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Discrete & Continuous Dynamical Systems - A
September 2010 , Volume 26 , Issue 3
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2010, 26(3): 781-794
doi: 10.3934/dcds.2010.26.781
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Abstract:
Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. In a companion paper [8] it is proved that every solution for the system with forcing blows up in finite time in the Sobolev $H^{5/6}$ norm. In this present paper, it is proved that after the blow-up time all solutions stay in $H^s$, $s < 5/6$ for almost all time. It is proved that the model system exhibits the phenomenon of anomalous (or turbulent) dissipation which was conjectured for the Euler equations by Onsager. As a consequence of this anomalous dissipation the unique equilibrium of the system is a global attractor.
Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. In a companion paper [8] it is proved that every solution for the system with forcing blows up in finite time in the Sobolev $H^{5/6}$ norm. In this present paper, it is proved that after the blow-up time all solutions stay in $H^s$, $s < 5/6$ for almost all time. It is proved that the model system exhibits the phenomenon of anomalous (or turbulent) dissipation which was conjectured for the Euler equations by Onsager. As a consequence of this anomalous dissipation the unique equilibrium of the system is a global attractor.
2010, 26(3): 795-804
doi: 10.3934/dcds.2010.26.795
+[Abstract](2171)
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Given a compact manifold $X,$ a continuous function $g:X\to \R{},$ and a map $T:X\to X,$ we study properties of the $T$-invariant Borel probability measures that maximize the integral of $g$.
We show that if $X$ is a $n$-dimensional connected Riemaniann manifold, with $n \geq 2$, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager.
We also show that, if $X$ is the circle, then the "topological size'' of the set of endomorphisms for which there are $g$ maximizing measures with support on a periodic orbit depends on properties of the function $g.$ In particular, if $g$ is $\mathcal{C}^1$, it has interior points.
Given a compact manifold $X,$ a continuous function $g:X\to \R{},$ and a map $T:X\to X,$ we study properties of the $T$-invariant Borel probability measures that maximize the integral of $g$.
We show that if $X$ is a $n$-dimensional connected Riemaniann manifold, with $n \geq 2$, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager.
We also show that, if $X$ is the circle, then the "topological size'' of the set of endomorphisms for which there are $g$ maximizing measures with support on a periodic orbit depends on properties of the function $g.$ In particular, if $g$ is $\mathcal{C}^1$, it has interior points.
2010, 26(3): 805-822
doi: 10.3934/dcds.2010.26.805
+[Abstract](1843)
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The problem of attraction of an infinitesimal particle by the gravitational force induced by a massive body of finite dimension on a plane is considered. We study the singularity problem of the solutions associated to this problem in the case when the massive body has the form of a straight segment, annulus disk or disk with constant linear mass density.
The problem of attraction of an infinitesimal particle by the gravitational force induced by a massive body of finite dimension on a plane is considered. We study the singularity problem of the solutions associated to this problem in the case when the massive body has the form of a straight segment, annulus disk or disk with constant linear mass density.
2010, 26(3): 823-837
doi: 10.3934/dcds.2010.26.823
+[Abstract](1887)
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Abstract:
This paper deals with the stability of steady states of the semilinear heat equation $u_t=$Δ$u+K(x)u^p+f(x)$ under proper assumptions on $K(x)$ and $f(x)$. We prove the weak asymptotic stability of positive steady states with respect to weighted uniform norms.
This paper deals with the stability of steady states of the semilinear heat equation $u_t=$Δ$u+K(x)u^p+f(x)$ under proper assumptions on $K(x)$ and $f(x)$. We prove the weak asymptotic stability of positive steady states with respect to weighted uniform norms.
2010, 26(3): 839-846
doi: 10.3934/dcds.2010.26.839
+[Abstract](1830)
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A divergence-free vector field satisfies the star property if any divergence-free vector field in some $C^1$-neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence-free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a $C^1$-structurally stable three-dimensional conservative flow is Anosov.
A divergence-free vector field satisfies the star property if any divergence-free vector field in some $C^1$-neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence-free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a $C^1$-structurally stable three-dimensional conservative flow is Anosov.
2010, 26(3): 847-856
doi: 10.3934/dcds.2010.26.847
+[Abstract](2006)
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Abstract:
Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff($\mathbb{C},0$). We also show that two nonlinearizable germs with the same rotation number are conjugate if and only if a hedgehog of one can be mapped conformally onto a hedgehog of the other. Thus the conjugacy class of a nonlinearizable germ is completely determined by its rotation number and the conformal class of its hedgehogs.
Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff($\mathbb{C},0$). We also show that two nonlinearizable germs with the same rotation number are conjugate if and only if a hedgehog of one can be mapped conformally onto a hedgehog of the other. Thus the conjugacy class of a nonlinearizable germ is completely determined by its rotation number and the conformal class of its hedgehogs.
2010, 26(3): 857-871
doi: 10.3934/dcds.2010.26.857
+[Abstract](2505)
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We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude $\epsilon$. The initial datum gives rise to a soliton when $\epsilon=0$. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of $\epsilon^{-2}$. We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than $T$, of the same order in $\epsilon$ and $T$. We obtain that the time scale is exactly the right one. We also study the similar probability for the exit from a neighborhood of the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton.
We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude $\epsilon$. The initial datum gives rise to a soliton when $\epsilon=0$. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of $\epsilon^{-2}$. We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than $T$, of the same order in $\epsilon$ and $T$. We obtain that the time scale is exactly the right one. We also study the similar probability for the exit from a neighborhood of the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton.
2010, 26(3): 873-899
doi: 10.3934/dcds.2010.26.873
+[Abstract](2212)
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Abstract:
For any integer $r\geq2$ and any real $\epsilon>0$, we construct an explicit example of $\mathcal{C}^r$ interval map $f$ with symbolic extension entropy $h_{sex}(f)\geq\frac{r}{r-1}\log\||f'\||_{\infty}-\epsilon$ and $\||f'\||_{\infty}\geq 2$. T.Downarawicz and A.Maass [10] proved that for $\mathcal{C}^r$ interval maps with $r>1$, the symbolic extension entropy was bounded above by $\frac{r}{r-1}\log\||f'\||_{\infty}$. So our example proves this bound is sharp. Similar examples had been already built by T.Downarowicz and S.Newhouse for diffeomorphisms in higher dimension by using generic arguments on homoclinic tangencies.
For any integer $r\geq2$ and any real $\epsilon>0$, we construct an explicit example of $\mathcal{C}^r$ interval map $f$ with symbolic extension entropy $h_{sex}(f)\geq\frac{r}{r-1}\log\||f'\||_{\infty}-\epsilon$ and $\||f'\||_{\infty}\geq 2$. T.Downarawicz and A.Maass [10] proved that for $\mathcal{C}^r$ interval maps with $r>1$, the symbolic extension entropy was bounded above by $\frac{r}{r-1}\log\||f'\||_{\infty}$. So our example proves this bound is sharp. Similar examples had been already built by T.Downarowicz and S.Newhouse for diffeomorphisms in higher dimension by using generic arguments on homoclinic tangencies.
2010, 26(3): 901-921
doi: 10.3934/dcds.2010.26.901
+[Abstract](2224)
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In this paper we study the dynamics of a single transition layer of a solution to a spatially inhomogeneous bistable reaction diffusion equation in one space dimension. The spatial inhomogeneity is given by a function $a(x)$. In particular, we consider the case where $a(x)$ is identically zero on an interval $I$ and study the dynamics of the transition layer on $I$. In this case the dynamics of the transition layer on $I$ becomes so-called very slow dynamics. In order to analyze such a dynamics, we construct an attractive local invariant manifold giving the dynamics of the transition layer and we derive an equation describing the flow on the manifold. We also give applications of our results to two well known nonlinearities of bistable type.
In this paper we study the dynamics of a single transition layer of a solution to a spatially inhomogeneous bistable reaction diffusion equation in one space dimension. The spatial inhomogeneity is given by a function $a(x)$. In particular, we consider the case where $a(x)$ is identically zero on an interval $I$ and study the dynamics of the transition layer on $I$. In this case the dynamics of the transition layer on $I$ becomes so-called very slow dynamics. In order to analyze such a dynamics, we construct an attractive local invariant manifold giving the dynamics of the transition layer and we derive an equation describing the flow on the manifold. We also give applications of our results to two well known nonlinearities of bistable type.
2010, 26(3): 923-947
doi: 10.3934/dcds.2010.26.923
+[Abstract](2376)
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Abstract:
Let $\g$ be a real semisimple Lie algebra and $G = \Int(\g)$. In this article, we relate the Jordan decomposition of $X \in \g$ (or $g \in G$) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by $X$ (or the discrete-time flow generated by $g$). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of $X$ is trivial, we compute the Conley indexes of all Morse components. Finally, we consider the dynamical aspects of linear differential equations with periodic coefficients in $\g$, which can be regarded as an extension of the dynamics generated by an element $X \in \g$. In this context, we generalize Floquet theory and extend our previous results to this case.
Let $\g$ be a real semisimple Lie algebra and $G = \Int(\g)$. In this article, we relate the Jordan decomposition of $X \in \g$ (or $g \in G$) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by $X$ (or the discrete-time flow generated by $g$). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of $X$ is trivial, we compute the Conley indexes of all Morse components. Finally, we consider the dynamical aspects of linear differential equations with periodic coefficients in $\g$, which can be regarded as an extension of the dynamics generated by an element $X \in \g$. In this context, we generalize Floquet theory and extend our previous results to this case.
2010, 26(3): 949-966
doi: 10.3934/dcds.2010.26.949
+[Abstract](2730)
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Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.
Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.
2010, 26(3): 967-987
doi: 10.3934/dcds.2010.26.967
+[Abstract](2450)
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Let $\Gamma$ be an amenable group and $V$ be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of l$^2(\Gamma;V)$ (with respect to $\Gamma$) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $\Gamma$-invariant linear subspaces $Y$ of l$^p(\Gamma;V)$ a real positive number dimlp Y (which is the von Neumann dimension when $p=2$). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective $\Gamma$-equivariant linear map of finite-type from l$^p(\Gamma;V) \to $l$^p(\Gamma; V')$ if $\dim V > \dim V'$. A generalization of the Ornstein-Weiss lemma is developed along the way.
Let $\Gamma$ be an amenable group and $V$ be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of l$^2(\Gamma;V)$ (with respect to $\Gamma$) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $\Gamma$-invariant linear subspaces $Y$ of l$^p(\Gamma;V)$ a real positive number dimlp Y (which is the von Neumann dimension when $p=2$). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective $\Gamma$-equivariant linear map of finite-type from l$^p(\Gamma;V) \to $l$^p(\Gamma; V')$ if $\dim V > \dim V'$. A generalization of the Ornstein-Weiss lemma is developed along the way.
2010, 26(3): 989-1006
doi: 10.3934/dcds.2010.26.989
+[Abstract](2264)
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We obtain a result of existence of solutions to the 2D-Navier-Stokes model with delays, when the forcing term containing the delay is sub-linear and only continuous. As a consequence of the continuity assumption the uniqueness of solutions does not hold in general. We use then the theory of multi-valued dynamical system to establish the existence of attractors for our problem in several senses and establish relations among them.
We obtain a result of existence of solutions to the 2D-Navier-Stokes model with delays, when the forcing term containing the delay is sub-linear and only continuous. As a consequence of the continuity assumption the uniqueness of solutions does not hold in general. We use then the theory of multi-valued dynamical system to establish the existence of attractors for our problem in several senses and establish relations among them.
2010, 26(3): 1007-1018
doi: 10.3934/dcds.2010.26.1007
+[Abstract](2382)
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Certain dynamical systems on the interval with indifferent fixed points admit invariant probability measures which are absolutely continuous with respect to Lebesgue measure. These maps are often used as a model of intermittent dynamics, and they exhibit sub-exponential decay of correlations (due to the absence of a spectral gap in the underlying transfer operator). This paper concerns a class of these maps which are expanding (with convex branches), but admit an indifferent fixed point with tangency of $O(x^{1+\alpha})$ at $x=0$ ($0<\alpha<1$). The main results show that invariant probability measures can be rigorously approximated by a finite calculation. More precisely: Ulam's method (a sequence of computable finite rank approximations to the transfer operator) exhibits $L^1$ - convergence; and the $n$th approximate invariant density is accurate to at least $O(n^{-(1-\alpha)^2})$. Explicitly given non-uniform Ulam methods can improve this rate to $O(n^{-(1-\alpha)})$.
Certain dynamical systems on the interval with indifferent fixed points admit invariant probability measures which are absolutely continuous with respect to Lebesgue measure. These maps are often used as a model of intermittent dynamics, and they exhibit sub-exponential decay of correlations (due to the absence of a spectral gap in the underlying transfer operator). This paper concerns a class of these maps which are expanding (with convex branches), but admit an indifferent fixed point with tangency of $O(x^{1+\alpha})$ at $x=0$ ($0<\alpha<1$). The main results show that invariant probability measures can be rigorously approximated by a finite calculation. More precisely: Ulam's method (a sequence of computable finite rank approximations to the transfer operator) exhibits $L^1$ - convergence; and the $n$th approximate invariant density is accurate to at least $O(n^{-(1-\alpha)^2})$. Explicitly given non-uniform Ulam methods can improve this rate to $O(n^{-(1-\alpha)})$.
2010, 26(3): 1019-1034
doi: 10.3934/dcds.2010.26.1019
+[Abstract](1723)
+[PDF](212.9KB)
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Electron magnetohydrodynamics (EMHD) models the flow of electrons in fast time scales in an inhomogeneous plasma, by considering the ions as stationary. A number of numerical studies has shown an interesting phenomenology, but so far no proof exists of existence of solutions for the main equation. This equation is similar to the vorticity equation of inviscid flows, but the techniques used to prove local existence in that case do not work well in a bounded domain. However, the formulation of the Cauchy solution for certain transport equations plus a number of estimates on the Hölder norm of the flux of a vector field are enough to provide a proof of existence of solutions in a certain time interval.
Electron magnetohydrodynamics (EMHD) models the flow of electrons in fast time scales in an inhomogeneous plasma, by considering the ions as stationary. A number of numerical studies has shown an interesting phenomenology, but so far no proof exists of existence of solutions for the main equation. This equation is similar to the vorticity equation of inviscid flows, but the techniques used to prove local existence in that case do not work well in a bounded domain. However, the formulation of the Cauchy solution for certain transport equations plus a number of estimates on the Hölder norm of the flux of a vector field are enough to provide a proof of existence of solutions in a certain time interval.
2010, 26(3): 1035-1054
doi: 10.3934/dcds.2010.26.1035
+[Abstract](1814)
+[PDF](313.6KB)
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We analyze parametrized families of multimodal $1D$ maps that arise as singular limits of parametrized families of rank one maps. For a generic $1$-parameter family of such maps that contains a Misiurewicz-like map, it has been shown that in a neighborhood of the Misiurewicz-like parameter, a subset of parameters of positive Lebesgue measure exhibits nonuniformly expanding dynamics characterized by the existence of a positive Lyapunov exponent and an absolutely continuous invariant measure. Under a mild combinatoric assumption, we prove that each such parameter is an accumulation point of the set of parameters admitting superstable periodic sinks.
We analyze parametrized families of multimodal $1D$ maps that arise as singular limits of parametrized families of rank one maps. For a generic $1$-parameter family of such maps that contains a Misiurewicz-like map, it has been shown that in a neighborhood of the Misiurewicz-like parameter, a subset of parameters of positive Lebesgue measure exhibits nonuniformly expanding dynamics characterized by the existence of a positive Lyapunov exponent and an absolutely continuous invariant measure. Under a mild combinatoric assumption, we prove that each such parameter is an accumulation point of the set of parameters admitting superstable periodic sinks.
2010, 26(3): 1055-1072
doi: 10.3934/dcds.2010.26.1055
+[Abstract](2104)
+[PDF](216.3KB)
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By variational methods, we construct infinitely many concentration solutions for a type of Paneitz problem under the condition that the Paneitz curvature has a sequence of strictly local maximum points moving to infinity.
By variational methods, we construct infinitely many concentration solutions for a type of Paneitz problem under the condition that the Paneitz curvature has a sequence of strictly local maximum points moving to infinity.
2010, 26(3): 1073-1100
doi: 10.3934/dcds.2010.26.1073
+[Abstract](1948)
+[PDF](355.2KB)
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In this work we prove that the global attractors for the flow of the equation
In this work we prove that the global attractors for the flow of the equation
$\frac{\partial m(r,t)}{\partial t}=-m(r,t)+ g(\beta J $∗$ m(r,t)+ \beta h),\ h ,\ \beta \geq 0,$
are continuous with respect to the parameters $h$ and $\beta$ if one assumes a property implying normal hyperbolicity for its (families of) equilibria.
2010, 26(3): 1101-1117
doi: 10.3934/dcds.2010.26.1101
+[Abstract](2114)
+[PDF](242.2KB)
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The global existence and pointwise estimates of the Cauchy problem for the Euler-Poisson equation with damping in multi-dimensions are considered. Based on the analysis of Green function, and using the special structure of the system together with weighted energy method, we obtain the global existence of the classical solution. What's more important, is that we derive a detailed, pointwise description of asymptotic behavior of the solutions of the Cauchy problem. Then we obtain the optimal $L^p(R^n)\ (p>\frac{n}{n-1})$ convergence rate of the solutions.
The global existence and pointwise estimates of the Cauchy problem for the Euler-Poisson equation with damping in multi-dimensions are considered. Based on the analysis of Green function, and using the special structure of the system together with weighted energy method, we obtain the global existence of the classical solution. What's more important, is that we derive a detailed, pointwise description of asymptotic behavior of the solutions of the Cauchy problem. Then we obtain the optimal $L^p(R^n)\ (p>\frac{n}{n-1})$ convergence rate of the solutions.
2010, 26(3): 1119-1120
doi: 10.3934/dcds.2010.26.1119
+[Abstract](1886)
+[PDF](54.9KB)
Abstract:
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