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1937-1632
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1937-1179
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Discrete & Continuous Dynamical Systems - S
December 2016 , Volume 9 , Issue 6
Issue dedicated to Professor Boling Guo on the occasion of his 80th birthday
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2016, 9(6): i-ii
doi: 10.3934/dcdss.201606i
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Abstract:
This special issue of DCDS is dedicated to Prof. Boling Guo on the occasion of his 80th birthday.
For more information please click the “Full Text” above.
This special issue of DCDS is dedicated to Prof. Boling Guo on the occasion of his 80th birthday.
For more information please click the “Full Text” above.
2016, 9(6): 1591-1611
doi: 10.3934/dcdss.2016065
+[Abstract](1863)
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Abstract:
This paper is concerned with the initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. We show that the system has a unique classical solution for $H^3$ initial data, and the non-slip boundary condition for velocity field and the perfectly conducting wall condition for magnetic field. In addition, we show that the kinetic energy is uniformly bounded in time.
This paper is concerned with the initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. We show that the system has a unique classical solution for $H^3$ initial data, and the non-slip boundary condition for velocity field and the perfectly conducting wall condition for magnetic field. In addition, we show that the kinetic energy is uniformly bounded in time.
2016, 9(6): 1613-1628
doi: 10.3934/dcdss.2016066
+[Abstract](1543)
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Abstract:
In this paper, we first give a sharp variational characterization to the smallest positive constant $C_{VGN}$ in the following Variant Gagliardo-Nirenberg interpolation inequality: $$ \int_{\mathbb{R}^N\times\mathbb{R}^N}{{|u(x)|^p|u(y)|^p}\over{|x-y|^\alpha}}dxdy\leq C_{VGN} \|\nabla u\|_{L^2}^{N(p-2)+\alpha} \|u\|_{L^2}^{2p-(N(p-2)+\alpha)}, $$ where $u\in W^{1,2}(\mathbb{R}^N)$ and $N\geq 1$. Then we use this characterization to determine the sharp threshold of $\|\varphi_0\|_{L^2}$ such that the solution of $i\varphi_t = - \triangle \varphi + |x|^2\varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$ with initial condition $\varphi(0, x) = \varphi_0$ exists globally or blows up in a finite time. We also outline some results on the applications of $C_{VGN}$ to the Cauchy problem of $i\varphi_t = - \triangle \varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.
In this paper, we first give a sharp variational characterization to the smallest positive constant $C_{VGN}$ in the following Variant Gagliardo-Nirenberg interpolation inequality: $$ \int_{\mathbb{R}^N\times\mathbb{R}^N}{{|u(x)|^p|u(y)|^p}\over{|x-y|^\alpha}}dxdy\leq C_{VGN} \|\nabla u\|_{L^2}^{N(p-2)+\alpha} \|u\|_{L^2}^{2p-(N(p-2)+\alpha)}, $$ where $u\in W^{1,2}(\mathbb{R}^N)$ and $N\geq 1$. Then we use this characterization to determine the sharp threshold of $\|\varphi_0\|_{L^2}$ such that the solution of $i\varphi_t = - \triangle \varphi + |x|^2\varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$ with initial condition $\varphi(0, x) = \varphi_0$ exists globally or blows up in a finite time. We also outline some results on the applications of $C_{VGN}$ to the Cauchy problem of $i\varphi_t = - \triangle \varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.
2016, 9(6): 1629-1645
doi: 10.3934/dcdss.2016067
+[Abstract](1478)
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Abstract:
In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.
In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.
2016, 9(6): 1647-1662
doi: 10.3934/dcdss.2016068
+[Abstract](1513)
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We investigate the 1D Swift-Hohenberg equation with dispersion $$u_t+2u_{\xi\xi}-\sigma u_{\xi\xi\xi}+u_{\xi\xi\xi\xi}=\alpha u+\beta u^2-\gamma u^3,$$ where $\sigma, \alpha, \beta$ and $\gamma$ are constants. Even if only the stationary solutions of this equation are considered, the dispersion term $-\sigma u_{\xi\xi\xi}$ destroys the spatial reversibility which plays an important role for studying localized patterns. In this paper, we focus on its traveling wave solutions and directly apply the dynamical approach to provide the first rigorous proof of existence of the periodic solutions and the homoclinic solutions bifurcating from the origin without the reversibility condition as the parameters are varied.
We investigate the 1D Swift-Hohenberg equation with dispersion $$u_t+2u_{\xi\xi}-\sigma u_{\xi\xi\xi}+u_{\xi\xi\xi\xi}=\alpha u+\beta u^2-\gamma u^3,$$ where $\sigma, \alpha, \beta$ and $\gamma$ are constants. Even if only the stationary solutions of this equation are considered, the dispersion term $-\sigma u_{\xi\xi\xi}$ destroys the spatial reversibility which plays an important role for studying localized patterns. In this paper, we focus on its traveling wave solutions and directly apply the dynamical approach to provide the first rigorous proof of existence of the periodic solutions and the homoclinic solutions bifurcating from the origin without the reversibility condition as the parameters are varied.
2016, 9(6): 1663-1685
doi: 10.3934/dcdss.2016069
+[Abstract](1670)
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Abstract:
The paper deals with the following equation of Kirchhoff type, \begin{align*} & -\left ( 1+b\left(\int_{\mathbb{R}^3}|\nabla u|^2dx\right)^r\, \right ) \Delta u+u=k(x)\left (|u|^{q-2}u+\theta g(u)\right )+\lambda h(x)u \end{align*} with $x\in \mathbb{R}^3$, where $u\in H^{1}(\mathbb{R}^3)$, $ b > 0, $ $0 < r < 2, q \in [2(r+1), 6) $, $\theta $ is a small constant, $\lambda$ is a parameter, and a weight function $h (x) \geq 0$. It is known that the linear operator $-\Delta u+u-\lambda h(x)u$ is coercive if $0<\lambda<\lambda_1(h)$ and is non-coercive if $\lambda>\lambda_1(h)$, where $\lambda_1(h)$ is the first eigenvalue of the operator $-\Delta u +u $ with the weight $h(x)$. Under suitable conditions on the functions $k(x)$ and $g(s)$, it is shown that the equation has a positive solution for any $\lambda\in(0,\lambda_1(h))$ and two positive solutions for $\lambda\in(\lambda_1(h), \lambda_1(h) + \tilde \delta )$ with $\tilde \delta > 0$ small. The conditions imposed on $k(x) $ and $g(s)$ are much weaker than those used before, thereby generalizing several existing results on the existence of positive solutions for this type of Kirchhoff equations.
The paper deals with the following equation of Kirchhoff type, \begin{align*} & -\left ( 1+b\left(\int_{\mathbb{R}^3}|\nabla u|^2dx\right)^r\, \right ) \Delta u+u=k(x)\left (|u|^{q-2}u+\theta g(u)\right )+\lambda h(x)u \end{align*} with $x\in \mathbb{R}^3$, where $u\in H^{1}(\mathbb{R}^3)$, $ b > 0, $ $0 < r < 2, q \in [2(r+1), 6) $, $\theta $ is a small constant, $\lambda$ is a parameter, and a weight function $h (x) \geq 0$. It is known that the linear operator $-\Delta u+u-\lambda h(x)u$ is coercive if $0<\lambda<\lambda_1(h)$ and is non-coercive if $\lambda>\lambda_1(h)$, where $\lambda_1(h)$ is the first eigenvalue of the operator $-\Delta u +u $ with the weight $h(x)$. Under suitable conditions on the functions $k(x)$ and $g(s)$, it is shown that the equation has a positive solution for any $\lambda\in(0,\lambda_1(h))$ and two positive solutions for $\lambda\in(\lambda_1(h), \lambda_1(h) + \tilde \delta )$ with $\tilde \delta > 0$ small. The conditions imposed on $k(x) $ and $g(s)$ are much weaker than those used before, thereby generalizing several existing results on the existence of positive solutions for this type of Kirchhoff equations.
2016, 9(6): 1687-1699
doi: 10.3934/dcdss.2016070
+[Abstract](1599)
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In this paper, we consider a fractional Schrödinger-KdV-Burgers system. First, the local existence and uniqueness of solution is obtained by contraction method. Then by some a priori estimates, global existence and uniqueness of smooth solution for this system is proved. Moreover, the regularity of the solution is improved.
In this paper, we consider a fractional Schrödinger-KdV-Burgers system. First, the local existence and uniqueness of solution is obtained by contraction method. Then by some a priori estimates, global existence and uniqueness of smooth solution for this system is proved. Moreover, the regularity of the solution is improved.
2016, 9(6): 1701-1715
doi: 10.3934/dcdss.2016071
+[Abstract](1556)
+[PDF](417.9KB)
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Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.
Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.
2016, 9(6): 1717-1752
doi: 10.3934/dcdss.2016072
+[Abstract](1804)
+[PDF](596.6KB)
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This paper is concerned with the Cauchy problem for compressible Navier-Stokes-Smoluchowski equations with vacuum in $\mathbb{R}^3$. We prove both existence and uniqueness of the local strong solution, and then obtain a local classical solution by deriving the smoothing effect of the strong solution for $t>0$.
This paper is concerned with the Cauchy problem for compressible Navier-Stokes-Smoluchowski equations with vacuum in $\mathbb{R}^3$. We prove both existence and uniqueness of the local strong solution, and then obtain a local classical solution by deriving the smoothing effect of the strong solution for $t>0$.
2016, 9(6): 1753-1773
doi: 10.3934/dcdss.2016073
+[Abstract](1667)
+[PDF](478.6KB)
Abstract:
We consider the Cauchy problem of the nonlinear Schrödinger equation with magnetic effect, and prove global existence of smooth solutions and decay estimates for suitably small initial data. The key step in our analysis is to exploit the null structures for the phases, which allow us to close our argument in the framework of space-time resonance method.
We consider the Cauchy problem of the nonlinear Schrödinger equation with magnetic effect, and prove global existence of smooth solutions and decay estimates for suitably small initial data. The key step in our analysis is to exploit the null structures for the phases, which allow us to close our argument in the framework of space-time resonance method.
2016, 9(6): 1775-1795
doi: 10.3934/dcdss.2016074
+[Abstract](1272)
+[PDF](434.7KB)
Abstract:
In this paper,we consider the so called generalized inhomogeneous Schrödinger flows from a closed Riemann surface $M$ into the standard 2-sphere $S^2$ associated with the energy functional given by \begin{align*} E_{f,P}(u)=\int_M\left(\frac{1}{2}f|\nabla u|^2+P(u_3)\right)dV_g. \end{align*} We showed the existence of special periodic solutions to the generalized inhomogeneous Schrödinger flows from $M$ with convolution symmetry (especially $M = S^2$) into $S^2$ when the function $f$ and $P$ satisfy certain conditions respectively. Especially, we show that the inhomogeneous Heisenberg spin chain system from a closed Riemann surface with convolution symmetry admits some special periodic solutions if the coupling function $f$ satisfies some suitable conditions. We also prove that there exist an infinite number of special periodic solutions to the Landau-Lifshitz system with an external magnetic field from $S^2$ into $S^2$.
In this paper,we consider the so called generalized inhomogeneous Schrödinger flows from a closed Riemann surface $M$ into the standard 2-sphere $S^2$ associated with the energy functional given by \begin{align*} E_{f,P}(u)=\int_M\left(\frac{1}{2}f|\nabla u|^2+P(u_3)\right)dV_g. \end{align*} We showed the existence of special periodic solutions to the generalized inhomogeneous Schrödinger flows from $M$ with convolution symmetry (especially $M = S^2$) into $S^2$ when the function $f$ and $P$ satisfy certain conditions respectively. Especially, we show that the inhomogeneous Heisenberg spin chain system from a closed Riemann surface with convolution symmetry admits some special periodic solutions if the coupling function $f$ satisfies some suitable conditions. We also prove that there exist an infinite number of special periodic solutions to the Landau-Lifshitz system with an external magnetic field from $S^2$ into $S^2$.
2016, 9(6): 1797-1851
doi: 10.3934/dcdss.2016075
+[Abstract](1700)
+[PDF](709.7KB)
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The Cauchy problem of the 3D Zakharov-Kuznetsov equation $$ u_{t}+\partial_{\tilde{x}_{*,1}}\Delta u +(u^2)_{\tilde{x}_{*,1}}=0, (x,t)\in \mathbb{R}^3 \times \mathbb{R}, \ x=(\tilde{x}_{*,1},\tilde{x}_{*,2},\tilde{x}_{*,3});$$ is considered. It is shown that it is globally well-posed in energy space $H^1(\mathbb{R}^3)$. It answer an open problem: Is it globally well-posed in energy space $H^1 (\mathbb{R}^3)$ for 3D Z-K equtation [10,12,13]?
Moreover, in 4-D and more higher dimension, it is shown that it is locally well-posed in $H^1(\mathbb{R}^n)$ with $n\geq 4$.
The method in this paper combine the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequality). We mainly extend the spaces $\mathbf{F}^s$ and $\mathbf{N}^s$ in one dimension [4] to higher dimension.
The Cauchy problem of the 3D Zakharov-Kuznetsov equation $$ u_{t}+\partial_{\tilde{x}_{*,1}}\Delta u +(u^2)_{\tilde{x}_{*,1}}=0, (x,t)\in \mathbb{R}^3 \times \mathbb{R}, \ x=(\tilde{x}_{*,1},\tilde{x}_{*,2},\tilde{x}_{*,3});$$ is considered. It is shown that it is globally well-posed in energy space $H^1(\mathbb{R}^3)$. It answer an open problem: Is it globally well-posed in energy space $H^1 (\mathbb{R}^3)$ for 3D Z-K equtation [10,12,13]?
Moreover, in 4-D and more higher dimension, it is shown that it is locally well-posed in $H^1(\mathbb{R}^n)$ with $n\geq 4$.
The method in this paper combine the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequality). We mainly extend the spaces $\mathbf{F}^s$ and $\mathbf{N}^s$ in one dimension [4] to higher dimension.
2016, 9(6): 1853-1898
doi: 10.3934/dcdss.2016076
+[Abstract](1679)
+[PDF](797.1KB)
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We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized problem and a local well-posedness result of smooth solutions to the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem. Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifies the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.
We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized problem and a local well-posedness result of smooth solutions to the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem. Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifies the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.
2016, 9(6): 1899-1912
doi: 10.3934/dcdss.2016077
+[Abstract](1573)
+[PDF](424.3KB)
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In this paper we study the threshold of global existence and blow-up for the solutions to the generalized 3D Davey-Stewartson equations \begin{equation*} \left\{ \begin{aligned} & iu_t + \Delta u + |u|^{p-1} u + E_1(|u|^2)u = 0, \quad t > 0, \ \ x\in \mathbb{R}^3, \\ & u(0,x) = u_0(x) \in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where $1 < p < \frac{7}{3}$ and the operator $E_1$ is given by $ E_1(f) = \mathcal {F}^{-1} \left( \frac{\xi_1^2}{|\xi|^2} \mathcal{F}(f) \right) $. We construct two kinds of invariant sets under the evolution flow by analyzing the property of the upper bound function of the energy. Then we show that the solution exists globally for the initial function $u_0$ in first kind of the invariant sets, while the solution blows up in finite time for $u_0$ in another kind. We remark that the exponent $ p $ is subcritical for the nonlinear Schrödinger equations for which blow-up solutions would not occur. The result shows that the occurrence of blow-up phenomenon is caused by the coupling mechanics of the Davey-Stewartson equations.
In this paper we study the threshold of global existence and blow-up for the solutions to the generalized 3D Davey-Stewartson equations \begin{equation*} \left\{ \begin{aligned} & iu_t + \Delta u + |u|^{p-1} u + E_1(|u|^2)u = 0, \quad t > 0, \ \ x\in \mathbb{R}^3, \\ & u(0,x) = u_0(x) \in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where $1 < p < \frac{7}{3}$ and the operator $E_1$ is given by $ E_1(f) = \mathcal {F}^{-1} \left( \frac{\xi_1^2}{|\xi|^2} \mathcal{F}(f) \right) $. We construct two kinds of invariant sets under the evolution flow by analyzing the property of the upper bound function of the energy. Then we show that the solution exists globally for the initial function $u_0$ in first kind of the invariant sets, while the solution blows up in finite time for $u_0$ in another kind. We remark that the exponent $ p $ is subcritical for the nonlinear Schrödinger equations for which blow-up solutions would not occur. The result shows that the occurrence of blow-up phenomenon is caused by the coupling mechanics of the Davey-Stewartson equations.
2016, 9(6): 1913-1937
doi: 10.3934/dcdss.2016078
+[Abstract](1264)
+[PDF](523.5KB)
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This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.
This paper is concerned with the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and uniqueness is established for the local strong solution with large initial data and also for the global strong solution with initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.
2016, 9(6): 1939-1957
doi: 10.3934/dcdss.2016079
+[Abstract](1624)
+[PDF](467.3KB)
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A global attractor in $L^2$ is shown for weakly dissipative $p$-Laplace equations on the entire Euclid space, where the weak dissipativeness means that the order of the source is lesser than $p-1$. Half-time decomposition and induction techniques are utilized to present the tail estimate outside a ball. It is also proved that the equations in both strongly and weakly dissipative cases possess an $(L^2,L^r)$-attractor for $r$ belonging to a special interval, which contains the critical exponent $p$. The obtained attractor is proved to be approximated by the corresponding attractor inside a ball in the sense of upper strictly and lower semicontinuity.
A global attractor in $L^2$ is shown for weakly dissipative $p$-Laplace equations on the entire Euclid space, where the weak dissipativeness means that the order of the source is lesser than $p-1$. Half-time decomposition and induction techniques are utilized to present the tail estimate outside a ball. It is also proved that the equations in both strongly and weakly dissipative cases possess an $(L^2,L^r)$-attractor for $r$ belonging to a special interval, which contains the critical exponent $p$. The obtained attractor is proved to be approximated by the corresponding attractor inside a ball in the sense of upper strictly and lower semicontinuity.
Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance
2016, 9(6): 1959-1974
doi: 10.3934/dcdss.2016080
+[Abstract](1707)
+[PDF](417.1KB)
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In this paper, we study a class of Kirchhoff type problems with resonance \begin{equation*} \begin{cases} -\left(a+b\displaystyle\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\nu u^{3}+ \lambda |u|^{q-1}u,&\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on} \ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^{3}$ is a bounded domain, $a,b,\nu,\lambda>0$ and $0< q <1$. By a minimizing method, we obtain the existence of positive ground state solutions for all $0<\nu\leq b\nu_{1}$ and $\lambda>0$. Furthermore, using the Nehari method, we obtain two positive solutions for all $\nu>b\nu_{1}$ and $0<\lambda<\tilde{\lambda},$ where $\nu_{1}$ is the first eigenvalue of problem (5) and $\tilde{\lambda}$ is a positive constant. And one of the two positive solutions is a ground state solution.
In this paper, we study a class of Kirchhoff type problems with resonance \begin{equation*} \begin{cases} -\left(a+b\displaystyle\int_{\Omega}|\nabla u|^2dx\right)\Delta u=\nu u^{3}+ \lambda |u|^{q-1}u,&\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on} \ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^{3}$ is a bounded domain, $a,b,\nu,\lambda>0$ and $0< q <1$. By a minimizing method, we obtain the existence of positive ground state solutions for all $0<\nu\leq b\nu_{1}$ and $\lambda>0$. Furthermore, using the Nehari method, we obtain two positive solutions for all $\nu>b\nu_{1}$ and $0<\lambda<\tilde{\lambda},$ where $\nu_{1}$ is the first eigenvalue of problem (5) and $\tilde{\lambda}$ is a positive constant. And one of the two positive solutions is a ground state solution.
2016, 9(6): 1975-2010
doi: 10.3934/dcdss.2016081
+[Abstract](1430)
+[PDF](841.5KB)
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In this paper, we reestablish the elementary Darboux transformation for Sasa-Satsuma equation with the aid of loop group method. Furthermore, the generalized Darboux transformation is given with the limit technique. As direct applications, we give the single solitonic solutions for the focusing and defocusing case. The general high order solution formulas with the determinant form are obtained through generalized DT and the formal series method.
In this paper, we reestablish the elementary Darboux transformation for Sasa-Satsuma equation with the aid of loop group method. Furthermore, the generalized Darboux transformation is given with the limit technique. As direct applications, we give the single solitonic solutions for the focusing and defocusing case. The general high order solution formulas with the determinant form are obtained through generalized DT and the formal series method.
2016, 9(6): 2011-2029
doi: 10.3934/dcdss.2016082
+[Abstract](1507)
+[PDF](482.6KB)
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The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang [14] on the two-dimensional Prandtl equations to the three-dimensional setting.
The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang [14] on the two-dimensional Prandtl equations to the three-dimensional setting.
2016, 9(6): 2031-2046
doi: 10.3934/dcdss.2016083
+[Abstract](1582)
+[PDF](683.2KB)
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This work is concerned the flow of a generalized Oldroyd-B fluid in a porous half-space with second-order slip effect. The fractional calculus approach is used to establish the constitutive relationship of the non-Newtonian fluid model. A new motion model is firstly proposed by modifying the boundary condition with second-order slip effect. Exact solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete inverse Laplace transform of the sequential fractional derivatives. The similar solutions for the generalized Oldroyd-B fluid with first-order slip or no slip, and the solutions for a generalized Oldroyd-B fluid in nonporous medium, are obtained as the limiting cases of our solutions. Furthermore, the behavior of various parameters on the corresponding flow characteristics is shown graphical through different diagrams.
This work is concerned the flow of a generalized Oldroyd-B fluid in a porous half-space with second-order slip effect. The fractional calculus approach is used to establish the constitutive relationship of the non-Newtonian fluid model. A new motion model is firstly proposed by modifying the boundary condition with second-order slip effect. Exact solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete inverse Laplace transform of the sequential fractional derivatives. The similar solutions for the generalized Oldroyd-B fluid with first-order slip or no slip, and the solutions for a generalized Oldroyd-B fluid in nonporous medium, are obtained as the limiting cases of our solutions. Furthermore, the behavior of various parameters on the corresponding flow characteristics is shown graphical through different diagrams.
2016, 9(6): 2047-2072
doi: 10.3934/dcdss.2016084
+[Abstract](1765)
+[PDF](487.9KB)
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This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces $H^{s}$ with $s > 3/2$, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker $H^{r}$-topology is used then it is shown that the solution map becomes Hölder continuous in $H^{s}$.
This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces $H^{s}$ with $s > 3/2$, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker $H^{r}$-topology is used then it is shown that the solution map becomes Hölder continuous in $H^{s}$.
2016, 9(6): 2073-2094
doi: 10.3934/dcdss.2016085
+[Abstract](1670)
+[PDF](550.9KB)
Abstract:
In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{t t}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\mathbb{R}^d)\times H_x^{s_c-1}(\mathbb{R}^d))$ with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. However, the scaling invariance is broken in the Klein-Gordon equation. We will utilize the concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the scenario: soliton-like solutions. And such solutions are precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.
In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{t t}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\mathbb{R}^d)\times H_x^{s_c-1}(\mathbb{R}^d))$ with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. However, the scaling invariance is broken in the Klein-Gordon equation. We will utilize the concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the scenario: soliton-like solutions. And such solutions are precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.
2016, 9(6): 2095-2111
doi: 10.3934/dcdss.2016086
+[Abstract](1343)
+[PDF](449.1KB)
Abstract:
The quasineutral limit of the three dimensional compressible Euler-Poisson (EP) system for ions in plasma under strong magnetic field is rigorously studied. It is proved that as the Debye length and the Larmor radius tend to zero, the solution of the compressible EP system converges strongly to the strong solution of the one-dimensional compressible Euler-equation in the external magnetic field direction. Higher order approximation and convergence rates are also given and detailed studied.
The quasineutral limit of the three dimensional compressible Euler-Poisson (EP) system for ions in plasma under strong magnetic field is rigorously studied. It is proved that as the Debye length and the Larmor radius tend to zero, the solution of the compressible EP system converges strongly to the strong solution of the one-dimensional compressible Euler-equation in the external magnetic field direction. Higher order approximation and convergence rates are also given and detailed studied.
2016, 9(6): 2113-2127
doi: 10.3934/dcdss.2016087
+[Abstract](1502)
+[PDF](405.6KB)
Abstract:
The purpose of this paper is to build the existence of time-spatial global regular solution to the coupled Navier-Stokes and Maxwell equations.
The purpose of this paper is to build the existence of time-spatial global regular solution to the coupled Navier-Stokes and Maxwell equations.
2016, 9(6): 2129-2148
doi: 10.3934/dcdss.2016088
+[Abstract](1393)
+[PDF](508.3KB)
Abstract:
In this paper, we discuss two main problems. In the first section, we establish a new global distributed exact controllability of the periodic two-component $\mu\rho$-Hunter-Saxton system on the circle by means of a distributed control. And in the second section, we present corresponding result of the asymptotic stabilization problem about the periodic two-component $\mu\rho$-Hunter-Saxton system. By presenting concrete form of the feedback law, an equivalent system is got.
In this paper, we discuss two main problems. In the first section, we establish a new global distributed exact controllability of the periodic two-component $\mu\rho$-Hunter-Saxton system on the circle by means of a distributed control. And in the second section, we present corresponding result of the asymptotic stabilization problem about the periodic two-component $\mu\rho$-Hunter-Saxton system. By presenting concrete form of the feedback law, an equivalent system is got.
2016, 9(6): 2149-2165
doi: 10.3934/dcdss.2016089
+[Abstract](1609)
+[PDF](415.9KB)
Abstract:
As we all know, wave breaking of the water wave is important and interesting to physicist and mathematician. In the article, we devote to the study of blow-up phenomena, the decay of solution and traveling wave solution to a shallow water wave equation. First, based on the blow-up scenario, some new blow-up phenomena is derived. By virtue of a weighted function, the persistent decay of solution is established. Finally, we explore the analytic solutions and traveling wave solutions.
As we all know, wave breaking of the water wave is important and interesting to physicist and mathematician. In the article, we devote to the study of blow-up phenomena, the decay of solution and traveling wave solution to a shallow water wave equation. First, based on the blow-up scenario, some new blow-up phenomena is derived. By virtue of a weighted function, the persistent decay of solution is established. Finally, we explore the analytic solutions and traveling wave solutions.
2016, 9(6): 2167-2179
doi: 10.3934/dcdss.2016090
+[Abstract](2065)
+[PDF](433.5KB)
Abstract:
In this paper, we investigate the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations in three dimensions. We obtain that if $ \nabla_{h}u,\nabla_{h}\omega\in L^{1}(0,T;\dot{B}^{0}_{\infty,\infty})$ or $ \nabla_{h}u,\nabla_{h}\omega\in L^{\frac{8}{3}}(0,T;\dot{B}^{-1}_{\infty,\infty})$ then the solution $(u,\omega)$ can be extended smoothly beyond $t=T$.
In this paper, we investigate the blow-up criteria of smooth solutions and the regularity of weak solutions to the micropolar fluid equations in three dimensions. We obtain that if $ \nabla_{h}u,\nabla_{h}\omega\in L^{1}(0,T;\dot{B}^{0}_{\infty,\infty})$ or $ \nabla_{h}u,\nabla_{h}\omega\in L^{\frac{8}{3}}(0,T;\dot{B}^{-1}_{\infty,\infty})$ then the solution $(u,\omega)$ can be extended smoothly beyond $t=T$.
2016, 9(6): 2181-2200
doi: 10.3934/dcdss.2016091
+[Abstract](1430)
+[PDF](373.9KB)
Abstract:
Consider the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations \begin{eqnarray*} \frac{\partial{\bf u}}{\partial t}-\alpha\triangle{\bf u}+({\bf u}\cdot\nabla){\bf u}+\nabla p={\bf f}({\bf x},t),\qquad {\bf u}({\bf x},0)={\bf u}_0({\bf x}). \end{eqnarray*} In this system, the dimension $n\geq 3$, ${\bf u}({\bf x},t)=(u_1({\bf x},t),u_2({\bf x},t),\cdots,u_n({\bf x},t))$ and ${\bf f}({\bf x},t)=(f_1({\bf x},t),f_2({\bf x},t),\cdots,f_n({\bf x},t))$ are real vector valued functions of ${\bf x}=(x_1,x_2,\cdots,x_n)$ and $t$. Additionally, $\alpha>0$ is a positive constant. Suppose that the initial function and the external force satisfy appropriate conditions.
The main purpose of this paper is to make complete use of the uniform energy estimates of the global smooth solutions and couple together a well known Gronwall's inequality to improve the Fourier splitting method to accomplish the decay estimates with sharp rates. The decay estimates with sharp rates of the global smooth solutions of the Cauchy problems for the $n$-dimensional magnetohydrodynamics equations may be established very similarly.
Consider the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations \begin{eqnarray*} \frac{\partial{\bf u}}{\partial t}-\alpha\triangle{\bf u}+({\bf u}\cdot\nabla){\bf u}+\nabla p={\bf f}({\bf x},t),\qquad {\bf u}({\bf x},0)={\bf u}_0({\bf x}). \end{eqnarray*} In this system, the dimension $n\geq 3$, ${\bf u}({\bf x},t)=(u_1({\bf x},t),u_2({\bf x},t),\cdots,u_n({\bf x},t))$ and ${\bf f}({\bf x},t)=(f_1({\bf x},t),f_2({\bf x},t),\cdots,f_n({\bf x},t))$ are real vector valued functions of ${\bf x}=(x_1,x_2,\cdots,x_n)$ and $t$. Additionally, $\alpha>0$ is a positive constant. Suppose that the initial function and the external force satisfy appropriate conditions.
The main purpose of this paper is to make complete use of the uniform energy estimates of the global smooth solutions and couple together a well known Gronwall's inequality to improve the Fourier splitting method to accomplish the decay estimates with sharp rates. The decay estimates with sharp rates of the global smooth solutions of the Cauchy problems for the $n$-dimensional magnetohydrodynamics equations may be established very similarly.
2016, 9(6): 2201-2211
doi: 10.3934/dcdss.2016092
+[Abstract](1317)
+[PDF](374.3KB)
Abstract:
Exact analytic solutions of the kink soliton equation obtained in a recent interesting study of the classical Skyrme model defined on a simple spherically symmetric background are presented. By a variational method, the existence of spherically symmetric monopole solutions are proved. In particular, all finite-energy kink solitons must be Bogomool'nyi--Prasad--Sommerfield are showed. Moreover, together with numerical analysis, we can clearly see the validity of our theoretical results.
Exact analytic solutions of the kink soliton equation obtained in a recent interesting study of the classical Skyrme model defined on a simple spherically symmetric background are presented. By a variational method, the existence of spherically symmetric monopole solutions are proved. In particular, all finite-energy kink solitons must be Bogomool'nyi--Prasad--Sommerfield are showed. Moreover, together with numerical analysis, we can clearly see the validity of our theoretical results.
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