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1534-0392
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Communications on Pure & Applied Analysis
November 2009 , Volume 8 , Issue 6
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2009, 8(6): 1725-1743
doi: 10.3934/cpaa.2009.8.1725
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Abstract:
We consider the initial value problem for the $L^2$-critical defocusing Hartree equation in $\mathbb{R}^n$, $n\ge 3$. We show that the problem is globally well posed in $H^s(\mathbb{R}^n)$ when $ 1 > s > \frac{2(n-2)}{3n-4}$. We use the "I-method" following [9] combined with a local in time Morawetz estimate for the smoothed out solution $I\phi$ as in [7].
We consider the initial value problem for the $L^2$-critical defocusing Hartree equation in $\mathbb{R}^n$, $n\ge 3$. We show that the problem is globally well posed in $H^s(\mathbb{R}^n)$ when $ 1 > s > \frac{2(n-2)}{3n-4}$. We use the "I-method" following [9] combined with a local in time Morawetz estimate for the smoothed out solution $I\phi$ as in [7].
2009, 8(6): 1745-1758
doi: 10.3934/cpaa.2009.8.1745
+[Abstract](3235)
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Abstract:
We consider the system
We consider the system
$-\varepsilon^{2} \Delta u +W(x)u=Q_{u}(u,v)$ in $\mathbb{R}^N,$
$-\varepsilon^{2} \Delta v +V(x)v=Q_{v}(u,v)$ in $\mathbb{R}^N, $
$u,v \in H^{1}(\mathbb{R}^N),u(x),v(x)>0$ for each $x \in \mathbb{R}^N, $
where $\varepsilon>0$, $W$ and $V$ are positive potentials and $Q$ is a homogeneous function with subcritical growth. We relate the number of solutions with the topology of the set where $W$ and $V$ attain their minimum values. In the proof we apply Ljusternik-Schnirelmann theory.
2009, 8(6): 1759-1777
doi: 10.3934/cpaa.2009.8.1759
+[Abstract](2481)
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Abstract:
We concerns here with the continuity on the geometry of the second Riemannian $L^p$-Sobolev best constant $B_0(p,g)$ associated to the AB program. Precisely, for $1 \leq p \leq 2$, we prove that $B_0(p,g)$ depends continuously on $g$ in the $C^2$-topology. Moreover, this topology is sharp for $p = 2$. From this discussion, we deduce some existence and $C^0$-compactness results on extremal functions.
We concerns here with the continuity on the geometry of the second Riemannian $L^p$-Sobolev best constant $B_0(p,g)$ associated to the AB program. Precisely, for $1 \leq p \leq 2$, we prove that $B_0(p,g)$ depends continuously on $g$ in the $C^2$-topology. Moreover, this topology is sharp for $p = 2$. From this discussion, we deduce some existence and $C^0$-compactness results on extremal functions.
2009, 8(6): 1779-1793
doi: 10.3934/cpaa.2009.8.1779
+[Abstract](2498)
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Abstract:
Stability properties for solutions of $-\Delta_m(u)=f(u)$ in $\mathbb{R}^N$ are investigated, where $N\geq 2$ and $m \geq 2$. The aim is to identify a critical dimension $N^\#$ so that every non-constant solution is linearly unstable whenever $2\leq N < N^\#$. For positive, increasing and convex nonlinearities $f(u)$, global bounds on $\frac{f \, f''}{(f')^2}$ allows us to find a dimension $N^\#$, which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with $C^1-$nonlinearities and the dimension $N^\#$ we find is still optimal.
Stability properties for solutions of $-\Delta_m(u)=f(u)$ in $\mathbb{R}^N$ are investigated, where $N\geq 2$ and $m \geq 2$. The aim is to identify a critical dimension $N^\#$ so that every non-constant solution is linearly unstable whenever $2\leq N < N^\#$. For positive, increasing and convex nonlinearities $f(u)$, global bounds on $\frac{f \, f''}{(f')^2}$ allows us to find a dimension $N^\#$, which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with $C^1-$nonlinearities and the dimension $N^\#$ we find is still optimal.
2009, 8(6): 1795-1801
doi: 10.3934/cpaa.2009.8.1795
+[Abstract](2341)
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Abstract:
In this paper, we apply the Mountain Pass Lemma of Ambrosetti-Rabinowitz [2] to study the existence of new periodic solutions with a prescribed energy for a class of second order Hamiltonian conservative systems.
In this paper, we apply the Mountain Pass Lemma of Ambrosetti-Rabinowitz [2] to study the existence of new periodic solutions with a prescribed energy for a class of second order Hamiltonian conservative systems.
2009, 8(6): 1803-1823
doi: 10.3934/cpaa.2009.8.1803
+[Abstract](3314)
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Abstract:
We give a unified treatment for a class of nonlinear Schrödinger (NLS) equations with non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness of the Cauchy problem in $L^2(\mathbb{R}^2)$, $H^1(\mathbb{R}^2)$, $H^2(\mathbb{R}^2)$ and in $\Sigma=H^1(\mathbb{R}^2)\cap L^2(|x|^2 dx)$. We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for $L^2(\mathbb{R}^2)$-solutions. Similarly, the energy and the momenta are conserved for the solutions in $H^1(\mathbb{R}^2)$. For the solutions in $\Sigma$, we show that the virial identity and the pseudo-conformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrödinger equation is of elliptic type. We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in $\Sigma$ in the defocusing case. In the focusing case, we show that when the initial energy of the solution in $\Sigma$ is negative then this solution blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.
We give a unified treatment for a class of nonlinear Schrödinger (NLS) equations with non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness of the Cauchy problem in $L^2(\mathbb{R}^2)$, $H^1(\mathbb{R}^2)$, $H^2(\mathbb{R}^2)$ and in $\Sigma=H^1(\mathbb{R}^2)\cap L^2(|x|^2 dx)$. We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for $L^2(\mathbb{R}^2)$-solutions. Similarly, the energy and the momenta are conserved for the solutions in $H^1(\mathbb{R}^2)$. For the solutions in $\Sigma$, we show that the virial identity and the pseudo-conformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrödinger equation is of elliptic type. We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in $\Sigma$ in the defocusing case. In the focusing case, we show that when the initial energy of the solution in $\Sigma$ is negative then this solution blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.
2009, 8(6): 1825-1839
doi: 10.3934/cpaa.2009.8.1825
+[Abstract](3985)
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Abstract:
In this work we consider a size-structured cannibalism model with the model ingredients (fertility, growth, and mortality rate) depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system, in particular on the effect of cannibalism on the long-term dynamics. To this end, we formally linearize the system about steady state and establish conditions in terms of the model ingredients which yield uniform exponential stability of the governing linear semigroup. We also show how the point spectrum of the linearized semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup.
In this work we consider a size-structured cannibalism model with the model ingredients (fertility, growth, and mortality rate) depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system, in particular on the effect of cannibalism on the long-term dynamics. To this end, we formally linearize the system about steady state and establish conditions in terms of the model ingredients which yield uniform exponential stability of the governing linear semigroup. We also show how the point spectrum of the linearized semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup.
2009, 8(6): 1841-1865
doi: 10.3934/cpaa.2009.8.1841
+[Abstract](2699)
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Abstract:
This paper is concerned with the global existence of cylindrical solution to an initial-boundary value problem for the magnetohydrodynamic equations in an exterior domain. The difficulty of the proof first lies in that the domain is unbounded and the coefficients tend to infinity as $x\to\infty$. Secondly, the additional nonlinear terms and nonlinear equations induced by magnetic field also make the problem more complicated than that for the compressible Navier-Stokes equations. To overcome such difficulties, we study approximate problems in bounded annular domains and assume that the heat conductivity satisfies certain physical growth condition. By virtue of the global (weighted) a priori estimates independent of the boundedness of the annular domain, letting the diameter of the annular domain go to infinity, we obtain the global existence theorem by the similar limit procedure as that in [23].
This paper is concerned with the global existence of cylindrical solution to an initial-boundary value problem for the magnetohydrodynamic equations in an exterior domain. The difficulty of the proof first lies in that the domain is unbounded and the coefficients tend to infinity as $x\to\infty$. Secondly, the additional nonlinear terms and nonlinear equations induced by magnetic field also make the problem more complicated than that for the compressible Navier-Stokes equations. To overcome such difficulties, we study approximate problems in bounded annular domains and assume that the heat conductivity satisfies certain physical growth condition. By virtue of the global (weighted) a priori estimates independent of the boundedness of the annular domain, letting the diameter of the annular domain go to infinity, we obtain the global existence theorem by the similar limit procedure as that in [23].
2009, 8(6): 1867-1894
doi: 10.3934/cpaa.2009.8.1867
+[Abstract](4039)
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Abstract:
We study a viscous two-phase liquid-gas model relevant for well and pipe flow modelling. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid leading to a pressure law which becomes singular when transition to single-phase liquid flow occurs. In order to handle this difficulty we reformulate the model in terms of Lagrangian variables and study the model in a free-boundary setting where the gas and liquid mass are of compact support initially and discontinuous at the boundaries. Then, by applying an appropriate variable transformation, point-wise control on masses can be obtained which guarantees that no single-phase regions will occur when the initial state represents a true mixture of both phases. This paves the way for deriving a global existence result for a class of weak solutions. The result requires that the viscous coefficient depends on the volume fraction in an appropriate manner. By assuming more regularity of the initial fluid velocity a uniqueness result is obtained for an appropriate (smaller) class of weak solutions.
We study a viscous two-phase liquid-gas model relevant for well and pipe flow modelling. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid leading to a pressure law which becomes singular when transition to single-phase liquid flow occurs. In order to handle this difficulty we reformulate the model in terms of Lagrangian variables and study the model in a free-boundary setting where the gas and liquid mass are of compact support initially and discontinuous at the boundaries. Then, by applying an appropriate variable transformation, point-wise control on masses can be obtained which guarantees that no single-phase regions will occur when the initial state represents a true mixture of both phases. This paves the way for deriving a global existence result for a class of weak solutions. The result requires that the viscous coefficient depends on the volume fraction in an appropriate manner. By assuming more regularity of the initial fluid velocity a uniqueness result is obtained for an appropriate (smaller) class of weak solutions.
2009, 8(6): 1895-1916
doi: 10.3934/cpaa.2009.8.1895
+[Abstract](2678)
+[PDF](282.6KB)
Abstract:
We examine a family of microscopic models of plasmas, with a parameter $\alpha$ comparing the typical distance between collisions to the strength of the grazing collisions. These microscopic models converge in distribution, in the weak coupling limit, to a velocity diffusion described by the linear Landau equation (also known as the Fokker-Planck equation). The present work extends and unifies previous results that handled the extremes of the parameter $\alpha$ to the whole range $(0, 1/2]$, by showing that clusters of overlapping obstacles are negligible in the limit. Additionally, we study the diffusion coefficient of the Landau equation and show it to be independent of the parameter.
We examine a family of microscopic models of plasmas, with a parameter $\alpha$ comparing the typical distance between collisions to the strength of the grazing collisions. These microscopic models converge in distribution, in the weak coupling limit, to a velocity diffusion described by the linear Landau equation (also known as the Fokker-Planck equation). The present work extends and unifies previous results that handled the extremes of the parameter $\alpha$ to the whole range $(0, 1/2]$, by showing that clusters of overlapping obstacles are negligible in the limit. Additionally, we study the diffusion coefficient of the Landau equation and show it to be independent of the parameter.
2009, 8(6): 1917-1924
doi: 10.3934/cpaa.2009.8.1917
+[Abstract](3049)
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Abstract:
In this short paper, we obtain the asymptotic behavior of the global solutions of a viscous Hamilton-Jacobi equation $u_t=\Delta u+|\nabla u|^p$ in $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and $u(x,t)=M$ on $\partial B_R$. It is proved that there exists a constant $M_c>0$ such that the problem admits a unique steady state if and only if $M\leq M_c$. When $M < M_c$, the global solution converges in $C^1(\overline{B_{r,R}})$ to the unique regular steady state. When $M=M_c$, the global solution converges in $C(\overline{B_{r,R}})$ to the unique singular steady state, and the grow-up rate of $||u_\nu(t)||_{L^\infty(\partial B_r)}$ in infinite time is obtained.
In this short paper, we obtain the asymptotic behavior of the global solutions of a viscous Hamilton-Jacobi equation $u_t=\Delta u+|\nabla u|^p$ in $B_{r,R}$, $u(x,t)=0$ on $\partial B_r$ and $u(x,t)=M$ on $\partial B_R$. It is proved that there exists a constant $M_c>0$ such that the problem admits a unique steady state if and only if $M\leq M_c$. When $M < M_c$, the global solution converges in $C^1(\overline{B_{r,R}})$ to the unique regular steady state. When $M=M_c$, the global solution converges in $C(\overline{B_{r,R}})$ to the unique singular steady state, and the grow-up rate of $||u_\nu(t)||_{L^\infty(\partial B_r)}$ in infinite time is obtained.
2009, 8(6): 1925-1932
doi: 10.3934/cpaa.2009.8.1925
+[Abstract](3062)
+[PDF](135.5KB)
Abstract:
In this paper, we consider radial symmetry of positive solutions for a system of three integral equations in $R^n$. Under some mild integrability conditions, we prove that all the solutions are radially symmetric and monotone decreasing about some point. This generalizes a recent result of Chen, Li, and Ou [4]. To establish the symmetry, we use an integral form of the method of moving planes which is quite different from the traditional method of moving planes for PDEs. We also generalize our result to a system containing any number of integral equations.
In this paper, we consider radial symmetry of positive solutions for a system of three integral equations in $R^n$. Under some mild integrability conditions, we prove that all the solutions are radially symmetric and monotone decreasing about some point. This generalizes a recent result of Chen, Li, and Ou [4]. To establish the symmetry, we use an integral form of the method of moving planes which is quite different from the traditional method of moving planes for PDEs. We also generalize our result to a system containing any number of integral equations.
2009, 8(6): 1933-1956
doi: 10.3934/cpaa.2009.8.1933
+[Abstract](2288)
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Abstract:
The dynamical system generated by a system describing nonlinear oscillations of two coupled Berger plates with nonlinear interior damping and clamped boundary is considered. The dependence of the long-time behavior of the system trajectories on the coupling parameter $\gamma$ is studied in the case of (i) same equations for both plates of the system and damping possibly degenerate at zero; and (ii) different equations and damping non-degenerate at any point. Ultimate synchronization at the level of attractors is proved for both cases, which means that the global attractor of the system approaches the diagonal of the phase space of the system as $\gamma\to\infty$. In case (ii) the structure of the upper limit of the attractor is studied. It coincides with the diagonal of the product of two samples of the attractor to the dynamical system generated by a single plate equation. If both the equations describing the plate dynamics are the same and the damping functions are non-degenerate at any point we prove the synchronization phenomenon for finite large $\gamma$. System synchronization rate is exponential in this case.
The dynamical system generated by a system describing nonlinear oscillations of two coupled Berger plates with nonlinear interior damping and clamped boundary is considered. The dependence of the long-time behavior of the system trajectories on the coupling parameter $\gamma$ is studied in the case of (i) same equations for both plates of the system and damping possibly degenerate at zero; and (ii) different equations and damping non-degenerate at any point. Ultimate synchronization at the level of attractors is proved for both cases, which means that the global attractor of the system approaches the diagonal of the phase space of the system as $\gamma\to\infty$. In case (ii) the structure of the upper limit of the attractor is studied. It coincides with the diagonal of the product of two samples of the attractor to the dynamical system generated by a single plate equation. If both the equations describing the plate dynamics are the same and the damping functions are non-degenerate at any point we prove the synchronization phenomenon for finite large $\gamma$. System synchronization rate is exponential in this case.
2009, 8(6): 1957-1974
doi: 10.3934/cpaa.2009.8.1957
+[Abstract](2486)
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Abstract:
In this paper we deal with a nonlinear Neumann problem driven by the $p$--Laplacian and with a potential function which asymptotically at infinity is $p$--linear. Using variational methods based on critical point theory coupled with suitable truncation techniques, we prove a theorem establishing the existence of at least three nontrivial smooth solutions for the Neumann problem. For the semilinear case (i.e., $p=2$) using Morse theory, we produce one more nontrivial smooth solution.
In this paper we deal with a nonlinear Neumann problem driven by the $p$--Laplacian and with a potential function which asymptotically at infinity is $p$--linear. Using variational methods based on critical point theory coupled with suitable truncation techniques, we prove a theorem establishing the existence of at least three nontrivial smooth solutions for the Neumann problem. For the semilinear case (i.e., $p=2$) using Morse theory, we produce one more nontrivial smooth solution.
2009, 8(6): 1975-1989
doi: 10.3934/cpaa.2009.8.1975
+[Abstract](4100)
+[PDF](245.4KB)
Abstract:
In the present paper, we establish the complete monotonicity of two functions involving divided differences of the digamma function $\psi$ and the trigamma function $\psi'$. Applying these monotonicity, we provide an alternative proof for the monotonicity and convexity of a function derived from bounding the ratio of two gamma functions, procure the logarithmically completely monotonic property of a function involving the ratio of two gamma functions, and obtain new bounds for the ratio of two gamma functions.
In the present paper, we establish the complete monotonicity of two functions involving divided differences of the digamma function $\psi$ and the trigamma function $\psi'$. Applying these monotonicity, we provide an alternative proof for the monotonicity and convexity of a function derived from bounding the ratio of two gamma functions, procure the logarithmically completely monotonic property of a function involving the ratio of two gamma functions, and obtain new bounds for the ratio of two gamma functions.
2009, 8(6): 1991-2012
doi: 10.3934/cpaa.2009.8.1991
+[Abstract](2868)
+[PDF](219.8KB)
Abstract:
This paper is concerned with the global existence and exponential stability of weak solutions in $H^4$ for a real viscous heat-conducting flow with shear viscosity in a bounded domain $\Omega=(0,1)$ . Some new ideas and more delicate estimates are introduced to prove these results.
This paper is concerned with the global existence and exponential stability of weak solutions in $H^4$ for a real viscous heat-conducting flow with shear viscosity in a bounded domain $\Omega=(0,1)$ . Some new ideas and more delicate estimates are introduced to prove these results.
2009, 8(6): 2013-2035
doi: 10.3934/cpaa.2009.8.2013
+[Abstract](2613)
+[PDF](259.3KB)
Abstract:
Using minimax methods we study the existence and multiplicity of solutions for a class of semilinear elliptic nonhomogeneous systems where the potentials can change sign and the nonlinearities may be unbounded in $x$ and behave like $\exp(\alpha s^2)$ when $|s|\rightarrow+\infty$. We establish the existence of two distinct solutions when the perturbations are suitably small.
Using minimax methods we study the existence and multiplicity of solutions for a class of semilinear elliptic nonhomogeneous systems where the potentials can change sign and the nonlinearities may be unbounded in $x$ and behave like $\exp(\alpha s^2)$ when $|s|\rightarrow+\infty$. We establish the existence of two distinct solutions when the perturbations are suitably small.
2009, 8(6): 2037-2053
doi: 10.3934/cpaa.2009.8.2037
+[Abstract](3357)
+[PDF](249.2KB)
Abstract:
In this work we consider the nonlocal stationary nonlinear problem $(J* u)(x) - u(x)= -\lambda u(x)+ a(x) u^p(x)$ in a domain $\Omega$, with the Dirichlet boundary condition $u(x)=0$ in $\mathbb{R}^N\setminus \Omega$ and $p>1$. The kernel $J$ involved in the convolution $(J*u) (x) = \int_{\mathbb{R}^N} J(x-y) u(y) dy$ is a smooth, compactly supported nonnegative function with unit integral, while the weight $a(x)$ is assumed to be nonnegative and is allowed to vanish in a smooth subdomain $\Omega_0$ of $\Omega$. Both when $a(x)$ is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter $\lambda$.
In this work we consider the nonlocal stationary nonlinear problem $(J* u)(x) - u(x)= -\lambda u(x)+ a(x) u^p(x)$ in a domain $\Omega$, with the Dirichlet boundary condition $u(x)=0$ in $\mathbb{R}^N\setminus \Omega$ and $p>1$. The kernel $J$ involved in the convolution $(J*u) (x) = \int_{\mathbb{R}^N} J(x-y) u(y) dy$ is a smooth, compactly supported nonnegative function with unit integral, while the weight $a(x)$ is assumed to be nonnegative and is allowed to vanish in a smooth subdomain $\Omega_0$ of $\Omega$. Both when $a(x)$ is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter $\lambda$.
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