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1078-0947
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Discrete & Continuous Dynamical Systems
September 2003 , Volume 9 , Issue 5
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2003, 9(5): 1081-1104
doi: 10.3934/dcds.2003.9.1081
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Abstract:
We study nonlinear degenerate parabolic equations where the flux function $f(x,t,u)$ does not depend Lipschitz continuously on the spatial location $x$. By properly adapting the "doubling of variables" device due to Kružkov [25] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form $k(x)f(u)$, where $k(x)$ is a vector-valued function and $f(u)$ is a scalar function.
We study nonlinear degenerate parabolic equations where the flux function $f(x,t,u)$ does not depend Lipschitz continuously on the spatial location $x$. By properly adapting the "doubling of variables" device due to Kružkov [25] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form $k(x)f(u)$, where $k(x)$ is a vector-valued function and $f(u)$ is a scalar function.
2003, 9(5): 1105-1132
doi: 10.3934/dcds.2003.9.1105
+[Abstract](2756)
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Abstract:
In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the heat equation on $\R^N $ and the asymptotic behavior as $|x|\to \infty $ of its initial value $u_0$. In particular, we show that, for a fixed $0$<$\sigma$<$N$, if the sequence of dilations $\lambda _n^\sigma u_0(\lambda _n\cdot)$ converges weakly to $z(\cdot)$ as $\lambda _n\to \infty $, then the rescaled solution $t^{\frac{\sigma}{2}}$ $u(t, \cdot\sqrt t)$ converges uniformly on $\R^N $ to $e^\Delta z$ along the subsequence $t_n=\lambda _n^2$. Moreover, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this fashion is a closed ball $B$ of a weighted $L^\infty $ space. The resulting "universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$.
In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the heat equation on $\R^N $ and the asymptotic behavior as $|x|\to \infty $ of its initial value $u_0$. In particular, we show that, for a fixed $0$<$\sigma$<$N$, if the sequence of dilations $\lambda _n^\sigma u_0(\lambda _n\cdot)$ converges weakly to $z(\cdot)$ as $\lambda _n\to \infty $, then the rescaled solution $t^{\frac{\sigma}{2}}$ $u(t, \cdot\sqrt t)$ converges uniformly on $\R^N $ to $e^\Delta z$ along the subsequence $t_n=\lambda _n^2$. Moreover, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this fashion is a closed ball $B$ of a weighted $L^\infty $ space. The resulting "universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$.
2003, 9(5): 1133-1148
doi: 10.3934/dcds.2003.9.1133
+[Abstract](2160)
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Abstract:
Here we show examples of homoclinic bifurcations which can be perturbed to produce invariant curves and attractors with high Hausdorff dimension.
Here we show examples of homoclinic bifurcations which can be perturbed to produce invariant curves and attractors with high Hausdorff dimension.
2003, 9(5): 1149-1173
doi: 10.3934/dcds.2003.9.1149
+[Abstract](2089)
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Abstract:
We consider the isosceles $3$--body problem with the third particle having a small mass which eventually tend to zero. Classical McGehee's blow up is not defined because the matrix of masses becomes degenerate. Following Elbialy [3] we perform the blow up using the Euclidean norm in the planar $3$--body problem. We then complete the phase portrait of the flow in the collision manifold giving the behavior of some branches of saddle points missing in [3]. The homothetic orbits within the fixed energy level then provide the necessary recurrence in order to build a symbolic dynamics. This is done following ideas of S. Kaplan [6] for the collinear $3$--body problem. Here the difficulty is the larger number of critical points.
We consider the isosceles $3$--body problem with the third particle having a small mass which eventually tend to zero. Classical McGehee's blow up is not defined because the matrix of masses becomes degenerate. Following Elbialy [3] we perform the blow up using the Euclidean norm in the planar $3$--body problem. We then complete the phase portrait of the flow in the collision manifold giving the behavior of some branches of saddle points missing in [3]. The homothetic orbits within the fixed energy level then provide the necessary recurrence in order to build a symbolic dynamics. This is done following ideas of S. Kaplan [6] for the collinear $3$--body problem. Here the difficulty is the larger number of critical points.
2003, 9(5): 1175-1184
doi: 10.3934/dcds.2003.9.1175
+[Abstract](2123)
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Abstract:
Let $T$ be a star and $\Omega(f)$ be the set of non-wandering points of a continuous map $f:T\rightarrow T$. For two distinct prime numbers $p$ and $q$, we prove: (1) $\Omega(f^p)\cup \Omega(f^q)=\Omega(f)$ for each $f \in C(T,T)$ if and only if $pq > End(T)$, (2) $\Omega(f^p)\cap \Omega(f^q)=\Omega(f^{p q})$ for each $f\in C(T,T)$ if and only if $p+q \ge End(T)$, where $End(T)$ is the number of the ends of $T$. Using (1)-(2) and the results in [3], we obtain a complete description of non-wandering sets of the powers of maps of 3-star and 4-star.
Let $T$ be a star and $\Omega(f)$ be the set of non-wandering points of a continuous map $f:T\rightarrow T$. For two distinct prime numbers $p$ and $q$, we prove: (1) $\Omega(f^p)\cup \Omega(f^q)=\Omega(f)$ for each $f \in C(T,T)$ if and only if $pq > End(T)$, (2) $\Omega(f^p)\cap \Omega(f^q)=\Omega(f^{p q})$ for each $f\in C(T,T)$ if and only if $p+q \ge End(T)$, where $End(T)$ is the number of the ends of $T$. Using (1)-(2) and the results in [3], we obtain a complete description of non-wandering sets of the powers of maps of 3-star and 4-star.
2003, 9(5): 1185-1192
doi: 10.3934/dcds.2003.9.1185
+[Abstract](2881)
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Abstract:
We show that if a mixing diffeomorphism of a compact manifold preserves an ergodic hyperbolic probability measure, then the measures supported by hyperbolic periodic points are dense in the set of invariant measures. This is a generalization of the result shown by Sigmund.
We show that if a mixing diffeomorphism of a compact manifold preserves an ergodic hyperbolic probability measure, then the measures supported by hyperbolic periodic points are dense in the set of invariant measures. This is a generalization of the result shown by Sigmund.
2003, 9(5): 1193-1200
doi: 10.3934/dcds.2003.9.1193
+[Abstract](2582)
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Abstract:
The Shigesada-Kawasaki-Teramoto model is a generalization of the classical Lotka-Volterra competition model for which the competing species undergo both diffusion, self-diffusion and cross-diffusion. Very few mathematical results are known for this model, especially in higher space dimensions. In this paper, we shall prove global existence of strong solutions in any space dimension for this model when the cross-diffusion coefficient in the first species is sufficiently small and when there is no self-diffusion or cross-diffusion in the second species.
The Shigesada-Kawasaki-Teramoto model is a generalization of the classical Lotka-Volterra competition model for which the competing species undergo both diffusion, self-diffusion and cross-diffusion. Very few mathematical results are known for this model, especially in higher space dimensions. In this paper, we shall prove global existence of strong solutions in any space dimension for this model when the cross-diffusion coefficient in the first species is sufficiently small and when there is no self-diffusion or cross-diffusion in the second species.
2003, 9(5): 1201-1222
doi: 10.3934/dcds.2003.9.1201
+[Abstract](2177)
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In this article we consider the Newton method for the numerical solution of a class of robust control problems in fluid mechanics recently studied in [5]. We prove the local convergence of the algorithm and we obtain the rate of convergence of the method.
In this article we consider the Newton method for the numerical solution of a class of robust control problems in fluid mechanics recently studied in [5]. We prove the local convergence of the algorithm and we obtain the rate of convergence of the method.
2003, 9(5): 1223-1241
doi: 10.3934/dcds.2003.9.1223
+[Abstract](2866)
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Abstract:
We prove a necessary and sufficient condition for the exponential stability of time-invariant linear systems on time scales in terms of the eigenvalues of the system matrix. In particular, this unifies the corresponding characterizations for finite-dimensional differential and difference equations. To this end we use a representation formula for the transition matrix of Jordan reducible systems in the regressive case. Also we give conditions under which the obtained characterizations can be exactly calculated and explicitly calculate the region of stability for several examples.
We prove a necessary and sufficient condition for the exponential stability of time-invariant linear systems on time scales in terms of the eigenvalues of the system matrix. In particular, this unifies the corresponding characterizations for finite-dimensional differential and difference equations. To this end we use a representation formula for the transition matrix of Jordan reducible systems in the regressive case. Also we give conditions under which the obtained characterizations can be exactly calculated and explicitly calculate the region of stability for several examples.
2003, 9(5): 1243-1262
doi: 10.3934/dcds.2003.9.1243
+[Abstract](2454)
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Abstract:
This paper is concerned with the large time behavior of global smooth solutions to the Cauchy problem of the $p-$system with relaxation. Former results in this direction indicate that such a problem possesses a global smooth solution provided that the first derivative of the solutions with respect to the space variable $x$ are sufficiently small. Under the same small assumption on the global smooth solution, we show that it converges to the corresponding nonlinear rarefaction wave and in our analysis, we do not ask the rarefaction wave to be weak and the initial error can also be chosen arbitrarily large.
This paper is concerned with the large time behavior of global smooth solutions to the Cauchy problem of the $p-$system with relaxation. Former results in this direction indicate that such a problem possesses a global smooth solution provided that the first derivative of the solutions with respect to the space variable $x$ are sufficiently small. Under the same small assumption on the global smooth solution, we show that it converges to the corresponding nonlinear rarefaction wave and in our analysis, we do not ask the rarefaction wave to be weak and the initial error can also be chosen arbitrarily large.
2003, 9(5): 1263-1275
doi: 10.3934/dcds.2003.9.1263
+[Abstract](1774)
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Abstract:
We consider semilinear hyperbolic problem associated with a second order partial differential operator in its divergence form. We prove a comparison theorem for the weak lower and upper solutions of the problem and then apply the method of generalized quasilinearization.
We consider semilinear hyperbolic problem associated with a second order partial differential operator in its divergence form. We prove a comparison theorem for the weak lower and upper solutions of the problem and then apply the method of generalized quasilinearization.
2003, 9(5): 1277-1292
doi: 10.3934/dcds.2003.9.1277
+[Abstract](2630)
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Abstract:
We consider various classes of superlinear parabolic problems, including reaction-diffusion systems and scalar reaction-diffusion equations with convective or dissipative gradient terms. For these problems we prove uniform a priori estimates for all nonnegative global solutions. The existence of an energy functional for these problems is not known, so that traditional methods for a priori estimates do not apply. We use a different approach based on scaling and Fujita-type theorems. In the case of reaction-diffusion systems, we also obtain some universal bounds, i.e. a priori estimates independent of the initial data.
We consider various classes of superlinear parabolic problems, including reaction-diffusion systems and scalar reaction-diffusion equations with convective or dissipative gradient terms. For these problems we prove uniform a priori estimates for all nonnegative global solutions. The existence of an energy functional for these problems is not known, so that traditional methods for a priori estimates do not apply. We use a different approach based on scaling and Fujita-type theorems. In the case of reaction-diffusion systems, we also obtain some universal bounds, i.e. a priori estimates independent of the initial data.
2003, 9(5): 1293-1322
doi: 10.3934/dcds.2003.9.1293
+[Abstract](2493)
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Abstract:
We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of Šil'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.
We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of Šil'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.
2003, 9(5): 1323-1327
doi: 10.3934/dcds.2003.9.1323
+[Abstract](2427)
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Abstract:
Developping ideas of S. Li [Tran. Amer. Math. Soc. 301 (1993), 243--249] concerning the notion of $\omega$-chaos we prove that any transitive continuous map $f$ of the interval is conjugate to a map $g$ of the interval which possesses an $\omega$-scrambled set $S$ of full Lebesgue measure. Thus, for any distinct $x, y$ in $S$, $\omega _g (x)\cap\omega _g(y)$ is non-empty, and $\omega _g(x) \setminus\omega _g(y)$ is uncountable.
Developping ideas of S. Li [Tran. Amer. Math. Soc. 301 (1993), 243--249] concerning the notion of $\omega$-chaos we prove that any transitive continuous map $f$ of the interval is conjugate to a map $g$ of the interval which possesses an $\omega$-scrambled set $S$ of full Lebesgue measure. Thus, for any distinct $x, y$ in $S$, $\omega _g (x)\cap\omega _g(y)$ is non-empty, and $\omega _g(x) \setminus\omega _g(y)$ is uncountable.
2003, 9(5): 1329-1341
doi: 10.3934/dcds.2003.9.1329
+[Abstract](2635)
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Abstract:
Let
Let
$\alpha( p,q,r) =$inf{$\frac{|| u'||_p}{||u||_q}:u\in W_{p e r}^{1,p}( -1,1) $\{$ 0$}, $\int_{-1}^1|u|^{r-2} u=0$} .
We show that
$\alpha( p,q,r )=\alpha ( p,q,q)$ if $q\leq rp+r-1$
$\alpha( p,q,r) <\alpha( p,q,q) $ if $q> ( 2r-1) p$
generalizing results of Dacorogna-Gangbo-Subía and others.
2003, 9(5): 1343-1357
doi: 10.3934/dcds.2003.9.1343
+[Abstract](2519)
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Abstract:
We consider a natural Lagrangian system on a torus and give sufficient conditions for the existence of chaotic trajectories for energy values slightly below the maximum of the potential energy. It turns out that chaotic trajectories always exist except when the system is "variationally separable", i.e. minimizers of the action functional behave like in a separable system. This gives some more support for an old conjecture that only separable natural Lagrangian systems on a torus are integrable.
We consider a natural Lagrangian system on a torus and give sufficient conditions for the existence of chaotic trajectories for energy values slightly below the maximum of the potential energy. It turns out that chaotic trajectories always exist except when the system is "variationally separable", i.e. minimizers of the action functional behave like in a separable system. This gives some more support for an old conjecture that only separable natural Lagrangian systems on a torus are integrable.
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