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1078-0947
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Discrete & Continuous Dynamical Systems - A
January 2007 , Volume 18 , Issue 1
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2007, 18(1): 1-14
doi: 10.3934/dcds.2007.18.1
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Abstract:
We make two observations concerning the generalised Korteweg de Vries equation $u_t + $uxxx$ = \mu ( |u|^{p-1} u )_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for $L^2$-critical equation ($p=5$) automatically implies an analogous scattering result for the $L^2$-critical nonlinear Schrödinger equation $iu_t + $uxx$ = \mu |u|^4 u$. Secondly, in the defocusing case $\mu > 0$ we present a new dispersion estimate which asserts, roughly speaking, that energy moves to the left faster than the mass, and hence strongly localised soliton-like behaviour at a fixed scale cannot persist for arbitrarily long times.
We make two observations concerning the generalised Korteweg de Vries equation $u_t + $uxxx$ = \mu ( |u|^{p-1} u )_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for $L^2$-critical equation ($p=5$) automatically implies an analogous scattering result for the $L^2$-critical nonlinear Schrödinger equation $iu_t + $uxx$ = \mu |u|^4 u$. Secondly, in the defocusing case $\mu > 0$ we present a new dispersion estimate which asserts, roughly speaking, that energy moves to the left faster than the mass, and hence strongly localised soliton-like behaviour at a fixed scale cannot persist for arbitrarily long times.
2007, 18(1): 15-38
doi: 10.3934/dcds.2007.18.15
+[Abstract](1453)
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Abstract:
This paper addresses a doubly nonlinear parabolic inclusion of the form
This paper addresses a doubly nonlinear parabolic inclusion of the form
$\mathcal A (u_t)+\mathcal B (u)$ ∋ f.
Existence of a solution is proved under suitable monotonicity, coercivity, and structure assumptions on the operators $\mathcal A $ and $\mathcal B$, which in particular are both supposed to be subdifferentials of functionals on $L^2(\Omega)$. Since unbounded operators $\mathcal A $ are included in the analysis, this theory partly extends Colli & Visintin's work [24]. Moreover, under additional hypotheses on $\mathcal B$, uniqueness of the solution is proved. Finally, a characterization of $\omega$-limit sets of solutions is given, and we investigate the convergence of trajectories to limit points.
2007, 18(1): 39-52
doi: 10.3934/dcds.2007.18.39
+[Abstract](1567)
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Abstract:
We construct new families of examples of (real) Anosov Lie algebras, starting with algebraic units. We also give examples of indecomposable Anosov Lie algebras (not a direct sum of proper Lie ideals) of dimension $13$ and $16$, and we conclude that for every $n \geq 6$ with $n \ne 7$ there exists an indecomposable Anosov Lie algebra of dimension $n$.
We construct new families of examples of (real) Anosov Lie algebras, starting with algebraic units. We also give examples of indecomposable Anosov Lie algebras (not a direct sum of proper Lie ideals) of dimension $13$ and $16$, and we conclude that for every $n \geq 6$ with $n \ne 7$ there exists an indecomposable Anosov Lie algebra of dimension $n$.
2007, 18(1): 53-70
doi: 10.3934/dcds.2007.18.53
+[Abstract](1168)
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Abstract:
For weakly damped non-autonomous hyperbolic equations, we introduce a new concept Condition (C*), denote the set of all functions satisfying Condition (C*) by L2 C* $(R;X)$ which are translation bounded but not translation compact in $L^2$ loc$(R;X)$, and show that there are many functions satisfying Condition (C*); then we study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class of time dependent external forces $g(x,t)\in $ L2 C* $(R;X)$ and prove the existence of the uniform attractors for the family of processes corresponding to the equation in $H^1_0\times L^2$ and $D(A)\times H^1_0$.
For weakly damped non-autonomous hyperbolic equations, we introduce a new concept Condition (C*), denote the set of all functions satisfying Condition (C*) by L2 C* $(R;X)$ which are translation bounded but not translation compact in $L^2$ loc$(R;X)$, and show that there are many functions satisfying Condition (C*); then we study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class of time dependent external forces $g(x,t)\in $ L2 C* $(R;X)$ and prove the existence of the uniform attractors for the family of processes corresponding to the equation in $H^1_0\times L^2$ and $D(A)\times H^1_0$.
2007, 18(1): 71-84
doi: 10.3934/dcds.2007.18.71
+[Abstract](1887)
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Abstract:
In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point case, the existence of a unique self-similar profile is proven. Moreover, by constructing a Lyapunov function, we prove the convergence of the solution to the unique self-similar solution as $t$ approaches the blow-up time.
In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point case, the existence of a unique self-similar profile is proven. Moreover, by constructing a Lyapunov function, we prove the convergence of the solution to the unique self-similar solution as $t$ approaches the blow-up time.
2007, 18(1): 85-106
doi: 10.3934/dcds.2007.18.85
+[Abstract](1523)
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Abstract:
In this paper we derive structure theorems which characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write an explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, especially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.
In this paper we derive structure theorems which characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write an explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, especially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.
2007, 18(1): 107-120
doi: 10.3934/dcds.2007.18.107
+[Abstract](1325)
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Abstract:
In this paper we study the equation $-u''+V(x)u=W(x)f(u),\ x\in\mathbb{R},$ where the nonlinear term $f$ has certain oscillatory behaviour. Via two different variational arguments, we show the existence of infinitely many homoclinic solutions whose norms in an appropriate functional space which involves the potential $V$ tend to zero (resp. at infinity) whenever $f$ oscillates at zero (resp. at infinity). Unlike in classical results, neither symmetry property on $f$ nor periodicity on the potentials $V$ and $W$ are required.
In this paper we study the equation $-u''+V(x)u=W(x)f(u),\ x\in\mathbb{R},$ where the nonlinear term $f$ has certain oscillatory behaviour. Via two different variational arguments, we show the existence of infinitely many homoclinic solutions whose norms in an appropriate functional space which involves the potential $V$ tend to zero (resp. at infinity) whenever $f$ oscillates at zero (resp. at infinity). Unlike in classical results, neither symmetry property on $f$ nor periodicity on the potentials $V$ and $W$ are required.
2007, 18(1): 121-134
doi: 10.3934/dcds.2007.18.121
+[Abstract](1973)
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Abstract:
We consider the conormal derivative problem for an elliptic system in divergence form with discontinuous coefficients in a more general geometric setting. We obtain the $L^{p}$, $1 < p <\infty$, regularity of the maximum order derivatives of the weak solutions for such a problem.
We consider the conormal derivative problem for an elliptic system in divergence form with discontinuous coefficients in a more general geometric setting. We obtain the $L^{p}$, $1 < p <\infty$, regularity of the maximum order derivatives of the weak solutions for such a problem.
2007, 18(1): 135-157
doi: 10.3934/dcds.2007.18.135
+[Abstract](1469)
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Abstract:
We investigate the evolution of the area of multi-dimensional surfaces along the flow of a dynamical system with known first integrals, and we formulate sufficient conditions for area contraction.
These results, together with known results about the Hausdorff dimension and the box-counting dimension of invariant sets, are applied to systems exhibiting almost global convergence/asymptotic stability. This leads to a generalization of a well-known result on almost global convergence of a system, based on the use of density functions. We conclude with an example.
We investigate the evolution of the area of multi-dimensional surfaces along the flow of a dynamical system with known first integrals, and we formulate sufficient conditions for area contraction.
These results, together with known results about the Hausdorff dimension and the box-counting dimension of invariant sets, are applied to systems exhibiting almost global convergence/asymptotic stability. This leads to a generalization of a well-known result on almost global convergence of a system, based on the use of density functions. We conclude with an example.
2007, 18(1): 159-186
doi: 10.3934/dcds.2007.18.159
+[Abstract](1711)
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Abstract:
We consider the viscous Camassa-Holm equation subject to an external force, where the viscosity term is given by second order differential operator in divergence form. We show that under some mild assumptions on the viscosity term, one has global well-posedness both in the periodic case and the case of the whole line. In the periodic case, we show the existence of global attractors in the energy space $H^1$, provided the external force is in the class $L^2(I)$. Moreover, we establish an asymptotic smoothing effect, which states that the elements of the attractor are in fact in the smoother Besov space B2 2, ∞$(I)$. Identical results (after adding an appropriate linear damping term) are obtained in the case of the whole line.
We consider the viscous Camassa-Holm equation subject to an external force, where the viscosity term is given by second order differential operator in divergence form. We show that under some mild assumptions on the viscosity term, one has global well-posedness both in the periodic case and the case of the whole line. In the periodic case, we show the existence of global attractors in the energy space $H^1$, provided the external force is in the class $L^2(I)$. Moreover, we establish an asymptotic smoothing effect, which states that the elements of the attractor are in fact in the smoother Besov space B2 2, ∞$(I)$. Identical results (after adding an appropriate linear damping term) are obtained in the case of the whole line.
2007, 18(1): 187-197
doi: 10.3934/dcds.2007.18.187
+[Abstract](2097)
+[PDF](174.6KB)
Abstract:
We prove that the conjugacies in the Grobman-Hartman theorem are always Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. We also consider the case of hyperbolic trajectories of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. All the results are obtained in Banach spaces. It is common knowledge that some authors claimed that the Hölder regularity of the conjugacies is well known, however without providing any reference. In fact, to the best of our knowledge, the proof appears nowhere in the published literature.
We prove that the conjugacies in the Grobman-Hartman theorem are always Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. We also consider the case of hyperbolic trajectories of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. All the results are obtained in Banach spaces. It is common knowledge that some authors claimed that the Hölder regularity of the conjugacies is well known, however without providing any reference. In fact, to the best of our knowledge, the proof appears nowhere in the published literature.
2007, 18(1): 199-217
doi: 10.3934/dcds.2007.18.199
+[Abstract](1114)
+[PDF](218.1KB)
Abstract:
In this paper we define random $\beta$-expansions with digits taken from a given set of real numbers $A= \{ a_1 , \ldots , a_m \}$. We study a generalization of the greedy and lazy expansion and define a function $K$ that generates essentially all $\beta$-expansions with digits belonging to the set $A$. We show that $K$ admits an invariant measure $\nu$ under which $K$ is isomorphic to the uniform Bernoulli shift on $A$.
In this paper we define random $\beta$-expansions with digits taken from a given set of real numbers $A= \{ a_1 , \ldots , a_m \}$. We study a generalization of the greedy and lazy expansion and define a function $K$ that generates essentially all $\beta$-expansions with digits belonging to the set $A$. We show that $K$ admits an invariant measure $\nu$ under which $K$ is isomorphic to the uniform Bernoulli shift on $A$.
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