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Discrete & Continuous Dynamical Systems
July 2015 , Volume 35 , Issue 7
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2015, 35(7): 2797-2815
doi: 10.3934/dcds.2015.35.2797
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Abstract:
We study the asymptotics of the scenery flow. We give corrected versions with proofs of a certain lemma by Hochman, and study some related phenomena.
We study the asymptotics of the scenery flow. We give corrected versions with proofs of a certain lemma by Hochman, and study some related phenomena.
2015, 35(7): 2817-2844
doi: 10.3934/dcds.2015.35.2817
+[Abstract](2696)
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Abstract:
For a general nonautonomous dynamics on a Banach space, we give a necessary and sufficient condition so that the existence of one-sided exponential dichotomies on the past and on the future gives rise to a two-sided exponential dichotomy. The condition is that the stable space of the future at the origin and the unstable space of the past at the origin generate the whole space. We consider the general cases of a noninvertible dynamics as well as of a nonuniform exponential dichotomy and a strong nonuniform exponential dichotomy (for the latter, besides the requirements for a nonuniform exponential dichotomy we need to have a minimal contraction and a maximal expansion). Both notions are ubiquitous in ergodic theory. Our approach consists in reducing the study of the dynamics to one with uniform exponential behavior with respect to a family of norms and then using the characterization of uniform hyperbolicity in terms of an admissibility property in order to show that the dynamics admits a two-sided exponential dichotomy. As an application, we give a complete characterization of the set of Lyapunov exponents of a Lyapunov regular dynamics, in an analogous manner to that in the Sacker--Sell theory.
For a general nonautonomous dynamics on a Banach space, we give a necessary and sufficient condition so that the existence of one-sided exponential dichotomies on the past and on the future gives rise to a two-sided exponential dichotomy. The condition is that the stable space of the future at the origin and the unstable space of the past at the origin generate the whole space. We consider the general cases of a noninvertible dynamics as well as of a nonuniform exponential dichotomy and a strong nonuniform exponential dichotomy (for the latter, besides the requirements for a nonuniform exponential dichotomy we need to have a minimal contraction and a maximal expansion). Both notions are ubiquitous in ergodic theory. Our approach consists in reducing the study of the dynamics to one with uniform exponential behavior with respect to a family of norms and then using the characterization of uniform hyperbolicity in terms of an admissibility property in order to show that the dynamics admits a two-sided exponential dichotomy. As an application, we give a complete characterization of the set of Lyapunov exponents of a Lyapunov regular dynamics, in an analogous manner to that in the Sacker--Sell theory.
2015, 35(7): 2845-2861
doi: 10.3934/dcds.2015.35.2845
+[Abstract](2695)
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Abstract:
In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.
In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.
2015, 35(7): 2863-2880
doi: 10.3934/dcds.2015.35.2863
+[Abstract](3691)
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Abstract:
We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < \alpha < 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-\alpha}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow map is not locally uniformly continuous.
We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < \alpha < 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-\alpha}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow map is not locally uniformly continuous.
2015, 35(7): 2881-2904
doi: 10.3934/dcds.2015.35.2881
+[Abstract](2781)
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Abstract:
We consider an abstract equation with memory of the form $$\partial_t x(t) + \int_{0}^\infty k(s) A x(t-s) ds + Bx(t)=0$$ where $A,B$ are operators acting on some Banach space, and the convolution kernel $k$ is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation $$\partial_{t t} u - h(0)\Delta u - \int_{0}^\infty h'(s) \Delta u(t-s) ds+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.
We consider an abstract equation with memory of the form $$\partial_t x(t) + \int_{0}^\infty k(s) A x(t-s) ds + Bx(t)=0$$ where $A,B$ are operators acting on some Banach space, and the convolution kernel $k$ is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation $$\partial_{t t} u - h(0)\Delta u - \int_{0}^\infty h'(s) \Delta u(t-s) ds+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.
2015, 35(7): 2905-2920
doi: 10.3934/dcds.2015.35.2905
+[Abstract](2406)
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We use Wasserstein metrics adapted to study the action of the flow of the BBM equation on probability measures. We prove the continuity of this flow and the stability of invariant measures for finite times.
We use Wasserstein metrics adapted to study the action of the flow of the BBM equation on probability measures. We prove the continuity of this flow and the stability of invariant measures for finite times.
2015, 35(7): 2921-2948
doi: 10.3934/dcds.2015.35.2921
+[Abstract](2792)
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Abstract:
We will focus on the existence and concentration of nodal solutions to the following critical nonlinear Schrödinger equations in $\Bbb R^2$ $$ -\epsilon^2\triangle u_{\epsilon}+V(x)u_{\epsilon}=K(x) |u_{\epsilon}|^{p-2}u_{\epsilon}e^{\alpha_{0}|u_{\epsilon}| ^{2}},\quad u_{\epsilon}\in H^1(\Bbb R^2), $$ where $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, and $\epsilon>0$ is a small constant. For the positive potential $V(x)$ which decays at infinity like $(1+|x|)^{-\alpha}$ with $0 < \alpha \le 2$, we will show that a nodal solution with one positive and one negative peaks exists, and concentrates around local minimum points of the related ground energy function $G(\xi)$ of the Schrödinger equation $ -\triangle u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{2}}$.
We will focus on the existence and concentration of nodal solutions to the following critical nonlinear Schrödinger equations in $\Bbb R^2$ $$ -\epsilon^2\triangle u_{\epsilon}+V(x)u_{\epsilon}=K(x) |u_{\epsilon}|^{p-2}u_{\epsilon}e^{\alpha_{0}|u_{\epsilon}| ^{2}},\quad u_{\epsilon}\in H^1(\Bbb R^2), $$ where $p>2$, $\alpha_{0}>0$, $V(x), K(x)>0$, and $\epsilon>0$ is a small constant. For the positive potential $V(x)$ which decays at infinity like $(1+|x|)^{-\alpha}$ with $0 < \alpha \le 2$, we will show that a nodal solution with one positive and one negative peaks exists, and concentrates around local minimum points of the related ground energy function $G(\xi)$ of the Schrödinger equation $ -\triangle u+V(\xi)u=K(\xi) |u|^{p-2}ue^{\alpha_{0}|u|^{2}}$.
2015, 35(7): 2949-2977
doi: 10.3934/dcds.2015.35.2949
+[Abstract](2841)
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Abstract:
Symmetries of the periodic Toda lattice are expresssed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Jacobi matrices. Using these symmetries, the phase space of the lattice with Dirichlet boundary conditions is embedded into the phase space of a higher-dimensional periodic lattice. As an application, we obtain a Birkhoff normal form and a KAM theorem for the lattice with Dirichlet boundary conditions.
Symmetries of the periodic Toda lattice are expresssed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Jacobi matrices. Using these symmetries, the phase space of the lattice with Dirichlet boundary conditions is embedded into the phase space of a higher-dimensional periodic lattice. As an application, we obtain a Birkhoff normal form and a KAM theorem for the lattice with Dirichlet boundary conditions.
2015, 35(7): 2979-2995
doi: 10.3934/dcds.2015.35.2979
+[Abstract](3319)
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We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set for a continuous map $f$ in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 0$. This characterization allows us to compute effectively the Lefschetz zeta functions for a wide class of continuous maps in the \(2\)-sphere, to obtain new results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree--\(d\) maps in the 2-sphere.
We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set for a continuous map $f$ in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 0$. This characterization allows us to compute effectively the Lefschetz zeta functions for a wide class of continuous maps in the \(2\)-sphere, to obtain new results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree--\(d\) maps in the 2-sphere.
2015, 35(7): 2997-3013
doi: 10.3934/dcds.2015.35.2997
+[Abstract](2559)
+[PDF](452.7KB)
Abstract:
Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algorithm is proposed to numerically compute the steplength threshold. For rational function type discretization methods we derive a valid steplength threshold for invariance preserving, which can be computed by using an analogous algorithm as in the first case. The relationship between the previous two types of discretization methods and the forward Euler method is studied. Finally, we show that, for the forward Euler method, the largest steplength threshold for invariance preserving can be computed by solving a finite number of linear optimization problems.
Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algorithm is proposed to numerically compute the steplength threshold. For rational function type discretization methods we derive a valid steplength threshold for invariance preserving, which can be computed by using an analogous algorithm as in the first case. The relationship between the previous two types of discretization methods and the forward Euler method is studied. Finally, we show that, for the forward Euler method, the largest steplength threshold for invariance preserving can be computed by solving a finite number of linear optimization problems.
2015, 35(7): 3015-3037
doi: 10.3934/dcds.2015.35.3015
+[Abstract](2903)
+[PDF](450.6KB)
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The governing equations in radiation hydrodynamics are derived from the conservation laws for macroscopic quantities, which have to be coupled with a radiative transfer equation to account for the radiative effects. In the present paper, we work with a mathematical model for the diffusion approximation of radiation hydrodynamics in the simplified framework of 1-D flows. We prove the existence, uniqueness and regularity of global solutions to an initial-boundary value problem with large data. The existence of global solution is proved by combining the local existence theorem with the global a priori estimates, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between photons and matter.
The governing equations in radiation hydrodynamics are derived from the conservation laws for macroscopic quantities, which have to be coupled with a radiative transfer equation to account for the radiative effects. In the present paper, we work with a mathematical model for the diffusion approximation of radiation hydrodynamics in the simplified framework of 1-D flows. We prove the existence, uniqueness and regularity of global solutions to an initial-boundary value problem with large data. The existence of global solution is proved by combining the local existence theorem with the global a priori estimates, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between photons and matter.
2015, 35(7): 3039-3057
doi: 10.3934/dcds.2015.35.3039
+[Abstract](3414)
+[PDF](478.9KB)
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We show that, for a given Hölder continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset \mathbb{R}^3 \times \mathbb{R}^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $\mathbb{R}^3$ which is regular outside this curve and singular on it. This is a solution of the homogeneous system outside the curve, however, as a distributional solution on $\mathbb{R}^3 \times \mathbb{R}^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.
We show that, for a given Hölder continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset \mathbb{R}^3 \times \mathbb{R}^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $\mathbb{R}^3$ which is regular outside this curve and singular on it. This is a solution of the homogeneous system outside the curve, however, as a distributional solution on $\mathbb{R}^3 \times \mathbb{R}^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.
2015, 35(7): 3059-3086
doi: 10.3934/dcds.2015.35.3059
+[Abstract](2556)
+[PDF](501.3KB)
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In this paper, we discuss the Cauchy problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. We establish the existence of a unique local regular solution with vacuum by the theory of quasi-linear symmetric hyperbolic systems and some techniques dealing with the complexity caused by the coupling between fluid and radiation field under some physical assumptions for the radiation quantities. Moreover, it is interesting to show the non-global existence of regular solutions caused by the effect of vacuum for polytropic gases with adiabatic exponent $1<\gamma\leq 3$ via some observations on the propagation of the radiation field. Compared with [11][15][20], some new initial conditions that will lead to the finite time blow-up for classical solutions have been introduced. These blow-up results tell us that the radiation effect on the fluid is not strong enough to prevent the formation of singularities caused by the appearance of vacuum.
In this paper, we discuss the Cauchy problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. We establish the existence of a unique local regular solution with vacuum by the theory of quasi-linear symmetric hyperbolic systems and some techniques dealing with the complexity caused by the coupling between fluid and radiation field under some physical assumptions for the radiation quantities. Moreover, it is interesting to show the non-global existence of regular solutions caused by the effect of vacuum for polytropic gases with adiabatic exponent $1<\gamma\leq 3$ via some observations on the propagation of the radiation field. Compared with [11][15][20], some new initial conditions that will lead to the finite time blow-up for classical solutions have been introduced. These blow-up results tell us that the radiation effect on the fluid is not strong enough to prevent the formation of singularities caused by the appearance of vacuum.
2015, 35(7): 3087-3102
doi: 10.3934/dcds.2015.35.3087
+[Abstract](2534)
+[PDF](433.7KB)
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The existence of multiple solutions $u\in H^1_0(\Omega)$ to a differential inclusion of the type $-\Delta u\in \partial J(x,u)$ in $\Omega$, where $\partial J(x,\cdot)$ denotes the generalized sub-differential of $J(x,\cdot)$, is investigated through critical point theorems for locally Lipschitz continuous functionals on closed convex sets of a Hilbert space.
The existence of multiple solutions $u\in H^1_0(\Omega)$ to a differential inclusion of the type $-\Delta u\in \partial J(x,u)$ in $\Omega$, where $\partial J(x,\cdot)$ denotes the generalized sub-differential of $J(x,\cdot)$, is investigated through critical point theorems for locally Lipschitz continuous functionals on closed convex sets of a Hilbert space.
2015, 35(7): 3103-3131
doi: 10.3934/dcds.2015.35.3103
+[Abstract](2908)
+[PDF](572.7KB)
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In this paper we consider the initial value problem for a family of shallow water equations on the line $\mathbb{R}$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as $x\to\pm\infty$ and prove their invariance with respect to the solution map.
In this paper we consider the initial value problem for a family of shallow water equations on the line $\mathbb{R}$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as $x\to\pm\infty$ and prove their invariance with respect to the solution map.
2015, 35(7): 3133-3154
doi: 10.3934/dcds.2015.35.3133
+[Abstract](2628)
+[PDF](453.5KB)
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We present a variational analysis for the semilinear equation of the vibrating plate $x_{t t}(t,y)+\Delta ^{2}x(t,y)+l(t,y,x(t,y))=0$ in a bounded domain and a free nonlinear boundary condition $\Delta x(t,y)=H_{x}(t,y,x(t,y))-Q_{x}(t,y,x(t,y))$. In this context new dual variational methods are developed. Applying a variational approach we discuss a stability of solutions with respect to initial conditions.
We present a variational analysis for the semilinear equation of the vibrating plate $x_{t t}(t,y)+\Delta ^{2}x(t,y)+l(t,y,x(t,y))=0$ in a bounded domain and a free nonlinear boundary condition $\Delta x(t,y)=H_{x}(t,y,x(t,y))-Q_{x}(t,y,x(t,y))$. In this context new dual variational methods are developed. Applying a variational approach we discuss a stability of solutions with respect to initial conditions.
2015, 35(7): 3155-3182
doi: 10.3934/dcds.2015.35.3155
+[Abstract](2970)
+[PDF](618.3KB)
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Dimension three is the lowest dimension where we can find chaotic behaviour for flows and it may be helpful to distinguish in advance ``equivalent'' complex dynamics. In this article, we give numerical invariants for the topological equivalence of vector fields on three-dimensional manifolds whose flows exhibit one-dimensional heteroclinic connections involving either saddle-foci or periodic solutions. Computed as an infinite limit time, these moduli of topological equivalence heavily rely on the behaviour near the invariant saddles. We also present an alternative proof of the Togawa's Theorem.
Dimension three is the lowest dimension where we can find chaotic behaviour for flows and it may be helpful to distinguish in advance ``equivalent'' complex dynamics. In this article, we give numerical invariants for the topological equivalence of vector fields on three-dimensional manifolds whose flows exhibit one-dimensional heteroclinic connections involving either saddle-foci or periodic solutions. Computed as an infinite limit time, these moduli of topological equivalence heavily rely on the behaviour near the invariant saddles. We also present an alternative proof of the Togawa's Theorem.
2015, 35(7): 3183-3201
doi: 10.3934/dcds.2015.35.3183
+[Abstract](3052)
+[PDF](442.9KB)
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In this paper we study the existence of positive multi-peak solutions to the semilinear equation \begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u + u= Q(x)u^{p-1}, \hskip0.5cm u >0, \hskip 0.2cm u\in H^{s}(\mathbb{R}^{N}) \end{eqnarray*} where $(-\Delta)^{s} $ stands for the fractional Laplacian, $s\in (0,1)$, $\varepsilon$ is a positive small parameter, $2 < p < \frac{2N}{N-2s}$, $Q(x)$ is a bounded positive continuous function. For any positive integer $k$, we prove the existence of a positive solution with $k$-peaks and concentrating near a given local minimum point of $Q$. For $s=1$ this corresponds to the result of [22].
In this paper we study the existence of positive multi-peak solutions to the semilinear equation \begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u + u= Q(x)u^{p-1}, \hskip0.5cm u >0, \hskip 0.2cm u\in H^{s}(\mathbb{R}^{N}) \end{eqnarray*} where $(-\Delta)^{s} $ stands for the fractional Laplacian, $s\in (0,1)$, $\varepsilon$ is a positive small parameter, $2 < p < \frac{2N}{N-2s}$, $Q(x)$ is a bounded positive continuous function. For any positive integer $k$, we prove the existence of a positive solution with $k$-peaks and concentrating near a given local minimum point of $Q$. For $s=1$ this corresponds to the result of [22].
2015, 35(7): 3203-3216
doi: 10.3934/dcds.2015.35.3203
+[Abstract](3041)
+[PDF](687.3KB)
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We study similarity solutions of a nonlinear partial differential equation that is a generalization of the heat equation. Substitution of the similarity ansatz reduces the partial differential equation to a nonlinear second-order ordinary differential equation on the half-line with Neumann boundary conditions at both boundaries. The existence and uniqueness of solutions is proven using Ważewski's Principle.
We study similarity solutions of a nonlinear partial differential equation that is a generalization of the heat equation. Substitution of the similarity ansatz reduces the partial differential equation to a nonlinear second-order ordinary differential equation on the half-line with Neumann boundary conditions at both boundaries. The existence and uniqueness of solutions is proven using Ważewski's Principle.
2015, 35(7): 3217-3238
doi: 10.3934/dcds.2015.35.3217
+[Abstract](2811)
+[PDF](482.4KB)
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In this paper, we study the nonlocal dispersal Logistic equation \begin{equation*} \begin{cases} u_t=Du+\lambda m(x)u-c(x)u^p &\text{ in }{\Omega}\times(0,+\infty),\\ u(x,0)=u_0(x)\geq0&\text{ in }{\Omega}, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $\lambda>0$ and $p>1$ are constants. $Du(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy$ represents the nonlocal dispersal operator with continuous and nonnegative dispersal kernel $J$, $m\in C(\bar{\Omega})$ and may change sign in $\Omega$. The function $c$ is nonnegative and has a degeneracy in some subdomain of $\Omega$. We establish the existence and uniqueness of positive stationary solution and also consider the effect of degeneracy of $c$ on the long-time behavior of positive solutions. Our results reveal that the necessary condition to guarantee a positive stationary solution and the asymptotic behaviour of solutions are quite different from those of the corresponding reaction-diffusion equation.
In this paper, we study the nonlocal dispersal Logistic equation \begin{equation*} \begin{cases} u_t=Du+\lambda m(x)u-c(x)u^p &\text{ in }{\Omega}\times(0,+\infty),\\ u(x,0)=u_0(x)\geq0&\text{ in }{\Omega}, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $\lambda>0$ and $p>1$ are constants. $Du(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy$ represents the nonlocal dispersal operator with continuous and nonnegative dispersal kernel $J$, $m\in C(\bar{\Omega})$ and may change sign in $\Omega$. The function $c$ is nonnegative and has a degeneracy in some subdomain of $\Omega$. We establish the existence and uniqueness of positive stationary solution and also consider the effect of degeneracy of $c$ on the long-time behavior of positive solutions. Our results reveal that the necessary condition to guarantee a positive stationary solution and the asymptotic behaviour of solutions are quite different from those of the corresponding reaction-diffusion equation.
2015, 35(7): 3239-3252
doi: 10.3934/dcds.2015.35.3239
+[Abstract](2749)
+[PDF](388.3KB)
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In this paper, we study a class of semilinear elliptic equations with the Hardy potential. By means of the super-subsolution method and the comparison principle, we explore the existence of a minimal positive solution and a maximal positive solution. Through a scaling technique, we obtain the asymptotic property of positive solutions near the origin. Finally, the nonexistence of a positive solution is proven when the parameter is larger than a critical value.
In this paper, we study a class of semilinear elliptic equations with the Hardy potential. By means of the super-subsolution method and the comparison principle, we explore the existence of a minimal positive solution and a maximal positive solution. Through a scaling technique, we obtain the asymptotic property of positive solutions near the origin. Finally, the nonexistence of a positive solution is proven when the parameter is larger than a critical value.
2015, 35(7): 3253-3276
doi: 10.3934/dcds.2015.35.3253
+[Abstract](3237)
+[PDF](547.8KB)
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In this paper, we study a competitive model involving two species separated by a free boundary by virtue of strong competition. When the initial data has positive lower bounds near $\pm\infty$, we prove that the solution converges, as $t\rightarrow \infty$, to a traveling wave solution and the free boundary moves to infinity with a constant speed.
In this paper, we study a competitive model involving two species separated by a free boundary by virtue of strong competition. When the initial data has positive lower bounds near $\pm\infty$, we prove that the solution converges, as $t\rightarrow \infty$, to a traveling wave solution and the free boundary moves to infinity with a constant speed.
2020
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