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1534-0392
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Communications on Pure & Applied Analysis
May 2010 , Volume 9 , Issue 3
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2010, 9(3): 583-610
doi: 10.3934/cpaa.2010.9.583
+[Abstract](2440)
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Abstract:
We study the bifurcations of limit cycles in a class of planar reversible quadratic systems whose critical points are a center, a saddle and two nodes, under small quadratic perturbations. By using the properties of related complete elliptic integrals and the geometry of some planar curves defined by them, we prove that at most two limit cycles bifurcate from the period annulus around the center. This bound is exact.
We study the bifurcations of limit cycles in a class of planar reversible quadratic systems whose critical points are a center, a saddle and two nodes, under small quadratic perturbations. By using the properties of related complete elliptic integrals and the geometry of some planar curves defined by them, we prove that at most two limit cycles bifurcate from the period annulus around the center. This bound is exact.
2010, 9(3): 611-624
doi: 10.3934/cpaa.2010.9.611
+[Abstract](2444)
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Abstract:
We give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigenmodes for a circular membrane. More exactly, we show that for each pair $(m,n) \in \{0,1,2, \cdots\}^2$ there exists a unique even eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves and, similarly, for each $(m,n) \in \{0,1,2, \cdots\}\times \{1,2, \cdots\}$ there exists a unique odd eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves. Our result is based on directly using the separation of variables method for the Helmholtz equation in elliptic coordinates and in proving that certain pairs of curves in the plane of parameters $a$ and $q$ cross each other at a single point. As side effects of our proof, a new and precise method for numerically calculating the eigenfrequencies of these modes is presented and also approximate formulae which explain rather well the qualitative asymptotic behavior of the eigenfrequencies for large eccentricities.
We give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigenmodes for a circular membrane. More exactly, we show that for each pair $(m,n) \in \{0,1,2, \cdots\}^2$ there exists a unique even eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves and, similarly, for each $(m,n) \in \{0,1,2, \cdots\}\times \{1,2, \cdots\}$ there exists a unique odd eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves. Our result is based on directly using the separation of variables method for the Helmholtz equation in elliptic coordinates and in proving that certain pairs of curves in the plane of parameters $a$ and $q$ cross each other at a single point. As side effects of our proof, a new and precise method for numerically calculating the eigenfrequencies of these modes is presented and also approximate formulae which explain rather well the qualitative asymptotic behavior of the eigenfrequencies for large eccentricities.
2010, 9(3): 625-642
doi: 10.3934/cpaa.2010.9.625
+[Abstract](2059)
+[PDF](255.5KB)
Abstract:
We study steady isothermal motions of a nonlinear weakly compressible viscoelastic fluids of Oldroyd type in a bounded domain $\Omega\subset\mathbb{R}^2$, with given non-zero velocities on the boundary of $\Omega$. We suppose that the pressure $p$ and the extra-stress tensor $\tau$ are prescribed on the part of the boundary corresponding to entering velocities. A uniqueness and existence result for the solution $(\mathbf u,p,\tau)$ is established in $W^{2,q}(\Omega)\times W^{1,q}(\Omega)\times W^{1,q}(\Omega)$ with $ 2 < q < 3$. The proof follows from an analysis of a linearized problem. The fixed point theorem is used to establish the existence of a solution. The solutions of two transport equations for $p$ and $\tau$ are obtained by integration along the streamlines.
We study steady isothermal motions of a nonlinear weakly compressible viscoelastic fluids of Oldroyd type in a bounded domain $\Omega\subset\mathbb{R}^2$, with given non-zero velocities on the boundary of $\Omega$. We suppose that the pressure $p$ and the extra-stress tensor $\tau$ are prescribed on the part of the boundary corresponding to entering velocities. A uniqueness and existence result for the solution $(\mathbf u,p,\tau)$ is established in $W^{2,q}(\Omega)\times W^{1,q}(\Omega)\times W^{1,q}(\Omega)$ with $ 2 < q < 3$. The proof follows from an analysis of a linearized problem. The fixed point theorem is used to establish the existence of a solution. The solutions of two transport equations for $p$ and $\tau$ are obtained by integration along the streamlines.
2010, 9(3): 643-654
doi: 10.3934/cpaa.2010.9.643
+[Abstract](1989)
+[PDF](168.4KB)
Abstract:
We consider linear equations $v'=A(t)v$ in a Banach space that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ determined by a function $\rho(t)$. The usual exponential behavior with $\rho(t)=t$ is included as a very special case. For other functions the Lyapunov exponents may be infinite (either $+\infty$ or $-\infty$), but we can still distinguish between different asymptotic rates. Our main objective is to establish the existence of center manifolds for a large class of nonlinear perturbations $v'=A(t)v+f(t,v)$ assuming that the linear equation has the above general asymptotic behavior. We also allow the stable, unstable and central components of $v'=A(t)v$ to exhibit a nonuniform exponential behavior. We emphasize that our results are new even in the very particular case of perturbations of uniform exponential trichotomies with arbitrary growth rates.
We consider linear equations $v'=A(t)v$ in a Banach space that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ determined by a function $\rho(t)$. The usual exponential behavior with $\rho(t)=t$ is included as a very special case. For other functions the Lyapunov exponents may be infinite (either $+\infty$ or $-\infty$), but we can still distinguish between different asymptotic rates. Our main objective is to establish the existence of center manifolds for a large class of nonlinear perturbations $v'=A(t)v+f(t,v)$ assuming that the linear equation has the above general asymptotic behavior. We also allow the stable, unstable and central components of $v'=A(t)v$ to exhibit a nonuniform exponential behavior. We emphasize that our results are new even in the very particular case of perturbations of uniform exponential trichotomies with arbitrary growth rates.
2010, 9(3): 655-666
doi: 10.3934/cpaa.2010.9.655
+[Abstract](2408)
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Abstract:
In this paper, we firstly determine the best constant of the three dimensional anisotropic Sobolev inequality [2]; then we use this best constant to investigate qualitative conditions for the uniform bound of the solution of the generalized Kadomtsev-Petviashvili (KP) I equation in three dimensions. The (KP) I equation is a model for the propagation of weakly nonlinear dispersive long waves on the surface of a fluid, when the wave motion is essentially one- directional and weak transverse effects are taken into account [11, 10]. Our results improve and optimize previous works [6, 12, 13, 14, 15].
In this paper, we firstly determine the best constant of the three dimensional anisotropic Sobolev inequality [2]; then we use this best constant to investigate qualitative conditions for the uniform bound of the solution of the generalized Kadomtsev-Petviashvili (KP) I equation in three dimensions. The (KP) I equation is a model for the propagation of weakly nonlinear dispersive long waves on the surface of a fluid, when the wave motion is essentially one- directional and weak transverse effects are taken into account [11, 10]. Our results improve and optimize previous works [6, 12, 13, 14, 15].
2010, 9(3): 667-684
doi: 10.3934/cpaa.2010.9.667
+[Abstract](2288)
+[PDF](278.1KB)
Abstract:
We consider the Boussinesq equations in either an exterior domain in $\mathbb{R}^{n}$, the whole space $\mathbb{R}^{n}$, the half space $\mathbb{R}_{+}^{n}$ or a bounded domain in $\mathbb{R}^{n}$, where the dimension $n$ satisfies $n \geq 3$. We give a class of stable steady solutions, which improves and complements the previous stability results. Our results give a complete answer to the stability problem for the Boussinesq equations in weak-$L^{p}$ spaces, in the sense that we only assume that the stable steady solution belongs to scaling invariant class $L_{\sigma }^{(n,\infty)}\times L^{(n,\infty)}$. Moreover, some considerations about the exponential decay (in bounded domains) and the uniqueness of the disturbance are done.
We consider the Boussinesq equations in either an exterior domain in $\mathbb{R}^{n}$, the whole space $\mathbb{R}^{n}$, the half space $\mathbb{R}_{+}^{n}$ or a bounded domain in $\mathbb{R}^{n}$, where the dimension $n$ satisfies $n \geq 3$. We give a class of stable steady solutions, which improves and complements the previous stability results. Our results give a complete answer to the stability problem for the Boussinesq equations in weak-$L^{p}$ spaces, in the sense that we only assume that the stable steady solution belongs to scaling invariant class $L_{\sigma }^{(n,\infty)}\times L^{(n,\infty)}$. Moreover, some considerations about the exponential decay (in bounded domains) and the uniqueness of the disturbance are done.
2010, 9(3): 685-702
doi: 10.3934/cpaa.2010.9.685
+[Abstract](2314)
+[PDF](257.0KB)
Abstract:
We prove that, under natural assumptions, the solution of the backward Euler scheme applied to a gradient flow converges to an equilibrium, as time goes to infinity. Optimal convergence rates are also obtained. As in the continuous case, the proof relies on the well known Lojasiewicz inequality. We extend these results to the $\theta$-scheme with $\theta\in [1/2, 1]$, and to the semilinear heat equation. Applications to semilinear parabolic equations, such as the Allen-Cahn or Cahn-Hilliard equation, are given
We prove that, under natural assumptions, the solution of the backward Euler scheme applied to a gradient flow converges to an equilibrium, as time goes to infinity. Optimal convergence rates are also obtained. As in the continuous case, the proof relies on the well known Lojasiewicz inequality. We extend these results to the $\theta$-scheme with $\theta\in [1/2, 1]$, and to the semilinear heat equation. Applications to semilinear parabolic equations, such as the Allen-Cahn or Cahn-Hilliard equation, are given
2010, 9(3): 703-719
doi: 10.3934/cpaa.2010.9.703
+[Abstract](1738)
+[PDF](254.1KB)
Abstract:
We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approx- imate model of the dynamics of dislocation densities in a bounded channel submitted to an exterior applied stress. The system of equations is written on a bounded interval with Dirichlet conditions and requires a special attention to the boundary. The proof of the global existence and uniqueness is done under the use of a certain comparison principle on the gradient of the solution.
We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approx- imate model of the dynamics of dislocation densities in a bounded channel submitted to an exterior applied stress. The system of equations is written on a bounded interval with Dirichlet conditions and requires a special attention to the boundary. The proof of the global existence and uniqueness is done under the use of a certain comparison principle on the gradient of the solution.
2010, 9(3): 721-730
doi: 10.3934/cpaa.2010.9.721
+[Abstract](2216)
+[PDF](178.7KB)
Abstract:
This article is focused on the solution semigroup in the history space framework arising from an abstract version of the boundary value problem with memory
This article is focused on the solution semigroup in the history space framework arising from an abstract version of the boundary value problem with memory
$\partial_{t t} u(t)-\Delta [u(t)+\int_0^\infty \mu(s)[u(t)-u(t-s)] ds ]=0,\quad u(t)_{|\partial\Omega}=0,$
modelling linear viscoelasticity. The exponential stability of the semigroup is discussed, establishing a necessary and sufficient condition involving the memory kernel $\mu$.
2010, 9(3): 731-740
doi: 10.3934/cpaa.2010.9.731
+[Abstract](2508)
+[PDF](195.7KB)
Abstract:
Consider the elliptic system $-\Delta u=f(x,v)$, $-\Delta v+v=g(x,u)$ in a bounded smooth domain $\Omega\subset\R^N$, complemented by the boundary conditions $u=\partial_\nu v = 0$ on $\partial\Omega$. Here $f,g$ are nonnegative Carathéodory functions satisfying the growth conditions $f\leq C(1+|v|^p)$, $g\leq C(1+|u|^q)$. We find necessary and sufficient conditions on $p,q$ guaranteeing that $u,v\in L^\infty(\Omega)$ for any very weak solution $(u,v)$. In addition, our conditions guarantee the a priori estimate $||u||_\infty+||v||_\infty\leq C$, where $C$ depends only on the norm of $(u,v)$ in $L^1_\delta(\Omega)\times L^1(\Omega)$.
Let us consider the borderline in the $(p,q)$-plane between the region where all very weak solutions are bounded and the region where unbounded solutions exist. It turns out that this borderline coincides with the corresponding borderline for the system with the Neumann boundary conditions $\partial_\nu u=\partial_\nu v = 0$ on $\partial\Omega$ if $p\leq N/(N-2)$, while it coincides with the borderline for the system with the Dirichlet boundary conditions $u=v=0$ on $\partial\Omega$ if $p\geq(N+1)/(N-2)$. If $p\in (N/(N-2),(N+1)/(N-2))$ then the borderline for the Dirichlet-Neumann problem lies strictly between the borderlines for the systems with pure Neumann and pure Dirichlet boundary conditions.
Our proofs are based on some new $L^p-L^q$ estimates in weighted $L^p$-spaces.
Consider the elliptic system $-\Delta u=f(x,v)$, $-\Delta v+v=g(x,u)$ in a bounded smooth domain $\Omega\subset\R^N$, complemented by the boundary conditions $u=\partial_\nu v = 0$ on $\partial\Omega$. Here $f,g$ are nonnegative Carathéodory functions satisfying the growth conditions $f\leq C(1+|v|^p)$, $g\leq C(1+|u|^q)$. We find necessary and sufficient conditions on $p,q$ guaranteeing that $u,v\in L^\infty(\Omega)$ for any very weak solution $(u,v)$. In addition, our conditions guarantee the a priori estimate $||u||_\infty+||v||_\infty\leq C$, where $C$ depends only on the norm of $(u,v)$ in $L^1_\delta(\Omega)\times L^1(\Omega)$.
Let us consider the borderline in the $(p,q)$-plane between the region where all very weak solutions are bounded and the region where unbounded solutions exist. It turns out that this borderline coincides with the corresponding borderline for the system with the Neumann boundary conditions $\partial_\nu u=\partial_\nu v = 0$ on $\partial\Omega$ if $p\leq N/(N-2)$, while it coincides with the borderline for the system with the Dirichlet boundary conditions $u=v=0$ on $\partial\Omega$ if $p\geq(N+1)/(N-2)$. If $p\in (N/(N-2),(N+1)/(N-2))$ then the borderline for the Dirichlet-Neumann problem lies strictly between the borderlines for the systems with pure Neumann and pure Dirichlet boundary conditions.
Our proofs are based on some new $L^p-L^q$ estimates in weighted $L^p$-spaces.
2010, 9(3): 741-750
doi: 10.3934/cpaa.2010.9.741
+[Abstract](2168)
+[PDF](175.0KB)
Abstract:
We prove the existence of radially symmetric ground--states for the system of Nonlinear Schrödinger equations
We prove the existence of radially symmetric ground--states for the system of Nonlinear Schrödinger equations
$-\Delta u+ u=f(u)+\beta u v^2$ in $R^3,$
$-\Delta v+ v=g(v)+\beta u^2 v$ in $R^3,$
under very weak assumptions on the two nonlinearities $f$ and $g$. In particular, no "Ambrosetti--Rabinowitz" condition is required.
2010, 9(3): 751-760
doi: 10.3934/cpaa.2010.9.751
+[Abstract](2120)
+[PDF](159.5KB)
Abstract:
In this paper we provide uniform estimates for $\lambda^{*}(N, \Omega, q, p, h, W)$ of nonlinear elliptic equations $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$ where $W$ may change sign. We use a variational technique. Still few general results are known for this type of estimates except [6] of Gazzola and Malchiodi, which provide uniform estimates for the extremal value in case $-\Delta u=\lambda (1+u)^{p}$.
In this paper we provide uniform estimates for $\lambda^{*}(N, \Omega, q, p, h, W)$ of nonlinear elliptic equations $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$ where $W$ may change sign. We use a variational technique. Still few general results are known for this type of estimates except [6] of Gazzola and Malchiodi, which provide uniform estimates for the extremal value in case $-\Delta u=\lambda (1+u)^{p}$.
2010, 9(3): 761-778
doi: 10.3934/cpaa.2010.9.761
+[Abstract](1937)
+[PDF](239.6KB)
Abstract:
We consider the sub- or supercritical Neumann elliptic problem $-\Delta u + \mu u = u^{\frac{N + 2}{N - 2} + \varepsilon}, u > 0 $ in $\Omega; \frac{\partial u}{\partial n} = 0 $ on $\partial \Omega, \Omega$ being a smooth bounded domain in $R^N, N \ge 4, \mu > 0 $ and $\varepsilon \ne 0$. Let $H(x)$ denote the mean curvature at $x$. We show that for slightly sub- or supercritical problem, if $\varepsilon \min_{x \in \partial\Omega} H(x) > 0$ then there always exist arbitrarily many solutions which blow up at the least curved part of the boundary as $\varepsilon$ goes to zero.
We consider the sub- or supercritical Neumann elliptic problem $-\Delta u + \mu u = u^{\frac{N + 2}{N - 2} + \varepsilon}, u > 0 $ in $\Omega; \frac{\partial u}{\partial n} = 0 $ on $\partial \Omega, \Omega$ being a smooth bounded domain in $R^N, N \ge 4, \mu > 0 $ and $\varepsilon \ne 0$. Let $H(x)$ denote the mean curvature at $x$. We show that for slightly sub- or supercritical problem, if $\varepsilon \min_{x \in \partial\Omega} H(x) > 0$ then there always exist arbitrarily many solutions which blow up at the least curved part of the boundary as $\varepsilon$ goes to zero.
2010, 9(3): 779-812
doi: 10.3934/cpaa.2010.9.779
+[Abstract](2011)
+[PDF](421.5KB)
Abstract:
An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in $\mathcal X$. Let $\rho$ be an admissible function on RD-space $\mathcal X$. The authors first introduce the localized spaces $BMO_\rho(\mathcal X)$ and $BLO_\rho(\mathcal X)$ and establish their basic properties, including the John-Nirenberg inequality for $BMO_\rho(\mathcal X)$, several equivalent characterizations for $BLO_\rho(\mathcal X)$, and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to $\rho$, and the Littlewood-Paley $g$-function associated to $\rho$, where the Littlewood-Paley $g$-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on $\mathbb R^d$, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.
An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in $\mathcal X$. Let $\rho$ be an admissible function on RD-space $\mathcal X$. The authors first introduce the localized spaces $BMO_\rho(\mathcal X)$ and $BLO_\rho(\mathcal X)$ and establish their basic properties, including the John-Nirenberg inequality for $BMO_\rho(\mathcal X)$, several equivalent characterizations for $BLO_\rho(\mathcal X)$, and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to $\rho$, and the Littlewood-Paley $g$-function associated to $\rho$, where the Littlewood-Paley $g$-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on $\mathbb R^d$, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.
2010, 9(3): 813-818
doi: 10.3934/cpaa.2010.9.813
+[Abstract](2385)
+[PDF](113.1KB)
Abstract:
We prove local in time existence and uniqueness of solutions of the ideal inhomogeneous magnetohydrodynamic equations.
We prove local in time existence and uniqueness of solutions of the ideal inhomogeneous magnetohydrodynamic equations.
2010, 9(3): 819-837
doi: 10.3934/cpaa.2010.9.819
+[Abstract](1997)
+[PDF](249.0KB)
Abstract:
This paper studies the entire solutions of a class of $p$-Laplace equation
This paper studies the entire solutions of a class of $p$-Laplace equation
- div$(|\nabla u|^{p-2}\nabla u)+a(x)W^'(u(x,y))=0, (x,y)\in R^2$
in the case $p>2$. where $a:\mathbf{R}\rightarrow \mathbf{R_{+}}$ is a periodic, positive function and $W:\mathbf{R}\rightarrow \mathbf{R}$ is a non-negative $C^{2}$ function. We look for the entire solutions of the above equation with asymptotic conditions $u(x,y)\rightarrow \pm 1 $ as $x\rightarrow\pm\infty$ uniformly with respect to $y\in \mathbf{R}$. Via variational methods we find layered solutions which depend on both x and y, i.e., solutions which do not exhibit one dimensional symmetries.
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