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Communications on Pure & Applied Analysis
November 2013 , Volume 12 , Issue 6
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2013, 12(6): 2331-2360
doi: 10.3934/cpaa.2013.12.2331
+[Abstract](2173)
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Abstract:
The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.
The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.
2013, 12(6): 2361-2380
doi: 10.3934/cpaa.2013.12.2361
+[Abstract](2294)
+[PDF](437.2KB)
Abstract:
In this paper, we consider the existence of nontrivial $1$-periodic solutions of the following Hamiltonian systems \begin{eqnarray} -J\dot{z}=H'(t,z), z\in R^{2N}, \end{eqnarray} where $J$ is the standard symplectic matrix of $2N\times 2N$, $H\in C^2 ( [0,1] \times R^{2N}, R)$ is $1$-periodic in its first variable and $H'(t,z)$ denotes the gradient of $H$ with respect to the variable $z$. Furthermore, $H'(t,z)$ is asymptotically linear both at origin and at infinity. Based on the precise computations of the critical groups, Maslov-type index theory and Galerkin approximation procedure, we obtain some existence results for nontrivial $1$-periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature.
In this paper, we consider the existence of nontrivial $1$-periodic solutions of the following Hamiltonian systems \begin{eqnarray} -J\dot{z}=H'(t,z), z\in R^{2N}, \end{eqnarray} where $J$ is the standard symplectic matrix of $2N\times 2N$, $H\in C^2 ( [0,1] \times R^{2N}, R)$ is $1$-periodic in its first variable and $H'(t,z)$ denotes the gradient of $H$ with respect to the variable $z$. Furthermore, $H'(t,z)$ is asymptotically linear both at origin and at infinity. Based on the precise computations of the critical groups, Maslov-type index theory and Galerkin approximation procedure, we obtain some existence results for nontrivial $1$-periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature.
2013, 12(6): 2381-2391
doi: 10.3934/cpaa.2013.12.2381
+[Abstract](2189)
+[PDF](390.2KB)
Abstract:
In this article, we prove the existence of multiple weak solutions to N-Laplace equation \begin{eqnarray} -\Delta_N u-\mu \frac{g(x)}{(|x| \log\frac{R}{|x|})^N }|u|^{N-2}u=\lambda f(x,u), \ in\ \Omega.\\ u =0, \ on\ \partial \Omega, \end{eqnarray} using Bonanno's three critical point theorem.
In this article, we prove the existence of multiple weak solutions to N-Laplace equation \begin{eqnarray} -\Delta_N u-\mu \frac{g(x)}{(|x| \log\frac{R}{|x|})^N }|u|^{N-2}u=\lambda f(x,u), \ in\ \Omega.\\ u =0, \ on\ \partial \Omega, \end{eqnarray} using Bonanno's three critical point theorem.
2013, 12(6): 2393-2408
doi: 10.3934/cpaa.2013.12.2393
+[Abstract](2349)
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Abstract:
In this paper, we consider the problem \begin{eqnarray} -\Delta u=|x|^\alpha u^{p-1}, x \in \Omega,\\ u>0, x \in \Omega,\\ \frac{\partial u}{\partial \nu }+\beta u=0, x\in \partial \Omega, \end{eqnarray} where $\Omega$ is the unit ball in $R^N$ centered at the origin with $N\geq 3$, $\alpha>0, \beta>\frac{N-2}{2}, p\geq 2$ and $\nu $ is the unit outward vector normal to $\partial \Omega$. We investigate the asymptotic behavior of the ground state solutions $u_p$ of (1) as $p\to \frac{2N}{N-2}$. We show that the ground state solutions $u_p$ has a unique maximum point $x_p\in \bar\Omega$. In addition, the ground state solutions is non-radial provided that $p\to \frac{2N}{N-2}$.
In this paper, we consider the problem \begin{eqnarray} -\Delta u=|x|^\alpha u^{p-1}, x \in \Omega,\\ u>0, x \in \Omega,\\ \frac{\partial u}{\partial \nu }+\beta u=0, x\in \partial \Omega, \end{eqnarray} where $\Omega$ is the unit ball in $R^N$ centered at the origin with $N\geq 3$, $\alpha>0, \beta>\frac{N-2}{2}, p\geq 2$ and $\nu $ is the unit outward vector normal to $\partial \Omega$. We investigate the asymptotic behavior of the ground state solutions $u_p$ of (1) as $p\to \frac{2N}{N-2}$. We show that the ground state solutions $u_p$ has a unique maximum point $x_p\in \bar\Omega$. In addition, the ground state solutions is non-radial provided that $p\to \frac{2N}{N-2}$.
2013, 12(6): 2409-2444
doi: 10.3934/cpaa.2013.12.2409
+[Abstract](2953)
+[PDF](507.6KB)
Abstract:
These notes are based on a series of lectures given by the author at the summer school of Partial Differential Equations at East China Normal University, Shanghai, July 18 through August 3, 2011. In these notes, we present information about linear oblique derivative problems for parabolic equations and nonlinear oblique derivative problems for elliptic equations. For the most part, all the theorems are true for both parabolic and elliptic problems provided we make some simple changes in the statements of the theorems to take into account the differences between the two types of equations, but we won't try to provide complete statements of results for the two classes of equations. Instead, we focus on presenting the basic techniques for these problems. Moreover, we only study second order equations, so that the maximum principle can be applied.
These notes are based on a series of lectures given by the author at the summer school of Partial Differential Equations at East China Normal University, Shanghai, July 18 through August 3, 2011. In these notes, we present information about linear oblique derivative problems for parabolic equations and nonlinear oblique derivative problems for elliptic equations. For the most part, all the theorems are true for both parabolic and elliptic problems provided we make some simple changes in the statements of the theorems to take into account the differences between the two types of equations, but we won't try to provide complete statements of results for the two classes of equations. Instead, we focus on presenting the basic techniques for these problems. Moreover, we only study second order equations, so that the maximum principle can be applied.
2013, 12(6): 2445-2464
doi: 10.3934/cpaa.2013.12.2445
+[Abstract](4424)
+[PDF](509.5KB)
Abstract:
The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities \begin{eqnarray} (-\Delta)^s u-\lambda u=|u|^{2^*-2}u, in \Omega \\ u=0, in R^n\setminus \Omega, \end{eqnarray} where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $R^n$, $n>2s$, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $R^n$ with $n\geq 4s$ and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when $2s < n < 4s$. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4]. In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators.
The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities \begin{eqnarray} (-\Delta)^s u-\lambda u=|u|^{2^*-2}u, in \Omega \\ u=0, in R^n\setminus \Omega, \end{eqnarray} where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $R^n$, $n>2s$, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $R^n$ with $n\geq 4s$ and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when $2s < n < 4s$. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4]. In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators.
2013, 12(6): 2465-2495
doi: 10.3934/cpaa.2013.12.2465
+[Abstract](2048)
+[PDF](540.0KB)
Abstract:
We investigate existence of reversed flow solutions of the Falkner-Skan equations by considering a system of two singular Hammerstein integral equations. We prove that the reversed flow solutions exist for each parameter in $(-1/6,0)$. This is an extension of results on nonexistence of reversed flow solutions obtained recently by the authors. As applications of our new results, we obtain existence of reversed flow similarity solutions of the boundary layer equations governing the flow of fluids over surfaces often arising from engineering problems.
We investigate existence of reversed flow solutions of the Falkner-Skan equations by considering a system of two singular Hammerstein integral equations. We prove that the reversed flow solutions exist for each parameter in $(-1/6,0)$. This is an extension of results on nonexistence of reversed flow solutions obtained recently by the authors. As applications of our new results, we obtain existence of reversed flow similarity solutions of the boundary layer equations governing the flow of fluids over surfaces often arising from engineering problems.
2013, 12(6): 2497-2514
doi: 10.3934/cpaa.2013.12.2497
+[Abstract](2131)
+[PDF](452.5KB)
Abstract:
In this paper, we prove that all the positive solutions for the PDE system \begin{eqnarray} (- \Delta)^k u_i = f_i(u_1, \cdots, u_m), \ x \in R^n, \ i = 1, 2, \cdots, m \ \ \ \ \ (1) \end{eqnarray} are super polyharmonic, i.e. \begin{eqnarray} (- \Delta)^j u_i > 0, \ j=1, 2, \cdots, k-1; \ i =1, 2, \cdots, m. \end{eqnarray}
To prove this important super polyharmonic property, we introduced a few new ideas and
derived some new estimates.
As an interesting application, we establish the equivalence between the integral system \begin{eqnarray} u_i(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} f_i(u_1(y), \cdots, u_m(y)) d y, \ x \in R^n \ \ \ \ \ (2) \end{eqnarray} and PDE system (1) when $\alpha = 2k < n.$
In the last few years, a series of results on qualitative properties for solutions of integral systems (2) have been obtained, since the introduction of a powerful tool--the method of moving planes in integral forms. Now due to the equivalence established here, all these properties can be applied to the corresponding PDE systems.
We say that systems (1) and (2) are equivalent, if whenever $u$ is a positive solution of (2), then $u$ is also a solution of \begin{eqnarray} (- \Delta)^k u_i = c f_i(u_1, \cdots, u_m), \ x \in R^n, \ i= 1,2, \cdots, m \end{eqnarray} with some constant $c$; and vice versa.
In this paper, we prove that all the positive solutions for the PDE system \begin{eqnarray} (- \Delta)^k u_i = f_i(u_1, \cdots, u_m), \ x \in R^n, \ i = 1, 2, \cdots, m \ \ \ \ \ (1) \end{eqnarray} are super polyharmonic, i.e. \begin{eqnarray} (- \Delta)^j u_i > 0, \ j=1, 2, \cdots, k-1; \ i =1, 2, \cdots, m. \end{eqnarray}
As an interesting application, we establish the equivalence between the integral system \begin{eqnarray} u_i(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} f_i(u_1(y), \cdots, u_m(y)) d y, \ x \in R^n \ \ \ \ \ (2) \end{eqnarray} and PDE system (1) when $\alpha = 2k < n.$
In the last few years, a series of results on qualitative properties for solutions of integral systems (2) have been obtained, since the introduction of a powerful tool--the method of moving planes in integral forms. Now due to the equivalence established here, all these properties can be applied to the corresponding PDE systems.
We say that systems (1) and (2) are equivalent, if whenever $u$ is a positive solution of (2), then $u$ is also a solution of \begin{eqnarray} (- \Delta)^k u_i = c f_i(u_1, \cdots, u_m), \ x \in R^n, \ i= 1,2, \cdots, m \end{eqnarray} with some constant $c$; and vice versa.
2013, 12(6): 2515-2542
doi: 10.3934/cpaa.2013.12.2515
+[Abstract](1797)
+[PDF](532.6KB)
Abstract:
We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonlocal elliptic operators, and an iteration mapping which possesses locally a unique fixed point. The approach reveals an intrinsic structure of the steady Euler system and subtle interactions of its elliptic and hyperbolic part.
We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonlocal elliptic operators, and an iteration mapping which possesses locally a unique fixed point. The approach reveals an intrinsic structure of the steady Euler system and subtle interactions of its elliptic and hyperbolic part.
2013, 12(6): 2543-2564
doi: 10.3934/cpaa.2013.12.2543
+[Abstract](2174)
+[PDF](442.1KB)
Abstract:
In this paper, we consider the properties of the vacuum states for weak solutions to one-dimensional full compressible Navier-Stokes system with viscosity and heat conductivities for general equation of states. Under weak conditions on initial data, we prove that if there is no vacuum initially then the vacuum states do not occur in a finite time. In particular, the temperature variation has no immediate effects on the formation of the vacuum. There are no assumptions on density in large sets. Furthermore, we prove that two initially non interacting vacuum regions will never touch in the future.
In this paper, we consider the properties of the vacuum states for weak solutions to one-dimensional full compressible Navier-Stokes system with viscosity and heat conductivities for general equation of states. Under weak conditions on initial data, we prove that if there is no vacuum initially then the vacuum states do not occur in a finite time. In particular, the temperature variation has no immediate effects on the formation of the vacuum. There are no assumptions on density in large sets. Furthermore, we prove that two initially non interacting vacuum regions will never touch in the future.
2013, 12(6): 2565-2575
doi: 10.3934/cpaa.2013.12.2565
+[Abstract](2330)
+[PDF](387.9KB)
Abstract:
Some critical Sobolev-Hardy inequalities with weight of distance function $d^{\frac{\alpha}{p}p^*}$ are established in a bounded domain $\Omega$, where $d$ is the distance to the boundary $\partial\Omega$. Using these inequalities we get the result that the embedding $\mathcal{D}^{1, 2}(\Omega, d^\alpha)\hookrightarrow L^q(\Omega, d^{\beta})$ is compact if $1\leq q<2^*$ and $\beta >\frac{\alpha}{2}q+\frac{q}{2^*}-1$. By the compactness result and critical-point theory about sign-changing solutions, we obtain infinitely many sign-changing solutions to a degenerate Dirichlet elliptic equation $-\hbox{div}(d^\alpha \nabla u)- \frac{(1-\alpha )^2}{4} d^{\alpha-2} u=f(x,u)$ provided that $f(x,u)$ satisfies suitable conditions.
Some critical Sobolev-Hardy inequalities with weight of distance function $d^{\frac{\alpha}{p}p^*}$ are established in a bounded domain $\Omega$, where $d$ is the distance to the boundary $\partial\Omega$. Using these inequalities we get the result that the embedding $\mathcal{D}^{1, 2}(\Omega, d^\alpha)\hookrightarrow L^q(\Omega, d^{\beta})$ is compact if $1\leq q<2^*$ and $\beta >\frac{\alpha}{2}q+\frac{q}{2^*}-1$. By the compactness result and critical-point theory about sign-changing solutions, we obtain infinitely many sign-changing solutions to a degenerate Dirichlet elliptic equation $-\hbox{div}(d^\alpha \nabla u)- \frac{(1-\alpha )^2}{4} d^{\alpha-2} u=f(x,u)$ provided that $f(x,u)$ satisfies suitable conditions.
2013, 12(6): 2577-2600
doi: 10.3934/cpaa.2013.12.2577
+[Abstract](1910)
+[PDF](516.2KB)
Abstract:
This paper deals with the existence of multiple positive solutions of a quasilinear elliptic equation \begin{eqnarray} -\Delta_p u+u^{p-1} = a(x)u^{q-1}+\lambda h(x) u^{r-1}, \text{in} R^N; \\ u\geq 0, \text{ a.e. }x \in R^N;\\ u \in W^{1,p}(R^N), \end{eqnarray} where $1 < p \leq 2$, $N>p$ and $1 < r < p$ $< q < p^* ( = \frac{pN}{N-p})$. A Nehari manifold is defined by a $C^1-$functional $I$ and is decomposed into two parts. Our work is to find four positive solutions of Eq. (1) when parameter $\lambda$ is sufficiently small.
This paper deals with the existence of multiple positive solutions of a quasilinear elliptic equation \begin{eqnarray} -\Delta_p u+u^{p-1} = a(x)u^{q-1}+\lambda h(x) u^{r-1}, \text{in} R^N; \\ u\geq 0, \text{ a.e. }x \in R^N;\\ u \in W^{1,p}(R^N), \end{eqnarray} where $1 < p \leq 2$, $N>p$ and $1 < r < p$ $< q < p^* ( = \frac{pN}{N-p})$. A Nehari manifold is defined by a $C^1-$functional $I$ and is decomposed into two parts. Our work is to find four positive solutions of Eq. (1) when parameter $\lambda$ is sufficiently small.
2013, 12(6): 2601-2613
doi: 10.3934/cpaa.2013.12.2601
+[Abstract](1953)
+[PDF](405.4KB)
Abstract:
Let $R^n_+$ be the $n$-dimensional upper half Euclidean space, $m$ be a positive integer. In this paper, we consider the following system of integral equations on $R^n_+$: \begin{eqnarray} u(x)=\int_{R^n_+}G(x,y)v^q(y)dy, \\ v(x)=\int_{R^n_+}G(x,y)u^p(y)dy, \end{eqnarray} where \begin{eqnarray} G(x,y)=\frac{c_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_ny_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{\frac{n}{2}}}dz \end{eqnarray} with $0 < 2m < n$ and $p,q>1$. Nonexistence of positive solution is proved by using the method of moving planes in integral forms. We also obtain the equivalence between the system of integral equations and corresponding partial differential equations.
Let $R^n_+$ be the $n$-dimensional upper half Euclidean space, $m$ be a positive integer. In this paper, we consider the following system of integral equations on $R^n_+$: \begin{eqnarray} u(x)=\int_{R^n_+}G(x,y)v^q(y)dy, \\ v(x)=\int_{R^n_+}G(x,y)u^p(y)dy, \end{eqnarray} where \begin{eqnarray} G(x,y)=\frac{c_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_ny_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{\frac{n}{2}}}dz \end{eqnarray} with $0 < 2m < n$ and $p,q>1$. Nonexistence of positive solution is proved by using the method of moving planes in integral forms. We also obtain the equivalence between the system of integral equations and corresponding partial differential equations.
2013, 12(6): 2615-2625
doi: 10.3934/cpaa.2013.12.2615
+[Abstract](1838)
+[PDF](358.0KB)
Abstract:
We consider the Stokes-Boussinesq (and the stationary Na\-vier-Stokes-Boussinesq) equations in a slanted, i.e. not aligned with the gravity's direction, 3d channel and with an arbitrary Rayleigh number. For the front-like initial data and under the no-slip boundary condition for the flow and no-flux boundary condition for the reactant temperature, we derive uniform estimates on the burning rate and the flow velocity, which can be interpreted as stability results for the laminar front.
We consider the Stokes-Boussinesq (and the stationary Na\-vier-Stokes-Boussinesq) equations in a slanted, i.e. not aligned with the gravity's direction, 3d channel and with an arbitrary Rayleigh number. For the front-like initial data and under the no-slip boundary condition for the flow and no-flux boundary condition for the reactant temperature, we derive uniform estimates on the burning rate and the flow velocity, which can be interpreted as stability results for the laminar front.
2013, 12(6): 2627-2644
doi: 10.3934/cpaa.2013.12.2627
+[Abstract](1978)
+[PDF](446.3KB)
Abstract:
We study a drift-diffusion system on bounded domain in two-space dimension. This model is provided with a hetero-separative and homo-aggregative feature subject to a gradient of physical or chemical potential which is proportional to their densities. We extend a criterion of global-in-time existence of the solution, especially for non-radially symmetric case. Then we perform the blowup analysis such as the formation of collapses and collapse mass separations. A slightly different model describing cross chemotaxis is also discussed.
We study a drift-diffusion system on bounded domain in two-space dimension. This model is provided with a hetero-separative and homo-aggregative feature subject to a gradient of physical or chemical potential which is proportional to their densities. We extend a criterion of global-in-time existence of the solution, especially for non-radially symmetric case. Then we perform the blowup analysis such as the formation of collapses and collapse mass separations. A slightly different model describing cross chemotaxis is also discussed.
2013, 12(6): 2645-2667
doi: 10.3934/cpaa.2013.12.2645
+[Abstract](1645)
+[PDF](433.1KB)
Abstract:
A linear transmission problem for a thermoelastic Timoshenko beam model with Fourier low of heat conduction which has a Kirchhoff part with hereditary heat conduction of Gurtin-Pipkin type is considered. We prove that the system is exponentially stable under certain conditions on its parameters. The same result for the problem with purely elastic Kirchhoff part is obtained.
A linear transmission problem for a thermoelastic Timoshenko beam model with Fourier low of heat conduction which has a Kirchhoff part with hereditary heat conduction of Gurtin-Pipkin type is considered. We prove that the system is exponentially stable under certain conditions on its parameters. The same result for the problem with purely elastic Kirchhoff part is obtained.
2013, 12(6): 2669-2684
doi: 10.3934/cpaa.2013.12.2669
+[Abstract](1729)
+[PDF](422.0KB)
Abstract:
The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from $L^2$ and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball in $H^s$, $s\in(0,1)$, whose radius depends only on $s$, the $L^2$ norm of the forcing term and the damping parameter. This gives a new proof for the existence of a smooth global attractor and provides quantitative information on the size of the attractor set in $H^s$. In addition we prove that higher order Sobolev norms are bounded for all positive times.
The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from $L^2$ and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball in $H^s$, $s\in(0,1)$, whose radius depends only on $s$, the $L^2$ norm of the forcing term and the damping parameter. This gives a new proof for the existence of a smooth global attractor and provides quantitative information on the size of the attractor set in $H^s$. In addition we prove that higher order Sobolev norms are bounded for all positive times.
2013, 12(6): 2685-2696
doi: 10.3934/cpaa.2013.12.2685
+[Abstract](2433)
+[PDF](387.7KB)
Abstract:
In this paper, we are concerned with properties of positive solutions of the following fractional elliptic system \begin{eqnarray} {(-\Delta+I)}^{\frac{\alpha}{2}}u=\frac{u^pv^q}{|x|^\beta}, \quad {(-\Delta+I)}^{\frac{\alpha}{2}}v=\frac{v^pu^q}{|x|^\beta}\quad in\quad R^n, \end{eqnarray} where $n \geq 3$, $0 \le \beta < \alpha < n$, $ p, q>1$ and $p+q<\frac{n+\alpha-\beta}{n-\alpha+\beta}$. We show that positive solutions of the system are radially symmetric and belong to $L^\infty(R^n)$, which possibly implies that the solutions are locally Hölder continuous. Moreover, if $ \alpha=2, \beta =0,p\le q$, we show that positive solution pair $(u,v)$ of the system is unique and $u=v = U$, where $U$ is the unique positive solution of the problem \begin{eqnarray} -\Delta u + u = u^{p+q}\quad {\rm in}\quad \mathbb{R}^n. \end{eqnarray}
In this paper, we are concerned with properties of positive solutions of the following fractional elliptic system \begin{eqnarray} {(-\Delta+I)}^{\frac{\alpha}{2}}u=\frac{u^pv^q}{|x|^\beta}, \quad {(-\Delta+I)}^{\frac{\alpha}{2}}v=\frac{v^pu^q}{|x|^\beta}\quad in\quad R^n, \end{eqnarray} where $n \geq 3$, $0 \le \beta < \alpha < n$, $ p, q>1$ and $p+q<\frac{n+\alpha-\beta}{n-\alpha+\beta}$. We show that positive solutions of the system are radially symmetric and belong to $L^\infty(R^n)$, which possibly implies that the solutions are locally Hölder continuous. Moreover, if $ \alpha=2, \beta =0,p\le q$, we show that positive solution pair $(u,v)$ of the system is unique and $u=v = U$, where $U$ is the unique positive solution of the problem \begin{eqnarray} -\Delta u + u = u^{p+q}\quad {\rm in}\quad \mathbb{R}^n. \end{eqnarray}
2013, 12(6): 2697-2713
doi: 10.3934/cpaa.2013.12.2697
+[Abstract](2077)
+[PDF](479.4KB)
Abstract:
In this article we establish the existence and nonexistence of a weak solution to singular elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions.
In this article we establish the existence and nonexistence of a weak solution to singular elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions.
2013, 12(6): 2715-2719
doi: 10.3934/cpaa.2013.12.2715
+[Abstract](2407)
+[PDF](305.0KB)
Abstract:
In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.
In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.
2013, 12(6): 2721-2737
doi: 10.3934/cpaa.2013.12.2721
+[Abstract](2178)
+[PDF](407.2KB)
Abstract:
This paper is concerned with integral systems involving the Bessel potentials. Such integral systems are helpful to understand the corresponding PDE systems, such as some static Shrödinger systems with the critical and the supercritical exponents. We use the lifting lemma on regularity to obtain an integrability interval of solutions. Since the Bessel kernel does not have singularity at infinity, we extend the integrability interval to the whole $[1,\infty]$. Next, we use the method of moving planes to prove the radial symmetry for the positive solution of the system. Based on these results, by an iteration we obtain the estimate of the exponential decay of those solutions near infinity. Finally, we discuss the uniqueness of the positive solution of PDE system under some assumption.
This paper is concerned with integral systems involving the Bessel potentials. Such integral systems are helpful to understand the corresponding PDE systems, such as some static Shrödinger systems with the critical and the supercritical exponents. We use the lifting lemma on regularity to obtain an integrability interval of solutions. Since the Bessel kernel does not have singularity at infinity, we extend the integrability interval to the whole $[1,\infty]$. Next, we use the method of moving planes to prove the radial symmetry for the positive solution of the system. Based on these results, by an iteration we obtain the estimate of the exponential decay of those solutions near infinity. Finally, we discuss the uniqueness of the positive solution of PDE system under some assumption.
2013, 12(6): 2739-2752
doi: 10.3934/cpaa.2013.12.2739
+[Abstract](2245)
+[PDF](365.5KB)
Abstract:
We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems of diagonal form. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409-421] suggests that one may achieve global smoothness even if the $C^1$ norm of the initial data is large, we prove that, if the $C^1$ norm and the BV norm of the boundary data are bounded but possibly large, then the solution remains $C^1$ globally in time. Applications include the equation of time-like extremal surfaces in Minkowski space $R^{1+(1+n)}$ and the one-dimensional Chaplygin gas equations.
We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems of diagonal form. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409-421] suggests that one may achieve global smoothness even if the $C^1$ norm of the initial data is large, we prove that, if the $C^1$ norm and the BV norm of the boundary data are bounded but possibly large, then the solution remains $C^1$ globally in time. Applications include the equation of time-like extremal surfaces in Minkowski space $R^{1+(1+n)}$ and the one-dimensional Chaplygin gas equations.
2013, 12(6): 2753-2772
doi: 10.3934/cpaa.2013.12.2753
+[Abstract](2757)
+[PDF](411.7KB)
Abstract:
This paper is concerned with general fractional Cauchy problems of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ in infinite-dimensional Banach spaces. A new notion, named general fractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ is developed. Some of its properties are obtained. Moreover, some sufficient conditions are presented to guarantee that the mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist. An illustrative example is presented.
This paper is concerned with general fractional Cauchy problems of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ in infinite-dimensional Banach spaces. A new notion, named general fractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ is developed. Some of its properties are obtained. Moreover, some sufficient conditions are presented to guarantee that the mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist. An illustrative example is presented.
2013, 12(6): 2773-2786
doi: 10.3934/cpaa.2013.12.2773
+[Abstract](2645)
+[PDF](369.6KB)
Abstract:
In the present paper, the existence and multiplicity of solutions for Kirchhoff type problem involving critical exponent with Dirichlet boundary value conditions are obtained via the variational method.
In the present paper, the existence and multiplicity of solutions for Kirchhoff type problem involving critical exponent with Dirichlet boundary value conditions are obtained via the variational method.
2013, 12(6): 2787-2795
doi: 10.3934/cpaa.2013.12.2787
+[Abstract](2379)
+[PDF](330.2KB)
Abstract:
In this paper, we study the convergence rates for the reiterated homogenization for equations of the form $-div(A(\frac{x}{\varepsilon},\frac{x}{\varepsilon^{2}})\nabla u_{\varepsilon})=f(x)$. As a consequence, we obtain the convergence rates in $L^{p}$ for solutions with Dirichlet boundary condition by a method based on the representation of elliptic equation solution by Green function. Meanwhile, the growth rate of Green function is found.
In this paper, we study the convergence rates for the reiterated homogenization for equations of the form $-div(A(\frac{x}{\varepsilon},\frac{x}{\varepsilon^{2}})\nabla u_{\varepsilon})=f(x)$. As a consequence, we obtain the convergence rates in $L^{p}$ for solutions with Dirichlet boundary condition by a method based on the representation of elliptic equation solution by Green function. Meanwhile, the growth rate of Green function is found.
2013, 12(6): 2797-2809
doi: 10.3934/cpaa.2013.12.2797
+[Abstract](2037)
+[PDF](381.6KB)
Abstract:
In the present paper we characterize the analytic integrability around the origin of a family of degenerate differential systems. Moreover, we study the analytic integrability of some degenerate systems through the orbital reversibility and from the existence of a Lie's symmetry for these systems. The results obtained for this family are similar to the results obtained when we characterize the analytic integrability of non-degenerate and nilpotent systems. The obtained results can be applied to compute the analytic integrable systems of any particular family of degenerate systems studied.
In the present paper we characterize the analytic integrability around the origin of a family of degenerate differential systems. Moreover, we study the analytic integrability of some degenerate systems through the orbital reversibility and from the existence of a Lie's symmetry for these systems. The results obtained for this family are similar to the results obtained when we characterize the analytic integrability of non-degenerate and nilpotent systems. The obtained results can be applied to compute the analytic integrable systems of any particular family of degenerate systems studied.
2013, 12(6): 2811-2827
doi: 10.3934/cpaa.2013.12.2811
+[Abstract](2399)
+[PDF](442.9KB)
Abstract:
The aim of this article is to study the existence and uniqueness of solutions for an equation of Allen-Cahn type and to prove the existence of the finite-dimensional global attractor as well as the existence of exponential attractors.
The aim of this article is to study the existence and uniqueness of solutions for an equation of Allen-Cahn type and to prove the existence of the finite-dimensional global attractor as well as the existence of exponential attractors.
2013, 12(6): 2829-2838
doi: 10.3934/cpaa.2013.12.2829
+[Abstract](2493)
+[PDF](349.1KB)
Abstract:
In this short note, we exploit the tools of multivalued dynamical systems to prove that the stationary statistical properties of the fully implicit Euler scheme converge, as the time-step parameter vanishes, to the stationary statistical properties of the two-dimensional Navier-Stokes equations.
In this short note, we exploit the tools of multivalued dynamical systems to prove that the stationary statistical properties of the fully implicit Euler scheme converge, as the time-step parameter vanishes, to the stationary statistical properties of the two-dimensional Navier-Stokes equations.
2013, 12(6): 2839-2872
doi: 10.3934/cpaa.2013.12.2839
+[Abstract](2086)
+[PDF](583.6KB)
Abstract:
In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.
In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.
2013, 12(6): 2873-2888
doi: 10.3934/cpaa.2013.12.2873
+[Abstract](2224)
+[PDF](382.7KB)
Abstract:
In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in $R^n(n=2,3)$. Firstly, we establish a regularity criterion in terms of the $BMO$ norm of the gradient of columns of the deformation tensor in two space dimensions; secondly, we obtain a Beale-Kato-Majda-type criterion in terms of vorticity with the $BMO$ norm in two and three space dimensions.
In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in $R^n(n=2,3)$. Firstly, we establish a regularity criterion in terms of the $BMO$ norm of the gradient of columns of the deformation tensor in two space dimensions; secondly, we obtain a Beale-Kato-Majda-type criterion in terms of vorticity with the $BMO$ norm in two and three space dimensions.
2013, 12(6): 2889-2922
doi: 10.3934/cpaa.2013.12.2889
+[Abstract](2335)
+[PDF](535.2KB)
Abstract:
We consider nonlinear Neumann and Dirichlet problems driven by a nonhomogeneous differential operator and a Caratheodory reaction. Our framework incorporates $p$-Laplacian equations and equations with the $(p,q)$-differential operator and with the generalized $p$-mean curvature operator. Using variational methods, together with truncation and comparison techniques and Morse theory, we prove multiplicity theorems, producing three, five or six nontrivial smooth solutions, all with sign information.
We consider nonlinear Neumann and Dirichlet problems driven by a nonhomogeneous differential operator and a Caratheodory reaction. Our framework incorporates $p$-Laplacian equations and equations with the $(p,q)$-differential operator and with the generalized $p$-mean curvature operator. Using variational methods, together with truncation and comparison techniques and Morse theory, we prove multiplicity theorems, producing three, five or six nontrivial smooth solutions, all with sign information.
2013, 12(6): 2923-2933
doi: 10.3934/cpaa.2013.12.2923
+[Abstract](2380)
+[PDF](490.3KB)
Abstract:
In this paper, we consider an abstract equation $F(\lambda,u)=0$ with one parameter $\lambda$, where $F\in C^p(\mathbb{R} \times X, Y)$, $p\geq 2$, is a nonlinear differentiable mapping, and $X,Y$ are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a "double" saddle-node bifurcation theorem with a two-dimensional kernel. Applications include a perturbed problem and a semilinear elliptic equation.
In this paper, we consider an abstract equation $F(\lambda,u)=0$ with one parameter $\lambda$, where $F\in C^p(\mathbb{R} \times X, Y)$, $p\geq 2$, is a nonlinear differentiable mapping, and $X,Y$ are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a "double" saddle-node bifurcation theorem with a two-dimensional kernel. Applications include a perturbed problem and a semilinear elliptic equation.
2013, 12(6): 2935-2946
doi: 10.3934/cpaa.2013.12.2935
+[Abstract](2526)
+[PDF](385.6KB)
Abstract:
This paper deals with the blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. The local existence and uniqueness of the solution are obtained. Furthermore, we prove that the solution of the equation blows up in finite time. Under appropriate hypotheses, we give the estimates of the blow-up rate, and obtain that the blow-up set is a single point $x=0$ for radially symmetric solution with a single maximum at the origin. Finally, some numerical experiments are performed, which illustrate our results.
This paper deals with the blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. The local existence and uniqueness of the solution are obtained. Furthermore, we prove that the solution of the equation blows up in finite time. Under appropriate hypotheses, we give the estimates of the blow-up rate, and obtain that the blow-up set is a single point $x=0$ for radially symmetric solution with a single maximum at the origin. Finally, some numerical experiments are performed, which illustrate our results.
2013, 12(6): 2947-2971
doi: 10.3934/cpaa.2013.12.2947
+[Abstract](2064)
+[PDF](470.4KB)
Abstract:
We study a single mode for the Kirchhoff string vibrating in space. In 3D a single mode is generally almost periodic in contrast to the 2D periodic case. In order to show a complete geometrical description of a single mode we prove some monotonicity properties of the almost periods of the solution, with respect to the mechanical energy and the momentum. As a consequence of these properties, we observe that a planar single mode in 3D is always unstable, while it is known that a single mode in 2D is stable (under a suitable definition of stability), if the energy is small.
We study a single mode for the Kirchhoff string vibrating in space. In 3D a single mode is generally almost periodic in contrast to the 2D periodic case. In order to show a complete geometrical description of a single mode we prove some monotonicity properties of the almost periods of the solution, with respect to the mechanical energy and the momentum. As a consequence of these properties, we observe that a planar single mode in 3D is always unstable, while it is known that a single mode in 2D is stable (under a suitable definition of stability), if the energy is small.
2013, 12(6): 2973-2996
doi: 10.3934/cpaa.2013.12.2973
+[Abstract](1968)
+[PDF](530.0KB)
Abstract:
We propose a temperature dependent model for fatigue accumulation in an oscillating elastoplastic beam. The full system consists of the momentum and energy balance equations, and an evolution equation for the fatigue rate. The main modeling hypothesis is that the fatigue accumulation rate is proportional to the dissipation rate. In nontrivial cases, the process develops a thermal singularity in finite time. The main result consists in proving the existence and uniqueness of a strong solution in a time interval depending on the size of the data.
We propose a temperature dependent model for fatigue accumulation in an oscillating elastoplastic beam. The full system consists of the momentum and energy balance equations, and an evolution equation for the fatigue rate. The main modeling hypothesis is that the fatigue accumulation rate is proportional to the dissipation rate. In nontrivial cases, the process develops a thermal singularity in finite time. The main result consists in proving the existence and uniqueness of a strong solution in a time interval depending on the size of the data.
2013, 12(6): 2997-3012
doi: 10.3934/cpaa.2013.12.2997
+[Abstract](1723)
+[PDF](541.7KB)
Abstract:
Variable population interactions with harvesting on one of the species are studied. Existence and stability of equilibria and existence of periodic solutions are established, existence of some bifurcation phenomena are analytically and numerically studied, explicit threshold values are computed to determine the kind of interaction (mutualism, competition, host-parasite) between the species, and several numerical examples are provided to illustrate the main results in this work. A brief discussion on the influence of the harvesting function on the dynamics of the model is also included. Hopf bifurcations and periodic solutions are found for the first time in this kind of models.
Variable population interactions with harvesting on one of the species are studied. Existence and stability of equilibria and existence of periodic solutions are established, existence of some bifurcation phenomena are analytically and numerically studied, explicit threshold values are computed to determine the kind of interaction (mutualism, competition, host-parasite) between the species, and several numerical examples are provided to illustrate the main results in this work. A brief discussion on the influence of the harvesting function on the dynamics of the model is also included. Hopf bifurcations and periodic solutions are found for the first time in this kind of models.
2013, 12(6): 3013-3026
doi: 10.3934/cpaa.2013.12.3013
+[Abstract](1946)
+[PDF](519.1KB)
Abstract:
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric. The semi-linear problems are studied in a framework where the associated functional is of class $C^1$ but not of class $C^2$.
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric. The semi-linear problems are studied in a framework where the associated functional is of class $C^1$ but not of class $C^2$.
2013, 12(6): 3027-3046
doi: 10.3934/cpaa.2013.12.3027
+[Abstract](2316)
+[PDF](464.0KB)
Abstract:
In the first part of this paper, we investigate the qualitative behavior of classical solutions for a one-dimensional parabolic system derived from a repulsive chemotaxis model on bounded domains. It is shown that classical solutions to the initial-boundary value problem exist globally in time for large data and converge to constant equilibrium states exponentially in time. The results indicate that repulsive chemotaxis exhibits a strong tendency against pattern formation. In the second part, we study diffusion limit and convergence rate of the model toward a non-diffusive problem studied in [11]. It is shown that when the chemical diffusion coefficient $\varepsilon$ tends to zero, the solution is convergent in $L^{\infty}$-norm with respect to $\varepsilon$ at order $O(\varepsilon)$.
In the first part of this paper, we investigate the qualitative behavior of classical solutions for a one-dimensional parabolic system derived from a repulsive chemotaxis model on bounded domains. It is shown that classical solutions to the initial-boundary value problem exist globally in time for large data and converge to constant equilibrium states exponentially in time. The results indicate that repulsive chemotaxis exhibits a strong tendency against pattern formation. In the second part, we study diffusion limit and convergence rate of the model toward a non-diffusive problem studied in [11]. It is shown that when the chemical diffusion coefficient $\varepsilon$ tends to zero, the solution is convergent in $L^{\infty}$-norm with respect to $\varepsilon$ at order $O(\varepsilon)$.
2013, 12(6): 3047-3071
doi: 10.3934/cpaa.2013.12.3047
+[Abstract](2218)
+[PDF](441.2KB)
Abstract:
We construct exponential pullback attractors for time continuous asymptotically compact evolution processes in Banach spaces and derive estimates on the fractal dimension of the attractors. We also discuss the corresponding results for autonomous processes.
We construct exponential pullback attractors for time continuous asymptotically compact evolution processes in Banach spaces and derive estimates on the fractal dimension of the attractors. We also discuss the corresponding results for autonomous processes.
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