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1078-0947
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Discrete & Continuous Dynamical Systems - A
March 2011 , Volume 31 , Issue 1
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2011, 31(1): 1-23
doi: 10.3934/dcds.2011.31.1
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Abstract:
This paper is concerned with traveling wavefronts in a Lotka-Volterra model with nonlocal delays for two cooperative species. By using comparison principle, some existence and nonexistence results are obtained. If the wave speed is larger than a threshold which can be formulated in terms of basic parameters, we prove the asymptotic stability of traveling wavefronts by the spectral analysis method together with squeezing technique.
This paper is concerned with traveling wavefronts in a Lotka-Volterra model with nonlocal delays for two cooperative species. By using comparison principle, some existence and nonexistence results are obtained. If the wave speed is larger than a threshold which can be formulated in terms of basic parameters, we prove the asymptotic stability of traveling wavefronts by the spectral analysis method together with squeezing technique.
2011, 31(1): 25-34
doi: 10.3934/dcds.2011.31.25
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Abstract:
We prove new results about the number of isolated periodic solutions of a first order differential equation with a polynomial nonlinearity. Such results are applied to bound the number of limit cycles of a family of planar polynomial vector fields which generalize the so-called rigid systems.
We prove new results about the number of isolated periodic solutions of a first order differential equation with a polynomial nonlinearity. Such results are applied to bound the number of limit cycles of a family of planar polynomial vector fields which generalize the so-called rigid systems.
2011, 31(1): 35-64
doi: 10.3934/dcds.2011.31.35
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Abstract:
The aim of the paper is to provide conditions ensuring the existence of non-trivial non-negative periodic solutions to a system of doubly degenerate parabolic equations containing delayed nonlocal terms and satisfying Dirichlet boundary conditions. The employed approach is based on the theory of the Leray-Schauder topological degree theory, thus a crucial purpose of the paper is to obtain a priori bounds in a convenient functional space, here $L^2(Q_T)$, on the solutions of certain homotopies. This is achieved under different assumptions on the sign of the kernels of the nonlocal terms. The considered system is a possible model of the interactions between two biological species sharing the same territory where such interactions are modeled by the kernels of the nonlocal terms. To this regard the obtained results can be viewed as coexistence results of the two biological populations under different intra and inter specific interferences on their natural growth rates.
The aim of the paper is to provide conditions ensuring the existence of non-trivial non-negative periodic solutions to a system of doubly degenerate parabolic equations containing delayed nonlocal terms and satisfying Dirichlet boundary conditions. The employed approach is based on the theory of the Leray-Schauder topological degree theory, thus a crucial purpose of the paper is to obtain a priori bounds in a convenient functional space, here $L^2(Q_T)$, on the solutions of certain homotopies. This is achieved under different assumptions on the sign of the kernels of the nonlocal terms. The considered system is a possible model of the interactions between two biological species sharing the same territory where such interactions are modeled by the kernels of the nonlocal terms. To this regard the obtained results can be viewed as coexistence results of the two biological populations under different intra and inter specific interferences on their natural growth rates.
2011, 31(1): 65-107
doi: 10.3934/dcds.2011.31.65
+[Abstract](2319)
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Abstract:
Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.
Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.
2011, 31(1): 109-118
doi: 10.3934/dcds.2011.31.109
+[Abstract](2571)
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Abstract:
We prove Strichartz estimates for the absolutely continuous evolution of a Schrödinger operator $H = (i\nabla + A)^2 + V$ in $R^n$, $n \ge 3$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which required extra conditions on ${\rm div}\,A$ or $|\nabla|^{\frac12}A$ in order to place the first order part of the perturbation within a suitable class of pseudo-differential operators.
We prove Strichartz estimates for the absolutely continuous evolution of a Schrödinger operator $H = (i\nabla + A)^2 + V$ in $R^n$, $n \ge 3$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which required extra conditions on ${\rm div}\,A$ or $|\nabla|^{\frac12}A$ in order to place the first order part of the perturbation within a suitable class of pseudo-differential operators.
2011, 31(1): 119-138
doi: 10.3934/dcds.2011.31.119
+[Abstract](2966)
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Abstract:
In this paper the long time behaviour of the solutions of the 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in $H_{0}^{1}(\Omega )\times L_{2}(\Omega )$ and then it is proved that this is also a global attractor in $(H^{2}(\Omega )\cap H_{0}^{1}(\Omega ))\times H_{0}^{1}(\Omega )$.
In this paper the long time behaviour of the solutions of the 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in $H_{0}^{1}(\Omega )\times L_{2}(\Omega )$ and then it is proved that this is also a global attractor in $(H^{2}(\Omega )\cap H_{0}^{1}(\Omega ))\times H_{0}^{1}(\Omega )$.
2011, 31(1): 139-164
doi: 10.3934/dcds.2011.31.139
+[Abstract](2366)
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Abstract:
We are concerned with a multiple boundary spike solution to the steady-state problem of a chemotaxis system: $P_t=\nabla \cdot \big( P\nabla ( \log \frac{P}{\Phi (W)})\big)$, $W_t=ε^2 \Delta W+F(P,W)$, in $\Omega \times (0,\infty)$, under the homogeneous Neumann boundary condition, where $\Omega\subset \mathbb{R}^N$ is a bounded domain with smooth boundary, $P(x,t)$ is a population density, $W(x,t)$ is a density of chemotaxis substance. We assume that $\Phi(W)=W^p$, $p>1$, and we are interested in the cases of $F(P,W)=-W+\frac{PW^q}{\alpha+\gamma W^q}$ and $F(P,W)=-W+\frac{P}{1+ k P}$ with $q>0, \alpha, \gamma, k\ge 0$, which has a saturating growth. Existence of a multiple spike stationary pattern is related to a weak saturation effect of $F(P,W)$ and the shape of the domain $\Omega$. In this paper, we assume that $\Omega$ is symmetric with respect to each hyperplane $\{ x_1=0\},\cdots ,\{ x_{N-1}=0\}$. For two classes of $F(P,W)$ above with saturation effect, we show the existence of multiple boundary spike stationary patterns on $\Omega$ under a weak saturation effect on parameters $\alpha,\gamma$ and $k$. Based on the method developed in [14] and [10], we shall present some technique to construct a multiple boundary spike solution to some reduced nonlocal problem on such domains systematically.
We are concerned with a multiple boundary spike solution to the steady-state problem of a chemotaxis system: $P_t=\nabla \cdot \big( P\nabla ( \log \frac{P}{\Phi (W)})\big)$, $W_t=ε^2 \Delta W+F(P,W)$, in $\Omega \times (0,\infty)$, under the homogeneous Neumann boundary condition, where $\Omega\subset \mathbb{R}^N$ is a bounded domain with smooth boundary, $P(x,t)$ is a population density, $W(x,t)$ is a density of chemotaxis substance. We assume that $\Phi(W)=W^p$, $p>1$, and we are interested in the cases of $F(P,W)=-W+\frac{PW^q}{\alpha+\gamma W^q}$ and $F(P,W)=-W+\frac{P}{1+ k P}$ with $q>0, \alpha, \gamma, k\ge 0$, which has a saturating growth. Existence of a multiple spike stationary pattern is related to a weak saturation effect of $F(P,W)$ and the shape of the domain $\Omega$. In this paper, we assume that $\Omega$ is symmetric with respect to each hyperplane $\{ x_1=0\},\cdots ,\{ x_{N-1}=0\}$. For two classes of $F(P,W)$ above with saturation effect, we show the existence of multiple boundary spike stationary patterns on $\Omega$ under a weak saturation effect on parameters $\alpha,\gamma$ and $k$. Based on the method developed in [14] and [10], we shall present some technique to construct a multiple boundary spike solution to some reduced nonlocal problem on such domains systematically.
2011, 31(1): 165-207
doi: 10.3934/dcds.2011.31.165
+[Abstract](2244)
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A sequence of "inner equations" attached to certain perturbations of the McMillan map was considered in [5], their solutions were used in that article to measure an exponentially small separatrix splitting. We prove here all the results relative to these equations which are necessary to complete the proof of the main result of [5]. The present work relies on ideas from resurgence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Écalle's alien derivations to measure the discrepancy between different Borel-Laplace sums.
A sequence of "inner equations" attached to certain perturbations of the McMillan map was considered in [5], their solutions were used in that article to measure an exponentially small separatrix splitting. We prove here all the results relative to these equations which are necessary to complete the proof of the main result of [5]. The present work relies on ideas from resurgence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Écalle's alien derivations to measure the discrepancy between different Borel-Laplace sums.
2011, 31(1): 209-220
doi: 10.3934/dcds.2011.31.209
+[Abstract](2562)
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Existence of global solutions to initial value problems for a discrete analogue of a $d$-dimensional semilinear heat equation is investigated. We prove that a parameter $\alpha$ in the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for $0<\alpha \le 2/d$, and, for $\alpha > 2/d$, existence of non-trivial global solutions for sufficiently small initial data.
Existence of global solutions to initial value problems for a discrete analogue of a $d$-dimensional semilinear heat equation is investigated. We prove that a parameter $\alpha$ in the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for $0<\alpha \le 2/d$, and, for $\alpha > 2/d$, existence of non-trivial global solutions for sufficiently small initial data.
2011, 31(1): 221-238
doi: 10.3934/dcds.2011.31.221
+[Abstract](2567)
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This article deals with the existence and orbital stability of a two--parameter family of periodic traveling-wave solutions for the Klein-Gordon-Schrödinger system with Yukawa interaction. The existence of such a family of periodic waves is deduced from the Implicit Function Theorem, and the orbital stability is obtained from arguments due to Benjamin, Bona, and Weinstein.
This article deals with the existence and orbital stability of a two--parameter family of periodic traveling-wave solutions for the Klein-Gordon-Schrödinger system with Yukawa interaction. The existence of such a family of periodic waves is deduced from the Implicit Function Theorem, and the orbital stability is obtained from arguments due to Benjamin, Bona, and Weinstein.
2011, 31(1): 239-252
doi: 10.3934/dcds.2011.31.239
+[Abstract](2798)
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In this paper, we show that the strong solution of the three-dimensional Navier-Stokes equations with damping $\alpha|u|^{\beta-1}u\ (\alpha>0, \frac{7}{2}\leq \beta\leq 5)$ has global attractors in $V$ and $H^2(\Omega)$ when initial data $u_0\in V$, where $\Omega\subset \mathbb{R}^3$ is bounded.
In this paper, we show that the strong solution of the three-dimensional Navier-Stokes equations with damping $\alpha|u|^{\beta-1}u\ (\alpha>0, \frac{7}{2}\leq \beta\leq 5)$ has global attractors in $V$ and $H^2(\Omega)$ when initial data $u_0\in V$, where $\Omega\subset \mathbb{R}^3$ is bounded.
2011, 31(1): 253-273
doi: 10.3934/dcds.2011.31.253
+[Abstract](2425)
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Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigorous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.
Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into overlapping finite elements, a special coupling condition between neighbouring elements preserves the self-adjoint dynamics and controls interelement interactions. When the interelement coupling parameter is small, an averaging method and an asymptotic expansion of the slow modes show that the macroscopic discrete model will be a family of coupled stochastic ordinary differential equations which describe the evolution of the grid values. This modelling shows the importance of subgrid scale interaction between noise and spatial diffusion and provides a new rigorous approach to constructing semi-discrete approximations to stochastic reaction-diffusion partial differential equations.
2011, 31(1): 275-299
doi: 10.3934/dcds.2011.31.275
+[Abstract](2368)
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Abstract:
In this paper, we propose a nondegenerate condition for a homoclinic orbit with respect to a parameter in delay differential equations. Based on this nondegeneracy we describe and investigate the regularity of the homoclinic orbit together with parameter. Then we show that a forward Euler method, when applied to a one-parameteric system of delay differential equations with a homoclinic orbit, also exhibits a closed loop of discrete homoclinic orbits. These discrete homoclinic orbits tend to the continuous one by the rate of $O(\varepsilon)$ as the step-size $\varepsilon$ goes to $0$. And the corresponding parameter varies periodically with respect to a phase parameter with period $\varepsilon$ while the orbit shifts its index after one revolution. We also show that at least two homoclinic tangencies occur on this loop. By numerical simulations, the theoretical results are illustrated, and the possibility of extending theoretical results to the implicit and higher order numerical schemes is discussed.
In this paper, we propose a nondegenerate condition for a homoclinic orbit with respect to a parameter in delay differential equations. Based on this nondegeneracy we describe and investigate the regularity of the homoclinic orbit together with parameter. Then we show that a forward Euler method, when applied to a one-parameteric system of delay differential equations with a homoclinic orbit, also exhibits a closed loop of discrete homoclinic orbits. These discrete homoclinic orbits tend to the continuous one by the rate of $O(\varepsilon)$ as the step-size $\varepsilon$ goes to $0$. And the corresponding parameter varies periodically with respect to a phase parameter with period $\varepsilon$ while the orbit shifts its index after one revolution. We also show that at least two homoclinic tangencies occur on this loop. By numerical simulations, the theoretical results are illustrated, and the possibility of extending theoretical results to the implicit and higher order numerical schemes is discussed.
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