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1078-0947
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Discrete and Continuous Dynamical Systems
November 2007 , Volume 18 , Issue 4
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2007, 18(4): 613-626
doi: 10.3934/dcds.2007.18.613
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Abstract:
We consider the singularly perturbed semilinear parabolic problem $u_t-d^2\Delta u+u=f(u)$ with homogeneous Neumann boundary conditions on a smoothly bounded domain $\Omega\subseteq \mathbb{R}^N$. Here $f$ is superlinear at $0$, and $\pm\infty$ and has subcritical growth. For small $d>0$ we construct a compact connected invariant set $X_d$ in the boundary of the domain of attraction of the asymptotically stable equilibrium $0$. The main features of $X_d$ are that it consists of positive functions that are pairwise non-comparable, and its topology is at least as rich as the topology of $\partial\Omega$ in a certain sense. If the number of equilibria in $X_d$ is finite, then this implies the existence of connecting orbits within $X_d$ that are not a consequence of a well known result by Matano.
We consider the singularly perturbed semilinear parabolic problem $u_t-d^2\Delta u+u=f(u)$ with homogeneous Neumann boundary conditions on a smoothly bounded domain $\Omega\subseteq \mathbb{R}^N$. Here $f$ is superlinear at $0$, and $\pm\infty$ and has subcritical growth. For small $d>0$ we construct a compact connected invariant set $X_d$ in the boundary of the domain of attraction of the asymptotically stable equilibrium $0$. The main features of $X_d$ are that it consists of positive functions that are pairwise non-comparable, and its topology is at least as rich as the topology of $\partial\Omega$ in a certain sense. If the number of equilibria in $X_d$ is finite, then this implies the existence of connecting orbits within $X_d$ that are not a consequence of a well known result by Matano.
2007, 18(4): 627-636
doi: 10.3934/dcds.2007.18.627
+[Abstract](2715)
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Abstract:
Existing approaches to the solution of the inverse scattering problems in two and three dimensions rely on linearization of the Helmholtz equation, which requires the knowledge of the Fr\'echet derivative of the far field with respect to the index of refraction. We present an efficient algorithm for this perturbational calculation in two dimensions. Our method is based on the merging and splitting procedures already established for the solution of the Lippmann-Schwinger equation [2], [3], [4]. For an $m$-by-$m$ wavelength problem, the algorithm obtains perturbations to scattered waves for $m$ distinct incident waves in $O(m^3)$ steps.
Existing approaches to the solution of the inverse scattering problems in two and three dimensions rely on linearization of the Helmholtz equation, which requires the knowledge of the Fr\'echet derivative of the far field with respect to the index of refraction. We present an efficient algorithm for this perturbational calculation in two dimensions. Our method is based on the merging and splitting procedures already established for the solution of the Lippmann-Schwinger equation [2], [3], [4]. For an $m$-by-$m$ wavelength problem, the algorithm obtains perturbations to scattered waves for $m$ distinct incident waves in $O(m^3)$ steps.
2007, 18(4): 637-642
doi: 10.3934/dcds.2007.18.637
+[Abstract](2847)
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Abstract:
We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region that shrinks to a point dynamically as the time, $t$, approaches a possible singularity time, $T$. The solution outside the inner core region is assumed to be regular, but it does not satisfy self-similar scaling. Under the assumption that the dynamically rescaled velocity profile converges to a limiting profile as $t \rightarrow T$ in $L^p$ for some $p \in (3,\infty )$, we prove that such a locally self-similar blow-up is not possible. We also obtain a simple but useful non-blowup criterion for the 3D Euler equations.
We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region that shrinks to a point dynamically as the time, $t$, approaches a possible singularity time, $T$. The solution outside the inner core region is assumed to be regular, but it does not satisfy self-similar scaling. Under the assumption that the dynamically rescaled velocity profile converges to a limiting profile as $t \rightarrow T$ in $L^p$ for some $p \in (3,\infty )$, we prove that such a locally self-similar blow-up is not possible. We also obtain a simple but useful non-blowup criterion for the 3D Euler equations.
2007, 18(4): 643-656
doi: 10.3934/dcds.2007.18.643
+[Abstract](3279)
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Abstract:
We show how global weak solutions of the Hunter-Saxton equation can be naturally constructed using the geometric interpretation of the equation as the Euler equation for the geodesic flow on an $L^2$-sphere. The approach involves forming a weak extension of the geodesic flow and relating it to a corresponding weak formulation of the Hunter-Saxton equation.
We show how global weak solutions of the Hunter-Saxton equation can be naturally constructed using the geometric interpretation of the equation as the Euler equation for the geodesic flow on an $L^2$-sphere. The approach involves forming a weak extension of the geodesic flow and relating it to a corresponding weak formulation of the Hunter-Saxton equation.
2007, 18(4): 657-675
doi: 10.3934/dcds.2007.18.657
+[Abstract](2654)
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Abstract:
In this paper we give a description of the asymptotic behavior, as $\varepsilon\to 0$, of the $\varepsilon$-gradient flow in the finite dimensional case. Under very general assumptions, we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic solutions of the gradient flow (fast dynamics).
In this paper we give a description of the asymptotic behavior, as $\varepsilon\to 0$, of the $\varepsilon$-gradient flow in the finite dimensional case. Under very general assumptions, we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic solutions of the gradient flow (fast dynamics).
2007, 18(4): 677-699
doi: 10.3934/dcds.2007.18.677
+[Abstract](2541)
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Abstract:
We consider reversible and equivariant dynamical systems in Banach spaces, either defined by maps or flows. We show that for a reversible (respectively, equivariant) system, the dynamics on any center manifold in a certain class of graphs (namely $C^1$ graphs with Lipschitz first derivative) is also reversible (respectively, equivariant). We consider the general case of center manifolds for a nonuniformly partially hyperbolic dynamics, corresponding to the existence of a nonuniform exponential trichotomy of the linear variational equation. We also consider the case of nonautonomous dynamics.
We consider reversible and equivariant dynamical systems in Banach spaces, either defined by maps or flows. We show that for a reversible (respectively, equivariant) system, the dynamics on any center manifold in a certain class of graphs (namely $C^1$ graphs with Lipschitz first derivative) is also reversible (respectively, equivariant). We consider the general case of center manifolds for a nonuniformly partially hyperbolic dynamics, corresponding to the existence of a nonuniform exponential trichotomy of the linear variational equation. We also consider the case of nonautonomous dynamics.
2007, 18(4): 701-707
doi: 10.3934/dcds.2007.18.701
+[Abstract](2757)
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Abstract:
In this note we outline some improvements to a result of Hilhorst, Peletier, Rotariu and Sivashinsky [5] on the $L_2$ boundedness of solutions to a non-local variant of the Kuramoto-Sivashinsky equation with additional stabilizing and destabilizing terms. We are able to make the following improvements: in the case of odd data we reduce the exponent in the estimate lim sup$_t\rightarrow \infty$ ||$u$ || $\le C L^{\nu}$ from $\nu = \frac{11}{5}$ to $\nu=\frac{3}{2}$, and for the case of general initial data we establish an estimate of the above form with $\nu = \frac{13}{6}$. We also remove the restrictions on the magnitudes of the parameters in the model and track the dependence of our estimates on these parameters, assuming they are at least $O(1)$.
In this note we outline some improvements to a result of Hilhorst, Peletier, Rotariu and Sivashinsky [5] on the $L_2$ boundedness of solutions to a non-local variant of the Kuramoto-Sivashinsky equation with additional stabilizing and destabilizing terms. We are able to make the following improvements: in the case of odd data we reduce the exponent in the estimate lim sup$_t\rightarrow \infty$ ||$u$ || $\le C L^{\nu}$ from $\nu = \frac{11}{5}$ to $\nu=\frac{3}{2}$, and for the case of general initial data we establish an estimate of the above form with $\nu = \frac{13}{6}$. We also remove the restrictions on the magnitudes of the parameters in the model and track the dependence of our estimates on these parameters, assuming they are at least $O(1)$.
2007, 18(4): 709-718
doi: 10.3934/dcds.2007.18.709
+[Abstract](2414)
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Abstract:
We consider a system of semilinear elliptic partial differential equations with exponential nonlinearities in $R^2$. We construct a solution of the system viewing the system as a perturbation of the decoupled Liouville equations and applying suitable implicit function theorem. As a byproduct we obtain very precise information on the asymptotic behaviors of the solutions near infinity.
We consider a system of semilinear elliptic partial differential equations with exponential nonlinearities in $R^2$. We construct a solution of the system viewing the system as a perturbation of the decoupled Liouville equations and applying suitable implicit function theorem. As a byproduct we obtain very precise information on the asymptotic behaviors of the solutions near infinity.
2007, 18(4): 719-746
doi: 10.3934/dcds.2007.18.719
+[Abstract](2291)
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Abstract:
A coupled system of dynamic hyperbolic equations in electroma- gnetic-elasticity theory in the exterior of an open bounded obstacle $\mathcal{O}$ in 3-D is considered. In the presence of dissipative effects we obtain uniform decay rates of the solution as $t \rightarrow +\infty$. We do not require geometric assumptions on the obstacle or extra assumptions on the initial data. We apply our results to study the above system with a nonlinear perturbation, showing that the solutions hold the same rate of decay provided the initial data is "small" in a suitable sense. Previous results of this type have recently been obtained for the scalar wave equation by M. Nakao [18, 19] and R. Ikehata [8].
A coupled system of dynamic hyperbolic equations in electroma- gnetic-elasticity theory in the exterior of an open bounded obstacle $\mathcal{O}$ in 3-D is considered. In the presence of dissipative effects we obtain uniform decay rates of the solution as $t \rightarrow +\infty$. We do not require geometric assumptions on the obstacle or extra assumptions on the initial data. We apply our results to study the above system with a nonlinear perturbation, showing that the solutions hold the same rate of decay provided the initial data is "small" in a suitable sense. Previous results of this type have recently been obtained for the scalar wave equation by M. Nakao [18, 19] and R. Ikehata [8].
2007, 18(4): 747-772
doi: 10.3934/dcds.2007.18.747
+[Abstract](3008)
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Abstract:
Let $a$ and $b$ be unbounded functions in $\mathbb R^N$ with $a$ sufficiently smooth. In this paper we prove that, under suitable growth assumptions on $a$ and $b$, the operator $Au=a\Delta u+b\cdot\nabla u$ admits realizations generating analytic semigroups in $L^p( R^N)$ for any $p\in [1,+\infty]$ and in $C_b( R^N)$. We also explicitly characterize the domain of the infinitesimal generator of such semigroups. Similar results are stated and proved when $R^N$ is replaced with a smooth exterior domain under general boundary conditions.
Let $a$ and $b$ be unbounded functions in $\mathbb R^N$ with $a$ sufficiently smooth. In this paper we prove that, under suitable growth assumptions on $a$ and $b$, the operator $Au=a\Delta u+b\cdot\nabla u$ admits realizations generating analytic semigroups in $L^p( R^N)$ for any $p\in [1,+\infty]$ and in $C_b( R^N)$. We also explicitly characterize the domain of the infinitesimal generator of such semigroups. Similar results are stated and proved when $R^N$ is replaced with a smooth exterior domain under general boundary conditions.
2007, 18(4): 773-791
doi: 10.3934/dcds.2007.18.773
+[Abstract](3649)
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Abstract:
In this paper we study exponential stability of the trivial solution of the state-dependent delay system $\dot x(t)=\sum_{i=1}^m A_{i}(t)x(t-\tau_{i}(t,x_t))$. We show that under mild assumptions, the trivial solution of the state-dependent system is exponentially stable if and only if the trivial solution of the corresponding linear time-dependent delay system $\dot y(t)=\sum_{i=1}^m A_{i}(t)y(t-\tau_{i}(t, 0))$ is exponentially stable. We also compare the order of the exponential stability of the nonlinear equation to that of its linearized equation. We show that in some cases, the two orders are equal. As an application of our main result, we formulate a necessary and sufficient condition for the exponential stability of the trivial solution of a threshold-type delay system.
In this paper we study exponential stability of the trivial solution of the state-dependent delay system $\dot x(t)=\sum_{i=1}^m A_{i}(t)x(t-\tau_{i}(t,x_t))$. We show that under mild assumptions, the trivial solution of the state-dependent system is exponentially stable if and only if the trivial solution of the corresponding linear time-dependent delay system $\dot y(t)=\sum_{i=1}^m A_{i}(t)y(t-\tau_{i}(t, 0))$ is exponentially stable. We also compare the order of the exponential stability of the nonlinear equation to that of its linearized equation. We show that in some cases, the two orders are equal. As an application of our main result, we formulate a necessary and sufficient condition for the exponential stability of the trivial solution of a threshold-type delay system.
2007, 18(4): 793-807
doi: 10.3934/dcds.2007.18.793
+[Abstract](2920)
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Abstract:
We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable $t$, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable.
We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable $t$, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable.
2007, 18(4): 809-816
doi: 10.3934/dcds.2007.18.809
+[Abstract](4856)
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Abstract:
In this paper, we are concerned with the following problem
In this paper, we are concerned with the following problem
(P) $ -\Delta u + V(x)u+\lambda \phi (x) u =f(x,u), x\in \mathbb{R}^3$
$ -\Delta\phi = u^2, \lim_{|x|\rightarrow +\infty}\phi(x)=0,$
where $\lambda >0$ is a parameter, the potential $V(x)$ may not be radially symmetric, and $f(x,s)$ is asymptotically linear with respect to $s$ at infinity. Under some simple assumptions on $V$ and $f$, we prove that the problem (P) has a positive solution for $\lambda$ small and has no any nontrivial solution for $\lambda$ large.
2007, 18(4): 817-827
doi: 10.3934/dcds.2007.18.817
+[Abstract](2761)
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Abstract:
In this paper the relationship between the return times set andseveral mixing properties in measure-theoretical dynamical systems(MDS) is investigated. For an MDS $T$ on a Lebesgue space$(X,$ß,$\mu)$, let ß$^+=\{B\in$ ß$:\mu(B)>0\}$ and$N(A,B)=\{n\in Z_+: \mu(A\cap T^{-n}B)>0\}$ for $A, B\in$ß$^+$. It turns out that $T$ is ergodic iff$N(A,B)$≠$\emptyset$ iff $N(A,B)$ is syndetic; $T$ is weaklymixing iff the lower Banach density of $N(A,B)$ is $1$ iff $N(A,B)$is thick; and $T$ is mildly mixing iff $N(A,B)$ is an $ IP^ * $-set iff$N(A,B)$ is an $(IP-IP)^*$-set for all $A,B\in$ ß$^+$ ifffor each $IP$-set $F$ and $A\in$ß$^+$, $\mu(\bigcup_{n\in{F}}T^{-n}A)=1$. Finally, it is shown that $T$ is intermixing iff$N(A,B)$ is cofinite for all $A,B\in$ß$^+$.
In this paper the relationship between the return times set andseveral mixing properties in measure-theoretical dynamical systems(MDS) is investigated. For an MDS $T$ on a Lebesgue space$(X,$ß,$\mu)$, let ß$^+=\{B\in$ ß$:\mu(B)>0\}$ and$N(A,B)=\{n\in Z_+: \mu(A\cap T^{-n}B)>0\}$ for $A, B\in$ß$^+$. It turns out that $T$ is ergodic iff$N(A,B)$≠$\emptyset$ iff $N(A,B)$ is syndetic; $T$ is weaklymixing iff the lower Banach density of $N(A,B)$ is $1$ iff $N(A,B)$is thick; and $T$ is mildly mixing iff $N(A,B)$ is an $ IP^ * $-set iff$N(A,B)$ is an $(IP-IP)^*$-set for all $A,B\in$ ß$^+$ ifffor each $IP$-set $F$ and $A\in$ß$^+$, $\mu(\bigcup_{n\in{F}}T^{-n}A)=1$. Finally, it is shown that $T$ is intermixing iff$N(A,B)$ is cofinite for all $A,B\in$ß$^+$.
2007, 18(4): 829-851
doi: 10.3934/dcds.2007.18.829
+[Abstract](3484)
+[PDF](227.9KB)
Abstract:
In [6], Cheng and Newhouse introduced and studied the new invariants - preimage entropies for deterministic dynamical systems. In this paper, the analogous notions, measure-theoretic preimage entropy and topological preimage entropy, are formulated for random dynamical systems. Analogues of many known results for entropies, such as the Shannon-McMillan-Breiman Theorem, the Kolmogorov-Sinai Theorem, the Abromov-Rokhlin formula and the power rule, are obtained for preimage entropies. In particular, a variational principle is given.
In [6], Cheng and Newhouse introduced and studied the new invariants - preimage entropies for deterministic dynamical systems. In this paper, the analogous notions, measure-theoretic preimage entropy and topological preimage entropy, are formulated for random dynamical systems. Analogues of many known results for entropies, such as the Shannon-McMillan-Breiman Theorem, the Kolmogorov-Sinai Theorem, the Abromov-Rokhlin formula and the power rule, are obtained for preimage entropies. In particular, a variational principle is given.
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