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1534-0392
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Communications on Pure & Applied Analysis
December 2007 , Volume 6 , Issue 4
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2007, 6(4): 917-936
doi: 10.3934/cpaa.2007.6.917
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Abstract:
We study the Osher-Solé-Vese model [11], which is the gradient flow of an energy consisting of the total variation functional plus an $H^{-1}$ fidelity term. A variational inequality weak formulation for this problem is proposed along the lines of that of Feng and Prohl [7] for the Rudin-Osher-Fatemi model [12]. A regularized energy is considered, and the minimization problems corresponding to both the original and regularized energies are shown to be well-posed. The Galerkin method of Lions [9] is used to prove the well-posedness of the weak problem corresponding to the regularized energy. By letting the regularization parameter $\epsilon$ tend to $0$, we recover the well-posedness of the weak problem corresponding to the original energy. Further, we show that for both energies the solution of the weak problem tends to the minimizer of the energy as $t \to \infty$. Finally, we find the rate of convergence of the weak solution of the regularized problem to that of the original one as $\epsilon \downarrow 0$.
We study the Osher-Solé-Vese model [11], which is the gradient flow of an energy consisting of the total variation functional plus an $H^{-1}$ fidelity term. A variational inequality weak formulation for this problem is proposed along the lines of that of Feng and Prohl [7] for the Rudin-Osher-Fatemi model [12]. A regularized energy is considered, and the minimization problems corresponding to both the original and regularized energies are shown to be well-posed. The Galerkin method of Lions [9] is used to prove the well-posedness of the weak problem corresponding to the regularized energy. By letting the regularization parameter $\epsilon$ tend to $0$, we recover the well-posedness of the weak problem corresponding to the original energy. Further, we show that for both energies the solution of the weak problem tends to the minimizer of the energy as $t \to \infty$. Finally, we find the rate of convergence of the weak solution of the regularized problem to that of the original one as $\epsilon \downarrow 0$.
2007, 6(4): 937-955
doi: 10.3934/cpaa.2007.6.937
+[Abstract](3096)
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The existence and finite fractal dimension of a pullback attractor in the space $V$ for a three dimensional system of the nonautonomous Globally Modified Navier-Stokes Equations on a bounded domain is established under appropriate properties on the time dependent forcing term. These equations were proposed recently by Caraballo et al and are obtained from the Navier- Stokes Equations by a global modification of the nonlinear advection term. The existence of the attractor is obtained via the flattening property, which is verified.
The existence and finite fractal dimension of a pullback attractor in the space $V$ for a three dimensional system of the nonautonomous Globally Modified Navier-Stokes Equations on a bounded domain is established under appropriate properties on the time dependent forcing term. These equations were proposed recently by Caraballo et al and are obtained from the Navier- Stokes Equations by a global modification of the nonlinear advection term. The existence of the attractor is obtained via the flattening property, which is verified.
2007, 6(4): 957-982
doi: 10.3934/cpaa.2007.6.957
+[Abstract](2309)
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Abstract:
In this paper, we are interested in some aspects of the biharmonic equation in the half-space $\mathbb R^N_+$, with $N\geq 2$. We study the regularity of generalized solutions in weighted Sobolev spaces, then we consider the question of singular boundary conditions. To finish, we envisage other sorts of boundary conditions.
In this paper, we are interested in some aspects of the biharmonic equation in the half-space $\mathbb R^N_+$, with $N\geq 2$. We study the regularity of generalized solutions in weighted Sobolev spaces, then we consider the question of singular boundary conditions. To finish, we envisage other sorts of boundary conditions.
2007, 6(4): 983-996
doi: 10.3934/cpaa.2007.6.983
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Abstract:
We consider the perturbed generalized equation $v \in f(x) +G(x)$ where $v$ is a perturbation parameter, $f$ is a function acting from a Banach space $X$ to a Banach space $Y$ while $G: X \rightarrow Y$ is a set-valued mapping. We associate to this generalized equation the following iterative procedure:
We consider the perturbed generalized equation $v \in f(x) +G(x)$ where $v$ is a perturbation parameter, $f$ is a function acting from a Banach space $X$ to a Banach space $Y$ while $G: X \rightarrow Y$ is a set-valued mapping. We associate to this generalized equation the following iterative procedure:
$ v \in f(x_n)+ \nabla f(x_n)(x_{n+1}-x_n) +\frac{1}{2}\nabla^2 f(x_n) (x_{n+1}-x_n)^2 +G(x_{n+1}).$ $\quad$ (*)
We investigate some stability properties of the method (*) and we study the behavior of the sequences that it generates, more precisely, we show that they inherit some regularity properties from the mapping $f+G$.
2007, 6(4): 997-1021
doi: 10.3934/cpaa.2007.6.997
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In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
2007, 6(4): 1023-1041
doi: 10.3934/cpaa.2007.6.1023
+[Abstract](2767)
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Abstract:
The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation in $\mathbb R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^s(\mathbb R^n )$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$. We use the "$I$-method" combined with a local in time Morawetz estimate.
The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation in $\mathbb R^n, n \geq 3$ is considered. We show that the problem is globally well posed in $H^s(\mathbb R^n )$ when $1>s>\frac{\sqrt{7}-1}{3}$ for $n=3$, and when $1>s> \frac{-(n-2)+\sqrt{(n-2)^2+8(n-2)}}{4}$ for $n \geq 4$. We use the "$I$-method" combined with a local in time Morawetz estimate.
2007, 6(4): 1043-1050
doi: 10.3934/cpaa.2007.6.1043
+[Abstract](3017)
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Abstract:
This paper concerns the orbital stability of solitary waves of the Schrödinger-Boussinesq equation
This paper concerns the orbital stability of solitary waves of the Schrödinger-Boussinesq equation
$ i\partial_t u+\partial_x^2 u+uv =0\qquad\qquad\qquad\qquad\qquad (0.1)$
$ \partial_t^2 v-\partial_x^2 v+\partial_x^4 v+\partial_x^2 (3v^2+|u|^2)=0.$
By applying the abstract results of Grillakis, Shatah and Strauss [11, 12] and detailed spectral analysis developed by Lopes in [17], we obtain the stability of solitary waves.
2007, 6(4): 1051-1074
doi: 10.3934/cpaa.2007.6.1051
+[Abstract](2710)
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Abstract:
$L^1$-estimates are established for the higher-order derivatives of classical solutions to the homogeneous Cauchy problem for linear second-order one-dimensional parabolic equations of general form. It is required that the initial data is uniformly continuous and of bounded total variation on some given bounded interval. If the latter condition holds on every bounded interval, then uniform $L^1$-estimates can be proved for the higher-order derivatives. In contrast to earlier findings, where the case of bounded initial data with a continuity modulus satisfying a Dini condition was considered, no constraints are imposed to such a continuity modulus in this paper. In particular, the initial data are allowed to be unbounded. Sets of initial data, in general discontinuous, are also considered.
$L^1$-estimates are established for the higher-order derivatives of classical solutions to the homogeneous Cauchy problem for linear second-order one-dimensional parabolic equations of general form. It is required that the initial data is uniformly continuous and of bounded total variation on some given bounded interval. If the latter condition holds on every bounded interval, then uniform $L^1$-estimates can be proved for the higher-order derivatives. In contrast to earlier findings, where the case of bounded initial data with a continuity modulus satisfying a Dini condition was considered, no constraints are imposed to such a continuity modulus in this paper. In particular, the initial data are allowed to be unbounded. Sets of initial data, in general discontinuous, are also considered.
2007, 6(4): 1075-1086
doi: 10.3934/cpaa.2007.6.1075
+[Abstract](3084)
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Abstract:
We consider the initial boundary value problem for the 3D convective Cahn - Hilliard equation with periodic boundary conditions. This gives rise to a continuous dynamical system on $\dot L^2(\Omega)$. Absorbing balls in $\dot L^2(\Omega), \dot H_{per}^1(\Omega)$ and $\dot H_{per}^2(\Omega)$ are shown to exist. Combining with the compactness property of the solution semigroup we conclude the existence of the global attractor. Restricting the dynamical system on the absorbing ball in $\dot H_{per}^2(\Omega)$ and using the general framework in Eden et. all. [5] the existence of an exponential attractor is guaranteed. This approach also gives an explicit upper estimate of the dimension of the exponential attractor, albeit of the global attractor.
We consider the initial boundary value problem for the 3D convective Cahn - Hilliard equation with periodic boundary conditions. This gives rise to a continuous dynamical system on $\dot L^2(\Omega)$. Absorbing balls in $\dot L^2(\Omega), \dot H_{per}^1(\Omega)$ and $\dot H_{per}^2(\Omega)$ are shown to exist. Combining with the compactness property of the solution semigroup we conclude the existence of the global attractor. Restricting the dynamical system on the absorbing ball in $\dot H_{per}^2(\Omega)$ and using the general framework in Eden et. all. [5] the existence of an exponential attractor is guaranteed. This approach also gives an explicit upper estimate of the dimension of the exponential attractor, albeit of the global attractor.
2007, 6(4): 1087-1111
doi: 10.3934/cpaa.2007.6.1087
+[Abstract](3321)
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Abstract:
This paper discusses the long time behavior of solutions for dissipative non-autonomous lattice dynamical systems. We first prove some sufficient and necessary conditions for the existence of a compact uniform attractor for the family of processes defined on a Hilbert space of infinite sequences, and then give an upper bound of the Kolmogorov $\varepsilon$-entropy for the uniform attractor. As an application, we consider the dissipative non-autonomous lattice Zakharov equations.
This paper discusses the long time behavior of solutions for dissipative non-autonomous lattice dynamical systems. We first prove some sufficient and necessary conditions for the existence of a compact uniform attractor for the family of processes defined on a Hilbert space of infinite sequences, and then give an upper bound of the Kolmogorov $\varepsilon$-entropy for the uniform attractor. As an application, we consider the dissipative non-autonomous lattice Zakharov equations.
Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux
2007, 6(4): 1113-1129
doi: 10.3934/cpaa.2007.6.1113
+[Abstract](2670)
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Abstract:
This paper deals with non-simultaneous blow-up for heat equations with positive-negative sources coupled via nonlinear boundary flux. At first, we establish the necessary-sufficient conditions for non-simultaneous blow-up of solutions under suitable initial data. Furthermore, the sufficient conditions are determined under which any blow-up of solutions to the model would be non-simultaneous. Comparing with those for a similar system possessing negative-negative sources obtained in a previous paper [19] we show clearly the contribution of the positive source in the present model to the non-simultaneous blowing up behavior of solutions.
This paper deals with non-simultaneous blow-up for heat equations with positive-negative sources coupled via nonlinear boundary flux. At first, we establish the necessary-sufficient conditions for non-simultaneous blow-up of solutions under suitable initial data. Furthermore, the sufficient conditions are determined under which any blow-up of solutions to the model would be non-simultaneous. Comparing with those for a similar system possessing negative-negative sources obtained in a previous paper [19] we show clearly the contribution of the positive source in the present model to the non-simultaneous blowing up behavior of solutions.
2007, 6(4): 1131-1143
doi: 10.3934/cpaa.2007.6.1131
+[Abstract](3090)
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Abstract:
A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.
A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.
2007, 6(4): 1145-1165
doi: 10.3934/cpaa.2007.6.1145
+[Abstract](2594)
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Abstract:
In this paper we study the existence of finite traveling wave solutions in a degenerate cross-diffusion system modeling the growth of bacteria colony. The importance of establishing the existence lies in the fact that the analysis of the stability of the wave front provides partial answers to the intriguing spatial patterns of the colony. There have been very few results on the finite traveling wave solutions of degenerate parabolic system. One reason is that the traditional method often leads to phase plane analysis on higher dimension which is usually a difficult task. Our method in this paper is based on Schauder fixed point theorem and shooting arguments.
In this paper we study the existence of finite traveling wave solutions in a degenerate cross-diffusion system modeling the growth of bacteria colony. The importance of establishing the existence lies in the fact that the analysis of the stability of the wave front provides partial answers to the intriguing spatial patterns of the colony. There have been very few results on the finite traveling wave solutions of degenerate parabolic system. One reason is that the traditional method often leads to phase plane analysis on higher dimension which is usually a difficult task. Our method in this paper is based on Schauder fixed point theorem and shooting arguments.
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