion, it is proved that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in $\Omega\times(0,\infty)$. Moreover, under the case of degenerate diffusion, we prove that the corresponding problem admits at least one nonnegative global bounded-in-time weak solution. Finally, under some additional conditions, we derive the temporal decay estimate of $w$.
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