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1534-0392
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Communications on Pure & Applied Analysis
September 2004 , Volume 3 , Issue 3
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2004, 3(3): 331-352
doi: 10.3934/cpaa.2004.3.331
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Abstract:
A model of a tumor cord cell population is analyzed in which individual cells are distinguished by cell age and radial position. The existence and asymptotic behavior of solutions are investigated. It is proved that solutions are asymptotically eventually periodic.
A model of a tumor cord cell population is analyzed in which individual cells are distinguished by cell age and radial position. The existence and asymptotic behavior of solutions are investigated. It is proved that solutions are asymptotically eventually periodic.
2004, 3(3): 353-366
doi: 10.3934/cpaa.2004.3.353
+[Abstract](2736)
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Abstract:
We consider the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on a bounded domain in $R^{3}$ with zero (no-slip) boundary conditions. With periodic boundary conditions on a box, these equations are also known as the Camassa-Holm equations. The (LANS-$\alpha$) model averages or coarse-grains the small, computationally unreasonable, scales of the Navier-Stokes equations; spatial scales smaller than $\alpha>0$ are averaged out. We establish the existence and uniqueness of local strong (i.e., regular) solutions with initial data in $H^{1//2}$, and then use the a priori estimate developed in [1] to conclude that these are global regular solutions. Our results extend those in [2] and [1], which show the global well-posedness of $H^{1}$ weak solutions in a periodic box and on a bounded domain with no-slip boundary conditions, respectively.
We consider the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on a bounded domain in $R^{3}$ with zero (no-slip) boundary conditions. With periodic boundary conditions on a box, these equations are also known as the Camassa-Holm equations. The (LANS-$\alpha$) model averages or coarse-grains the small, computationally unreasonable, scales of the Navier-Stokes equations; spatial scales smaller than $\alpha>0$ are averaged out. We establish the existence and uniqueness of local strong (i.e., regular) solutions with initial data in $H^{1//2}$, and then use the a priori estimate developed in [1] to conclude that these are global regular solutions. Our results extend those in [2] and [1], which show the global well-posedness of $H^{1}$ weak solutions in a periodic box and on a bounded domain with no-slip boundary conditions, respectively.
2004, 3(3): 367-393
doi: 10.3934/cpaa.2004.3.367
+[Abstract](2108)
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Abstract:
In this paper we study the asymptotic behaviour of the solutions of the system coupling Landau-Lifschitz equations and Maxwell equations as the exchange coefficient tends to zero. We prove that it appears a boundary layer described by a BKW method.
In this paper we study the asymptotic behaviour of the solutions of the system coupling Landau-Lifschitz equations and Maxwell equations as the exchange coefficient tends to zero. We prove that it appears a boundary layer described by a BKW method.
2004, 3(3): 395-415
doi: 10.3934/cpaa.2004.3.395
+[Abstract](3293)
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Abstract:
We prove a Strong Maximum Principle for upper semicontinuous viscosity subsolutions to fully nonlinear degenerate parabolic pde's. We also describe the set of propagation of maxima in the case of second order Hamilton-Jacobi-Bellman equations which are either convex or concave with respect to the $(u,Du,D^2 u)$ variables and we derive the Strong Maximum Principle in some cases, including a class of nonlinear operators which are not strictly parabolic.
We prove a Strong Maximum Principle for upper semicontinuous viscosity subsolutions to fully nonlinear degenerate parabolic pde's. We also describe the set of propagation of maxima in the case of second order Hamilton-Jacobi-Bellman equations which are either convex or concave with respect to the $(u,Du,D^2 u)$ variables and we derive the Strong Maximum Principle in some cases, including a class of nonlinear operators which are not strictly parabolic.
2004, 3(3): 417-431
doi: 10.3934/cpaa.2004.3.417
+[Abstract](2751)
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Abstract:
We study some questions related to the well-posedness for the initial value problem associated to the system
We study some questions related to the well-posedness for the initial value problem associated to the system
$u_{t}+u_{x x x}+a_3 v_{x x x}+u u_{x}+a_1 v v_{x}+a_2(uv)_x =0,$
$b_1 v_{t}+v_{x x x}+b_2 a_3 u_{x x x}+v v_{x}+b_2 a_2 u u_{x}+b_2 a_1(uv)_x=0.$
Using recent methods, we prove a sharp local result in Sobolev spaces. We also prove global result under some conditions on the coefficients.
2004, 3(3): 433-446
doi: 10.3934/cpaa.2004.3.433
+[Abstract](2927)
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Abstract:
In this paper we study the asymptotic dynamics of the 2-dimensional Navier-Stokes equations on a bounded domain $\Omega \subset R^2$ with the mixed-free boundary conditions. We prove that there exists a system of reaction-diffusion equations which possesses exactly the same asymptotic dynamics as the Navier-Stokes equations.
In this paper we study the asymptotic dynamics of the 2-dimensional Navier-Stokes equations on a bounded domain $\Omega \subset R^2$ with the mixed-free boundary conditions. We prove that there exists a system of reaction-diffusion equations which possesses exactly the same asymptotic dynamics as the Navier-Stokes equations.
2004, 3(3): 447-464
doi: 10.3934/cpaa.2004.3.447
+[Abstract](2559)
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Abstract:
For each Mathieu characteristic number of integer order (MCN) we construct sequences of upper and lower bounds both converging to the MCN. The bounds arise as zeros of polynomials in sequences generated by recursion. This result is based on a constructive proof of convergence for Ince's continued fractions. An important role is also played by the fact that the continued fractions define meromorphic functions.
For each Mathieu characteristic number of integer order (MCN) we construct sequences of upper and lower bounds both converging to the MCN. The bounds arise as zeros of polynomials in sequences generated by recursion. This result is based on a constructive proof of convergence for Ince's continued fractions. An important role is also played by the fact that the continued fractions define meromorphic functions.
2004, 3(3): 465-474
doi: 10.3934/cpaa.2004.3.465
+[Abstract](2987)
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Abstract:
Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, su$p_{s\geq 1} f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive solutions to the equation $\Delta u+|\nabla u|=p(x)f(u)$ in $\Omega,$ where $\Omega$ is either a smooth bounded domain or $\Omega=\mathbb R^N$. If $\Omega$ is bounded we prove that the above problem has never a blow-up boundary solution. Since $f$ does not satisfy the Keller-Osserman growth condition at infinity, we supply in the case $\Omega=\mathbb R^N$ a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.
Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, su$p_{s\geq 1} f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive solutions to the equation $\Delta u+|\nabla u|=p(x)f(u)$ in $\Omega,$ where $\Omega$ is either a smooth bounded domain or $\Omega=\mathbb R^N$. If $\Omega$ is bounded we prove that the above problem has never a blow-up boundary solution. Since $f$ does not satisfy the Keller-Osserman growth condition at infinity, we supply in the case $\Omega=\mathbb R^N$ a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.
2004, 3(3): 475-490
doi: 10.3934/cpaa.2004.3.475
+[Abstract](2857)
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Abstract:
In this paper we consider the existence, nonexistence and the asymptotic behavior of the global solutions of the quasilinear parabolic equation of the following form:
In this paper we consider the existence, nonexistence and the asymptotic behavior of the global solutions of the quasilinear parabolic equation of the following form:
$u_t-\Delta_pu=|u|^{q-2}u, \quad (x,t)\in\Omega\times (0,T),$
$u(x,t)=0,\quad (x,t)\in\partial\Omega\times (0,T), $
$ u(x,0)=u_0(x), \quad u_0(x)\geq 0, u_0(x)$ ≠ $0, $
where $\Omega$ is a smooth bounded domain in $R^N(N\geq 3)$, $\Delta_pu=$ div$(|\nabla u|^{p-2}\nabla u )$, $\frac{2N}{N+2}$ < $p$ < $N$, $q=p^\star=\frac{pN}{N-p}$ is the critical Sobolev exponent. In particular, we employ the concentration-compactness principle to prove that the global solutions with the initial data in "stable set" converge strongly to zero in $W_0^{1,p}(\Omega)$.
2004, 3(3): 491-500
doi: 10.3934/cpaa.2004.3.491
+[Abstract](3148)
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Abstract:
For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations. Results are also obtained for variational problems with higher-order derivatives.
For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the calculus of variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBois-Reymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the calculus of variations. Results are also obtained for variational problems with higher-order derivatives.
2004, 3(3): 501-508
doi: 10.3934/cpaa.2004.3.501
+[Abstract](2914)
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Abstract:
We establish the local well-posedness for the periodic generalized Camassa-Holm equation. We also give the precise blow-up scenario and prove that the equation has smooth solutions that blow up in finite time.
We establish the local well-posedness for the periodic generalized Camassa-Holm equation. We also give the precise blow-up scenario and prove that the equation has smooth solutions that blow up in finite time.
2004, 3(3): 509-514
doi: 10.3934/cpaa.2004.3.509
+[Abstract](2370)
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Abstract:
We prove the existence of oscillatory solutions of the nonlinear wave equation, under irrationality conditions stronger than the usual Diophantine one, by perturbative techniques inspired by the Lindstedt series method originally introduced in classical mechanics to study the existence of invariant tori in quasi-integrable Hamiltonian systems.
We prove the existence of oscillatory solutions of the nonlinear wave equation, under irrationality conditions stronger than the usual Diophantine one, by perturbative techniques inspired by the Lindstedt series method originally introduced in classical mechanics to study the existence of invariant tori in quasi-integrable Hamiltonian systems.
2004, 3(3): 515-526
doi: 10.3934/cpaa.2004.3.515
+[Abstract](2445)
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Abstract:
In this paper, we report the existence of twelve small limit cycles in a planar system with 3rd-degree polynomial functions. The system has $Z_2$-symmetry, with a saddle point, or a node, or a focus point at the origin, and two focus points which are symmetric about the origin. It is shown that such a $Z_2$-equivariant vector field can have twelve small limit cycles. Fourteen or sixteen small limit cycles, as expected before, cannot not exist.
In this paper, we report the existence of twelve small limit cycles in a planar system with 3rd-degree polynomial functions. The system has $Z_2$-symmetry, with a saddle point, or a node, or a focus point at the origin, and two focus points which are symmetric about the origin. It is shown that such a $Z_2$-equivariant vector field can have twelve small limit cycles. Fourteen or sixteen small limit cycles, as expected before, cannot not exist.
2004, 3(3): 527-543
doi: 10.3934/cpaa.2004.3.527
+[Abstract](2446)
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Abstract:
This paper is a continuation of the paper ([1] ), which is called paper (I) afterward. In the present paper, which we shall call paper (II), we study the interfacial instability property of the side interface of a growing disc-like crystal at the early stage of growth by using the approach developed in the interfacial wave (IFW) theory of dendritic growth. Our analysis show that the system allows two types of unstable modes over the side-interface: (1). The axi-symmetric $(m=0)$ modes. The most dangerous axi-symmetric mode is the base mode $A_0$, which is responsible for formation of the axi-symmetric pattern over the side-interface, anti-symmetric about the central plane of the disc; (2). The non-axi-symmetric modes, which are responsible for non-axi-symmetric pattern formation around the edge of the disc. The growth rates of these non-axi-symmetric modes are much smaller than the growth rate of the base mode $A_0$. During the course of disc growth, the unstable $A_0$-mode merges first. It leads to the formation of anti-symmetric pattern about the central plane over the side-interface. Following the onset of unstable base mode $A_0$, a set of non-axi-symmetric growing modes also appear. However, due to the smallness of growth rate of these unstable modes, the non-axi-symmetric pattern around the edge of the disc becomes observable, only after a sufficiently long time. Our theoretical predictions are in good agreement with the available experimental data.
This paper is a continuation of the paper ([1] ), which is called paper (I) afterward. In the present paper, which we shall call paper (II), we study the interfacial instability property of the side interface of a growing disc-like crystal at the early stage of growth by using the approach developed in the interfacial wave (IFW) theory of dendritic growth. Our analysis show that the system allows two types of unstable modes over the side-interface: (1). The axi-symmetric $(m=0)$ modes. The most dangerous axi-symmetric mode is the base mode $A_0$, which is responsible for formation of the axi-symmetric pattern over the side-interface, anti-symmetric about the central plane of the disc; (2). The non-axi-symmetric modes, which are responsible for non-axi-symmetric pattern formation around the edge of the disc. The growth rates of these non-axi-symmetric modes are much smaller than the growth rate of the base mode $A_0$. During the course of disc growth, the unstable $A_0$-mode merges first. It leads to the formation of anti-symmetric pattern about the central plane over the side-interface. Following the onset of unstable base mode $A_0$, a set of non-axi-symmetric growing modes also appear. However, due to the smallness of growth rate of these unstable modes, the non-axi-symmetric pattern around the edge of the disc becomes observable, only after a sufficiently long time. Our theoretical predictions are in good agreement with the available experimental data.
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