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1534-0392
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Communications on Pure & Applied Analysis
June 2007 , Volume 6 , Issue 2
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2007, 6(2): 311-333
doi: 10.3934/cpaa.2007.6.311
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Abstract:
We consider an Ackerberg-O'Malley singular perturbation problem $\epsilon y'' + f(x,\epsilon)y' + g(x,\epsilon)y=0, y(a)=A, y(b)=B$ with a single turning point and study the nature of resonant solutions $y=\varphi(x,\epsilon)$, i.e. solutions for which $\varphi(x,\epsilon)$ tends to a nontrivial solution of $f(x,0)y'+ g(x,0)y=0$ as $\epsilon\to 0$. Many techniques have been applied to the study of this problem (WKBJ, invariant manifolds, asymptotic methods, spectral methods, variational techniques) and they have been successful in characterizing these resonant solutions when $f(x,0)$ has a simple zero at the origin. When the order of zero is higher the increase in complexity of the problem is significant. The existence of a nonzero formal power series solution is no longer necessary for resonance and resonant solutions are in general not smooth at the origin. We apply the method of blow up to study the nature of resonant solutions in this setting, using techniques from invariant manifold theory and planar singular perturbation theory. The main result is the sufficiency of the Matkowsky condition for turning points of arbitrary order (based on Gevrey-asymptotics), but we also give a characterization of the location of the boundary layer in resonant solutions.
We consider an Ackerberg-O'Malley singular perturbation problem $\epsilon y'' + f(x,\epsilon)y' + g(x,\epsilon)y=0, y(a)=A, y(b)=B$ with a single turning point and study the nature of resonant solutions $y=\varphi(x,\epsilon)$, i.e. solutions for which $\varphi(x,\epsilon)$ tends to a nontrivial solution of $f(x,0)y'+ g(x,0)y=0$ as $\epsilon\to 0$. Many techniques have been applied to the study of this problem (WKBJ, invariant manifolds, asymptotic methods, spectral methods, variational techniques) and they have been successful in characterizing these resonant solutions when $f(x,0)$ has a simple zero at the origin. When the order of zero is higher the increase in complexity of the problem is significant. The existence of a nonzero formal power series solution is no longer necessary for resonance and resonant solutions are in general not smooth at the origin. We apply the method of blow up to study the nature of resonant solutions in this setting, using techniques from invariant manifold theory and planar singular perturbation theory. The main result is the sufficiency of the Matkowsky condition for turning points of arbitrary order (based on Gevrey-asymptotics), but we also give a characterization of the location of the boundary layer in resonant solutions.
2007, 6(2): 335-366
doi: 10.3934/cpaa.2007.6.335
+[Abstract](2711)
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Abstract:
The main scope of this article is to define the concept of principal eigenvalue for fully non linear second order operators in bounded domains that are elliptic, homogenous with lower order terms. In particular we prove maximum and comparison principle, Hölder and Lipschitz regularity. This leads to the existence of a first eigenvalue and eigenfunction and to the existence of solutions of Dirichlet problems within this class of operators.
The main scope of this article is to define the concept of principal eigenvalue for fully non linear second order operators in bounded domains that are elliptic, homogenous with lower order terms. In particular we prove maximum and comparison principle, Hölder and Lipschitz regularity. This leads to the existence of a first eigenvalue and eigenfunction and to the existence of solutions of Dirichlet problems within this class of operators.
2007, 6(2): 367-387
doi: 10.3934/cpaa.2007.6.367
+[Abstract](2800)
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Abstract:
This paper deals with a singular integro-differential PDE system describing phase transitions in terms of nonlinear evolution equations for micromotions and for the entropy. The model is derived from a non-convex free energy functional, possibly accounting for thermal memory effects. After recovering a global existence result for a related initial and boundary value problem, the long-time behaviour of the solutions is investigated. In particular, it is proved that the elements of the $\omega$-limit set (i.e. the cluster points) of the solution trajectories solve the steady state system which is naturally associated to the evolution problem.
This paper deals with a singular integro-differential PDE system describing phase transitions in terms of nonlinear evolution equations for micromotions and for the entropy. The model is derived from a non-convex free energy functional, possibly accounting for thermal memory effects. After recovering a global existence result for a related initial and boundary value problem, the long-time behaviour of the solutions is investigated. In particular, it is proved that the elements of the $\omega$-limit set (i.e. the cluster points) of the solution trajectories solve the steady state system which is naturally associated to the evolution problem.
2007, 6(2): 389-410
doi: 10.3934/cpaa.2007.6.389
+[Abstract](2307)
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Abstract:
This paper discusses viscosity solutions of general Hamilton-Jacobi equations in the time periodic case. Existence results are presented under usual hypotheses. The main idea is to reduce the study of time periodic problems to the study of stationary problems obtained by averaging the source term over a period. These results hold also for almost-periodic viscosity solutions.
This paper discusses viscosity solutions of general Hamilton-Jacobi equations in the time periodic case. Existence results are presented under usual hypotheses. The main idea is to reduce the study of time periodic problems to the study of stationary problems obtained by averaging the source term over a period. These results hold also for almost-periodic viscosity solutions.
2007, 6(2): 411-428
doi: 10.3934/cpaa.2007.6.411
+[Abstract](2138)
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Abstract:
In this paper we prove some improved Hardy type inequalities with respect to the Gaussian measure. We show that they are strictly related to the well-known Gross Logarithmic Sobolev inequality. Some applications to elliptic P.D.E.'s are also given.
In this paper we prove some improved Hardy type inequalities with respect to the Gaussian measure. We show that they are strictly related to the well-known Gross Logarithmic Sobolev inequality. Some applications to elliptic P.D.E.'s are also given.
2007, 6(2): 429-440
doi: 10.3934/cpaa.2007.6.429
+[Abstract](2196)
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Abstract:
In this paper, we discuss the Liouville integrability of the Burgers-Korteweg-de Vries equation under certain parametric condition. An approximate solution is obtained by means of the Adomian decomposition method.
In this paper, we discuss the Liouville integrability of the Burgers-Korteweg-de Vries equation under certain parametric condition. An approximate solution is obtained by means of the Adomian decomposition method.
2007, 6(2): 441-451
doi: 10.3934/cpaa.2007.6.441
+[Abstract](1991)
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Abstract:
Multiple positive solutions for semi-positone Hammerstein integral equations are investigated. This provides a general framework for studying the existence of positive solutions for some semi-positone boundary value problems which can be transferred into the Hammerstein integral equations. We apply the new results on the existence of one or two positive solutions of the semi-positone integral equations to treat the semi-positone conjugate boundary value problems (BVPs) as illustrations although there are many other BVPs which can be treated in the similar way. We provide two explicit examples of semi-positone conjugate BVPs to exhibit applications of our results on the existence of one or two positive solutions.
Multiple positive solutions for semi-positone Hammerstein integral equations are investigated. This provides a general framework for studying the existence of positive solutions for some semi-positone boundary value problems which can be transferred into the Hammerstein integral equations. We apply the new results on the existence of one or two positive solutions of the semi-positone integral equations to treat the semi-positone conjugate boundary value problems (BVPs) as illustrations although there are many other BVPs which can be treated in the similar way. We provide two explicit examples of semi-positone conjugate BVPs to exhibit applications of our results on the existence of one or two positive solutions.
2007, 6(2): 453-464
doi: 10.3934/cpaa.2007.6.453
+[Abstract](2572)
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Abstract:
We consider a system of Euler-Lagrange equations associated with the weighted Hardy-Littlewood-Sobolev inequality in $R^n$. We demonstrate that the positive solutions of the system of Euler-Lagrange equations are asymptotic to certain forms of limit around the center and near infinity, respectively. The results are proven using the optimal integrability conditions for the positive solutions of the system of equations.
We consider a system of Euler-Lagrange equations associated with the weighted Hardy-Littlewood-Sobolev inequality in $R^n$. We demonstrate that the positive solutions of the system of Euler-Lagrange equations are asymptotic to certain forms of limit around the center and near infinity, respectively. The results are proven using the optimal integrability conditions for the positive solutions of the system of equations.
2007, 6(2): 465-480
doi: 10.3934/cpaa.2007.6.465
+[Abstract](2372)
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Abstract:
In this paper, we consider an initial-boundary problem for a fourth-order nonlinear parabolic equations. The problem as a model shares the scaling properties of the thin film equation, or as a model arises in epitaxial growth of nanoscale thin films. Our approach lies in the combination of the energy techniques with some methods based on the framework of Campanato spaces. Based on the uniform estimates for the approximate solutions, we establish the existence of weak solutions.
In this paper, we consider an initial-boundary problem for a fourth-order nonlinear parabolic equations. The problem as a model shares the scaling properties of the thin film equation, or as a model arises in epitaxial growth of nanoscale thin films. Our approach lies in the combination of the energy techniques with some methods based on the framework of Campanato spaces. Based on the uniform estimates for the approximate solutions, we establish the existence of weak solutions.
2007, 6(2): 481-486
doi: 10.3934/cpaa.2007.6.481
+[Abstract](3173)
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Abstract:
In this note, we establish a general result on the existence of global attractors for semigroups $S(t)$ of operators acting on a Banach space $\mathcal X$, where the strong continuity $S(t)\in C(\mathcal X,\mathcal X)$ is replaced by the much weaker requirement that $S(t)$ be a closed map.
In this note, we establish a general result on the existence of global attractors for semigroups $S(t)$ of operators acting on a Banach space $\mathcal X$, where the strong continuity $S(t)\in C(\mathcal X,\mathcal X)$ is replaced by the much weaker requirement that $S(t)$ be a closed map.
2007, 6(2): 487-503
doi: 10.3934/cpaa.2007.6.487
+[Abstract](2227)
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Abstract:
This paper deals with the blow-up properties of solutions to a degenerate parabolic system coupled via nonlinear boundary flux. Firstly, we construct the self-similar supersolution and subsolution to obtain the critical global existence curve. Secondly, we establish the precise blow-up rate estimates for solutions which blow up in a finite time. Finally, we investigate the localization of blow-up points. The critical curve of Fujita type is conjectured with the aid of some new results.
This paper deals with the blow-up properties of solutions to a degenerate parabolic system coupled via nonlinear boundary flux. Firstly, we construct the self-similar supersolution and subsolution to obtain the critical global existence curve. Secondly, we establish the precise blow-up rate estimates for solutions which blow up in a finite time. Finally, we investigate the localization of blow-up points. The critical curve of Fujita type is conjectured with the aid of some new results.
2007, 6(2): 505-520
doi: 10.3934/cpaa.2007.6.505
+[Abstract](1834)
+[PDF](187.9KB)
Abstract:
We consider a scalar conservation law of Burgers' type: $u_t+(u^2/2)_x = \varepsilon u_{x x}-\delta u_{x x x}+\gamma u_{x x x x x}$ ($(x, t)\in \mathbf R \times$ (0, ∞)). We prove that if $\varepsilon$, $\delta=\delta(\varepsilon)$, $\gamma=\gamma(\varepsilon)$ tend to $0$, then for $q\in (2, 16/5)$, the sequence {$u^\varepsilon$} of solutions converges in $L^k(0, T^\star; L^p(\mathbf R)) (k< $ ∞, p$<$q) to the unique entropy solution $u\in L^\infty (0, T^\star; L^q(\mathbf R))$ to the inviscid Burgers equation $u_t+(u^2/2)_x = 0$. More precisely we show that, under the condition $\delta=O(\varepsilon^{3/(3-q)})$ and $\gamma=O(\varepsilon^4$ $\delta^{(8q-7)/9})$ for $q\in(2,3)$ or $\delta=O(\varepsilon^{12/(19-4q)}$ $\gamma^{3/(19-4q)})$ and $\gamma=O(\varepsilon^{4}$ $\delta^{(8q-7)/9})$ for $q\in[3,16/5)$, the limit of the sequence is the entropy solution. Moreover if we assume the uniform boundedness of {$u^\varepsilon(\cdot,t)$} in $L^q(\mathbf R)$ for $q>2$, the condition $\delta=o(\varepsilon^3)$ and $\gamma=o(\varepsilon^4\delta)$ is sufficient to establish the conclusion. We derive new a priori estimates which enable to use the technique of the compensated compactness, the Young measures and the entropy measure-valued solutions.
We consider a scalar conservation law of Burgers' type: $u_t+(u^2/2)_x = \varepsilon u_{x x}-\delta u_{x x x}+\gamma u_{x x x x x}$ ($(x, t)\in \mathbf R \times$ (0, ∞)). We prove that if $\varepsilon$, $\delta=\delta(\varepsilon)$, $\gamma=\gamma(\varepsilon)$ tend to $0$, then for $q\in (2, 16/5)$, the sequence {$u^\varepsilon$} of solutions converges in $L^k(0, T^\star; L^p(\mathbf R)) (k< $ ∞, p$<$q) to the unique entropy solution $u\in L^\infty (0, T^\star; L^q(\mathbf R))$ to the inviscid Burgers equation $u_t+(u^2/2)_x = 0$. More precisely we show that, under the condition $\delta=O(\varepsilon^{3/(3-q)})$ and $\gamma=O(\varepsilon^4$ $\delta^{(8q-7)/9})$ for $q\in(2,3)$ or $\delta=O(\varepsilon^{12/(19-4q)}$ $\gamma^{3/(19-4q)})$ and $\gamma=O(\varepsilon^{4}$ $\delta^{(8q-7)/9})$ for $q\in[3,16/5)$, the limit of the sequence is the entropy solution. Moreover if we assume the uniform boundedness of {$u^\varepsilon(\cdot,t)$} in $L^q(\mathbf R)$ for $q>2$, the condition $\delta=o(\varepsilon^3)$ and $\gamma=o(\varepsilon^4\delta)$ is sufficient to establish the conclusion. We derive new a priori estimates which enable to use the technique of the compensated compactness, the Young measures and the entropy measure-valued solutions.
2007, 6(2): 521-529
doi: 10.3934/cpaa.2007.6.521
+[Abstract](2235)
+[PDF](137.2KB)
Abstract:
By a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems Δ$u\pm|\nabla u|^q=b(x)e^u,\ x \in \Omega, \ u|_{\partial \Omega}=\infty, $ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$, $q \geq 0$, $b$ is non-negative in $\Omega$ and singular on $\partial\Omega$.
By a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems Δ$u\pm|\nabla u|^q=b(x)e^u,\ x \in \Omega, \ u|_{\partial \Omega}=\infty, $ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$, $q \geq 0$, $b$ is non-negative in $\Omega$ and singular on $\partial\Omega$.
2007, 6(2): 531-540
doi: 10.3934/cpaa.2007.6.531
+[Abstract](2485)
+[PDF](140.3KB)
Abstract:
In this paper, we study positive solution of the following system of quasilinear elliptic equations
In this paper, we study positive solution of the following system of quasilinear elliptic equations
div$(|\nabla u|^{p-2}\nabla u)=u^{m_1}v^{n_1},$ in $\Omega$
div$(|\nabla v|^{q-2}\nabla v)=u^{m_2}v^{n_2},$ in $\Omega,$ $\qquad\qquad\qquad\qquad$ (0.1)
where $m_1>p-1,n_2>q-1, m_2,n_1>0$, and $\Omega\subset R^N$ is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: $u=\lambda, v=\mu$ or $u=v=+\infty$ or $u=+\infty, v=\mu$ on $\partial\Omega$, where $\lambda, \mu>0$. Under several hypotheses on the parameters $m_1,n_1,m_2,n_2$, we show that the existence of positive solutions. We further provide the asymptotic behaviors of the solutions near $\partial\Omega$. Some more general related problems are also studied.
2007, 6(2): 541-547
doi: 10.3934/cpaa.2007.6.541
+[Abstract](2301)
+[PDF](113.4KB)
Abstract:
This article deals with the Poincaré mapping of some nonautonomous differential systems by reflecting function. The results are applied to discuss the existence and stability of the periodic solutions of these systems.
This article deals with the Poincaré mapping of some nonautonomous differential systems by reflecting function. The results are applied to discuss the existence and stability of the periodic solutions of these systems.
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