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1534-0392
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Communications on Pure & Applied Analysis
September 2009 , Volume 8 , Issue 5
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2009, 8(5): 1469-1492
doi: 10.3934/cpaa.2009.8.1469
+[Abstract](2211)
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Abstract:
We investigate the spectral properties of a class of weighted shift semigroups $(\mathcal{U}(t))_{t \geq 0}$ associated to abstract transport equations with a Lipschitz continuous vector field $\mathcal{F}$ and no--reentry boundary conditions. Generalizing the results of [25], we prove that the semigroup $(\mathcal{U}(t))_{t \geq 0}$ admits a canonical decomposition into three $C_0$-semigroups $(\mathcal{U}_1(t))_{t \geq 0}$, $(\mathcal{U}_2(t))_{t \geq 0}$ and $(\mathcal{U}_3(t))_{t \geq 0}$ with independent dynamics. A complete description of the spectra of the semigroups $(\mathcal{U}_i(t))_{t \geq 0}$ and their generators $\mathcal{T}_i$, $i=1,2$ is given. In particular, we prove that the spectrum of $\mathcal{T}_i$ is a left-half plane and that the Spectral Mapping Theorem holds: $\mathfrak{S}(\mathcal{U}_i(t))=\exp$ {$t \mathfrak{S}(\mathcal{T}_i)$}, $i=1,2$. Moreover, the semigroup $(\mathcal{U}_3(t))_{t \geq 0}$ extends to a $C_0$-group and its spectral properties are investigated by means of abstract results from positive semigroups theory. The properties of the flow associated to $\mathcal{F}$ are particularly relevant here and we investigate separately the cases of periodic and aperiodic flows. In particular, we show that, for periodic flow, the Spectral Mapping Theorem fails in general but $(\mathcal{U}_3(t))_{t \geq 0}$ and its generator $\mathcal{T}_3$ satisfy the so-called Annular Hull Theorem. We illustrate our results with various examples taken from collisionless kinetic theory.
We investigate the spectral properties of a class of weighted shift semigroups $(\mathcal{U}(t))_{t \geq 0}$ associated to abstract transport equations with a Lipschitz continuous vector field $\mathcal{F}$ and no--reentry boundary conditions. Generalizing the results of [25], we prove that the semigroup $(\mathcal{U}(t))_{t \geq 0}$ admits a canonical decomposition into three $C_0$-semigroups $(\mathcal{U}_1(t))_{t \geq 0}$, $(\mathcal{U}_2(t))_{t \geq 0}$ and $(\mathcal{U}_3(t))_{t \geq 0}$ with independent dynamics. A complete description of the spectra of the semigroups $(\mathcal{U}_i(t))_{t \geq 0}$ and their generators $\mathcal{T}_i$, $i=1,2$ is given. In particular, we prove that the spectrum of $\mathcal{T}_i$ is a left-half plane and that the Spectral Mapping Theorem holds: $\mathfrak{S}(\mathcal{U}_i(t))=\exp$ {$t \mathfrak{S}(\mathcal{T}_i)$}, $i=1,2$. Moreover, the semigroup $(\mathcal{U}_3(t))_{t \geq 0}$ extends to a $C_0$-group and its spectral properties are investigated by means of abstract results from positive semigroups theory. The properties of the flow associated to $\mathcal{F}$ are particularly relevant here and we investigate separately the cases of periodic and aperiodic flows. In particular, we show that, for periodic flow, the Spectral Mapping Theorem fails in general but $(\mathcal{U}_3(t))_{t \geq 0}$ and its generator $\mathcal{T}_3$ satisfy the so-called Annular Hull Theorem. We illustrate our results with various examples taken from collisionless kinetic theory.
2009, 8(5): 1493-1501
doi: 10.3934/cpaa.2009.8.1493
+[Abstract](1976)
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Abstract:
We study the number of limit cycles (isolated periodic solutions in the set of all periodic solutions) for the generalized Abel equation $x'=a(t)x^{n_a}+b(t)x^{n_b}+c(t)x^{n_c}+d(t)x$, where $n_a > n_b > n_c > 1$, $a(t),b(t),c(t), d(t)$ are $2\pi$-periodic continuous functions, and two of $a(t),b(t),c(t)$ have definite sign.
We obtain examples with at least seven limit cycles, and some sufficient conditions for the equation to have at most one or at most two positive limit cycles.
We study the number of limit cycles (isolated periodic solutions in the set of all periodic solutions) for the generalized Abel equation $x'=a(t)x^{n_a}+b(t)x^{n_b}+c(t)x^{n_c}+d(t)x$, where $n_a > n_b > n_c > 1$, $a(t),b(t),c(t), d(t)$ are $2\pi$-periodic continuous functions, and two of $a(t),b(t),c(t)$ have definite sign.
We obtain examples with at least seven limit cycles, and some sufficient conditions for the equation to have at most one or at most two positive limit cycles.
2009, 8(5): 1503-1520
doi: 10.3934/cpaa.2009.8.1503
+[Abstract](3063)
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Abstract:
The equations of an incompressible, homogeneous fluid occupying a bounded domain in $\mathbb R^3$ are considered.
The stress tensor has a general polynomial dependence on the symmetric velocity gradient. The goal is to estimate the dimension of the global attractor in terms of relevant physical constants.
The equations of an incompressible, homogeneous fluid occupying a bounded domain in $\mathbb R^3$ are considered.
The stress tensor has a general polynomial dependence on the symmetric velocity gradient. The goal is to estimate the dimension of the global attractor in terms of relevant physical constants.
2009, 8(5): 1521-1539
doi: 10.3934/cpaa.2009.8.1521
+[Abstract](2116)
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Abstract:
We study the local well-posedness of the initial-value problem for the nonlinear generalized Boussinesq equation with data in $H^s(\mathbb R^n) \times H^s(\mathbb R^n)$, $s\geq 0$. Under some assumption on the nonlinearity $f$, local existence results are proved for $H^s(\mathbb R^n)$-solutions using an auxiliary space of Lebesgue type. Furthermore, under certain hypotheses on $s$, $n$ and the growth rate of $f$ these auxiliary conditions can be eliminated.
We study the local well-posedness of the initial-value problem for the nonlinear generalized Boussinesq equation with data in $H^s(\mathbb R^n) \times H^s(\mathbb R^n)$, $s\geq 0$. Under some assumption on the nonlinearity $f$, local existence results are proved for $H^s(\mathbb R^n)$-solutions using an auxiliary space of Lebesgue type. Furthermore, under certain hypotheses on $s$, $n$ and the growth rate of $f$ these auxiliary conditions can be eliminated.
2009, 8(5): 1541-1554
doi: 10.3934/cpaa.2009.8.1541
+[Abstract](2233)
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Abstract:
In this paper, we apply a cross-constrained variational approach for the nonlinear Klein-Gordon equations with an inverse square potential in three space dimensions (which is a representative of the class of equations of interest) based on the relationship between a type of cross-constrained variational problem and energy. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we first derive a sharp threshold for global existence and blow-up of solutions to the Cauchy problem for the equations under study. On the other hand, we get an answer of the question: how small are the initial data, the global solutions exist?
In this paper, we apply a cross-constrained variational approach for the nonlinear Klein-Gordon equations with an inverse square potential in three space dimensions (which is a representative of the class of equations of interest) based on the relationship between a type of cross-constrained variational problem and energy. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we first derive a sharp threshold for global existence and blow-up of solutions to the Cauchy problem for the equations under study. On the other hand, we get an answer of the question: how small are the initial data, the global solutions exist?
2009, 8(5): 1555-1575
doi: 10.3934/cpaa.2009.8.1555
+[Abstract](2084)
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Abstract:
We prove that the weakly damped Davey-Stewartson system (DS), considered as an infinite-dimensional dynamical system in $H^1(\mathbb R^2)$, has a compact global attractor that is actually a compact subset of $H^2(\mathbb R^2)$.
We prove that the weakly damped Davey-Stewartson system (DS), considered as an infinite-dimensional dynamical system in $H^1(\mathbb R^2)$, has a compact global attractor that is actually a compact subset of $H^2(\mathbb R^2)$.
2009, 8(5): 1577-1583
doi: 10.3934/cpaa.2009.8.1577
+[Abstract](2131)
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Abstract:
We show an optimal Hölder continuity for the solutions of the equation $- \Delta_A u=\mu$ provided that $\mu (B(x,r)) \leq C r^{n-1} $ for any ball $B(x,r)\subset \Omega$, with $r\leq 1$.
We show an optimal Hölder continuity for the solutions of the equation $- \Delta_A u=\mu$ provided that $\mu (B(x,r)) \leq C r^{n-1} $ for any ball $B(x,r)\subset \Omega$, with $r\leq 1$.
2009, 8(5): 1585-1606
doi: 10.3934/cpaa.2009.8.1585
+[Abstract](2145)
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Abstract:
In this paper, we construct small amplitude quasi-periodic solutions for one dimensional nonlinear Schrödinger equation
In this paper, we construct small amplitude quasi-periodic solutions for one dimensional nonlinear Schrödinger equation
i$u_t=u_{x x}-mu-f(\beta t,x)|u|^2 u,$
with the boundary conditions
$u(t,0)=u(t,a\pi)=0, \ -\infty < t < \infty,$
where $m$ is real and $f(\beta t,x)$ is real analytic and quasi-periodic on $t$ satisfying the non-degeneracy condition
$\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^Tf(\beta t,x)dt\equiv f_0=$ const., $\quad 0\ne f_0 \in\mathbb R,$
with $\beta\in\mathbb R^b$ a fixed Diophantine vector.
2009, 8(5): 1607-1618
doi: 10.3934/cpaa.2009.8.1607
+[Abstract](1837)
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Abstract:
We prove existence and nonexistence of nonnegative entire large solutions for the semilinear elliptic equation $\Delta u = p(x)f(u) + q(x)g(u)$ in which $f$ and $g$ are nondecreasing and vanish at the origin. The locally Hölder continuous functions $p$ and $q$ are nonnegative. We extend results previously obtained for special cases of $f$ and $g$.
We prove existence and nonexistence of nonnegative entire large solutions for the semilinear elliptic equation $\Delta u = p(x)f(u) + q(x)g(u)$ in which $f$ and $g$ are nondecreasing and vanish at the origin. The locally Hölder continuous functions $p$ and $q$ are nonnegative. We extend results previously obtained for special cases of $f$ and $g$.
2009, 8(5): 1619-1636
doi: 10.3934/cpaa.2009.8.1619
+[Abstract](2340)
+[PDF](227.2KB)
Abstract:
In this work, we study the Dirichlet problem for a class of semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy's inequality, and the nonlinearity is controlled by Sobolev's inequality. We prove the existence of a nontrivial solution for a homogenous Dirichlet problem.
In this work, we study the Dirichlet problem for a class of semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy's inequality, and the nonlinearity is controlled by Sobolev's inequality. We prove the existence of a nontrivial solution for a homogenous Dirichlet problem.
2009, 8(5): 1637-1645
doi: 10.3934/cpaa.2009.8.1637
+[Abstract](2124)
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Abstract:
Let $\pi = (\Phi, \sigma)$ be an exponentially bounded, strongly continuous cocycle over a continuous semiflow $\sigma$. We prove that $\pi = (\Phi, \sigma)$ is uniformly exponentially stable if and only if there exist $T>0$ and $c \in(0,1)$, such that for each $\theta \in \Theta$ and $x \in X$ there exists $\tau_{\theta,x} \in (0,T]$ with the property that
Let $\pi = (\Phi, \sigma)$ be an exponentially bounded, strongly continuous cocycle over a continuous semiflow $\sigma$. We prove that $\pi = (\Phi, \sigma)$ is uniformly exponentially stable if and only if there exist $T>0$ and $c \in(0,1)$, such that for each $\theta \in \Theta$ and $x \in X$ there exists $\tau_{\theta,x} \in (0,T]$ with the property that
$||\Phi(\theta, \tau_{\theta,x})x|| \leq c||x||.$
As a consequence of the above result we obtain generalizations, in both continuous-time and discrete-time, of the the well-known theorems of Datko-Pazy, Rolewicz and Zabczyk for an exponentially bounded, strongly continuous cocycle over a semiflow $\sigma$. A version of the above theorems for the case of the exponential instability is also obtained.
2009, 8(5): 1647-1668
doi: 10.3934/cpaa.2009.8.1647
+[Abstract](2222)
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Abstract:
In this paper we prove the global existence and uniqueness of classical solution to the Boltzmann equation with external force near a stationary solution for hard potentials. The optimal time decay to the stationary solution is also obtained.
In this paper we prove the global existence and uniqueness of classical solution to the Boltzmann equation with external force near a stationary solution for hard potentials. The optimal time decay to the stationary solution is also obtained.
2009, 8(5): 1669-1688
doi: 10.3934/cpaa.2009.8.1669
+[Abstract](2107)
+[PDF](277.3KB)
Abstract:
In this paper we study a moving boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. The problem contains two coupled reaction-diffusion equations and one elliptic equation defined on a strip-like domain in $R^n$, with one part of the boundary moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation, with the surface tension effect taken into consideration. Local existence and asymptotic behavior of solutions to this problem are investigated. The analysis is based on the employment of the functional analysis method combing with the well-posedness and geometric theory for parabolic differential equations in Banach spaces.
In this paper we study a moving boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. The problem contains two coupled reaction-diffusion equations and one elliptic equation defined on a strip-like domain in $R^n$, with one part of the boundary moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation, with the surface tension effect taken into consideration. Local existence and asymptotic behavior of solutions to this problem are investigated. The analysis is based on the employment of the functional analysis method combing with the well-posedness and geometric theory for parabolic differential equations in Banach spaces.
2009, 8(5): 1689-1707
doi: 10.3934/cpaa.2009.8.1689
+[Abstract](1740)
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Abstract:
In this paper we formulate a mathematical model using the predator-prey paradigm for study and analysis of the relationships between a complex four-dimensional system consisting of agriculture, two industries and the ecosphere. Mathematical analysis of the model equations with regard to invariance of nonnegativity, dissipativity (i.e. eventually boundedness) of solutions, and a complete local and global analysis of the system’s equilibria are done. We establish the existence of a positive interior equilibrium and give conditions under which such a system exhibits uniform persistence. The analysis is carried out both analytically and numerically.
In this paper we formulate a mathematical model using the predator-prey paradigm for study and analysis of the relationships between a complex four-dimensional system consisting of agriculture, two industries and the ecosphere. Mathematical analysis of the model equations with regard to invariance of nonnegativity, dissipativity (i.e. eventually boundedness) of solutions, and a complete local and global analysis of the system’s equilibria are done. We establish the existence of a positive interior equilibrium and give conditions under which such a system exhibits uniform persistence. The analysis is carried out both analytically and numerically.
2009, 8(5): 1709-1723
doi: 10.3934/cpaa.2009.8.1709
+[Abstract](2248)
+[PDF](203.1KB)
Abstract:
We consider the class of radial solutions of semilinear equations $\Delta^2 u=\lambda f(u)$ in the unit ball of $\mathbb R^N$. It is the class of stable solutions which includes minimal solutions and extremal solution. We establish the regularity of this extremal solution for $N\leq 9$. Our regularity results do not depend on the specific nonlinearity $f$.
We consider the class of radial solutions of semilinear equations $\Delta^2 u=\lambda f(u)$ in the unit ball of $\mathbb R^N$. It is the class of stable solutions which includes minimal solutions and extremal solution. We establish the regularity of this extremal solution for $N\leq 9$. Our regularity results do not depend on the specific nonlinearity $f$.
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