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1534-0392
eISSN:
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Communications on Pure & Applied Analysis
March 2015 , Volume 14 , Issue 2
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2015, 14(2): 361-371
doi: 10.3934/cpaa.2015.14.361
+[Abstract](1707)
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Abstract:
In this paper by using the minimal principle and Morse theory, we prove the existence of solutions to the following Kirchhoff nonlocal fractional equation: \begin{eqnarray} & M \left (\int_{\mathbb{R}^n\times \mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y \right) (- \Delta)^s u = f (x, u (x)),\quad \textrm{in}\;\; \Omega,\\ & u = 0, \quad \textrm{in}\;\; \mathbb{R}^n \setminus \Omega, \end{eqnarray} where $(- \Delta)^s$ is the fractional Laplace operator, $s \in (0, 1)$ is a fix, $\Omega$ an open bounded subset of $\mathbb{R}^n$, $n > 2 s$, with Lipschitz boundary, $f: \Omega \times \mathbb{R} \to \mathbb{R}$ Carathéodory function and $M : \mathbb{R}^+ \to \mathbb{R}^+$ is a function that satisfy some suitable conditions.
In this paper by using the minimal principle and Morse theory, we prove the existence of solutions to the following Kirchhoff nonlocal fractional equation: \begin{eqnarray} & M \left (\int_{\mathbb{R}^n\times \mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y \right) (- \Delta)^s u = f (x, u (x)),\quad \textrm{in}\;\; \Omega,\\ & u = 0, \quad \textrm{in}\;\; \mathbb{R}^n \setminus \Omega, \end{eqnarray} where $(- \Delta)^s$ is the fractional Laplace operator, $s \in (0, 1)$ is a fix, $\Omega$ an open bounded subset of $\mathbb{R}^n$, $n > 2 s$, with Lipschitz boundary, $f: \Omega \times \mathbb{R} \to \mathbb{R}$ Carathéodory function and $M : \mathbb{R}^+ \to \mathbb{R}^+$ is a function that satisfy some suitable conditions.
2015, 14(2): 373-382
doi: 10.3934/cpaa.2015.14.373
+[Abstract](1879)
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Abstract:
In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian. This covers a critical case left open in [9].
In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian. This covers a critical case left open in [9].
2015, 14(2): 383-396
doi: 10.3934/cpaa.2015.14.383
+[Abstract](2104)
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Abstract:
We study a Keller-Segel type chemotaxis model with a modified sensitivity function in a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq2$. The global existence of classical solutions to the fully parabolic system is established provided that the ratio of the chemotactic coefficient to the motility of cells is not too large.
We study a Keller-Segel type chemotaxis model with a modified sensitivity function in a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq2$. The global existence of classical solutions to the fully parabolic system is established provided that the ratio of the chemotactic coefficient to the motility of cells is not too large.
2015, 14(2): 397-406
doi: 10.3934/cpaa.2015.14.397
+[Abstract](1873)
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Abstract:
We analyze the asymptotic stability of solutions of linear Volterra integral equations with general continuous convolution kernels and vanishing delays. The analysis is based on an extension of the variation-of-parameter formula for non-delay Volterra integral equations and on energy function techniques. The delay integral equations studied in this paper will be of interest in the (still open) stability analysis of numerical methods (e.g. collocation and Runge-Kutta-type methods) for Volterra integral equations with vanishing delays.
We analyze the asymptotic stability of solutions of linear Volterra integral equations with general continuous convolution kernels and vanishing delays. The analysis is based on an extension of the variation-of-parameter formula for non-delay Volterra integral equations and on energy function techniques. The delay integral equations studied in this paper will be of interest in the (still open) stability analysis of numerical methods (e.g. collocation and Runge-Kutta-type methods) for Volterra integral equations with vanishing delays.
2015, 14(2): 407-419
doi: 10.3934/cpaa.2015.14.407
+[Abstract](1494)
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Abstract:
We study second order elliptic operators whose diffusion coefficients degenerate at the boundary in first order and whose drift term strongly points outward. It is shown that these operators generate analytic semigroups in $L^2$ where they are equipped with their natural domain without boundary conditions. Hence, the corresponding parabolic problem can be solved with optimal regularity. In a previous work we had treated the case of inward pointing drift terms.
We study second order elliptic operators whose diffusion coefficients degenerate at the boundary in first order and whose drift term strongly points outward. It is shown that these operators generate analytic semigroups in $L^2$ where they are equipped with their natural domain without boundary conditions. Hence, the corresponding parabolic problem can be solved with optimal regularity. In a previous work we had treated the case of inward pointing drift terms.
2015, 14(2): 421-437
doi: 10.3934/cpaa.2015.14.421
+[Abstract](1626)
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Abstract:
This paper focuses on quasi-periodic perturbation of four dimensional nonlinear quasi-periodic system. Using the KAM method, the perturbed system can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.
This paper focuses on quasi-periodic perturbation of four dimensional nonlinear quasi-periodic system. Using the KAM method, the perturbed system can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.
2015, 14(2): 439-455
doi: 10.3934/cpaa.2015.14.439
+[Abstract](2000)
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Abstract:
We study the following elliptic equation with two Sobolev-Hardy critical exponents \begin{eqnarray} & -\Delta u=\mu\frac{|u|^{2^{*}(s_1)-2}u}{|x|^{s_1}}+\frac{|u|^{2^{*}(s_2)-2}u}{|x|^{s_2}} \quad x\in \Omega, \\ & u=0, \quad x\in \partial\Omega, \end{eqnarray} where $\Omega\subset R^N (N\geq3)$ is a bounded smooth domain, $0\in\partial\Omega$, $0\leq s_2 < s_1 \leq 2$ and $2^*(s):=\frac{2(N-s)}{N-2}$. In this paper, by means of variational methods, we obtain the existence of sign-changing solutions if $H(0)<0$, where $H(0)$ denote the mean curvature of $\partial\Omega$ at $0$.
We study the following elliptic equation with two Sobolev-Hardy critical exponents \begin{eqnarray} & -\Delta u=\mu\frac{|u|^{2^{*}(s_1)-2}u}{|x|^{s_1}}+\frac{|u|^{2^{*}(s_2)-2}u}{|x|^{s_2}} \quad x\in \Omega, \\ & u=0, \quad x\in \partial\Omega, \end{eqnarray} where $\Omega\subset R^N (N\geq3)$ is a bounded smooth domain, $0\in\partial\Omega$, $0\leq s_2 < s_1 \leq 2$ and $2^*(s):=\frac{2(N-s)}{N-2}$. In this paper, by means of variational methods, we obtain the existence of sign-changing solutions if $H(0)<0$, where $H(0)$ denote the mean curvature of $\partial\Omega$ at $0$.
2015, 14(2): 457-491
doi: 10.3934/cpaa.2015.14.457
+[Abstract](2099)
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Abstract:
In this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given.
In this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given.
2015, 14(2): 493-515
doi: 10.3934/cpaa.2015.14.493
+[Abstract](1787)
+[PDF](442.0KB)
Abstract:
This paper examines systems of poly-harmonic equations of the Hardy--Sobolev type and the closely related weighted systems of integral equations involving Riesz potentials. Namely, it is shown that the two systems are equivalent under some appropriate conditions. Then a sharp criterion for the existence and non-existence of positive solutions is determined for both differential and integral versions of a Hardy--Sobolev type system with variable coefficients. In the constant coefficient case, Liouville type theorems for positive radial solutions are also established using radial decay estimates and Pohozaev type identities in integral form.
This paper examines systems of poly-harmonic equations of the Hardy--Sobolev type and the closely related weighted systems of integral equations involving Riesz potentials. Namely, it is shown that the two systems are equivalent under some appropriate conditions. Then a sharp criterion for the existence and non-existence of positive solutions is determined for both differential and integral versions of a Hardy--Sobolev type system with variable coefficients. In the constant coefficient case, Liouville type theorems for positive radial solutions are also established using radial decay estimates and Pohozaev type identities in integral form.
2015, 14(2): 517-525
doi: 10.3934/cpaa.2015.14.517
+[Abstract](1568)
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Abstract:
The Monge-Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry in recent years. Shen and Zheng propose a probability method to transform the celebrated Monge-Kantorovich problem in a bounded region of the Euclidean plane into a Dirichlet boundary problem associated to a nonlinear elliptic equation. Their results are original and sound, however, their arguments leading to the main results are skipped and difficult to follow. In the present paper, we adopt a different approach and give a short and easy-followed detailed proof for their main results.
The Monge-Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry in recent years. Shen and Zheng propose a probability method to transform the celebrated Monge-Kantorovich problem in a bounded region of the Euclidean plane into a Dirichlet boundary problem associated to a nonlinear elliptic equation. Their results are original and sound, however, their arguments leading to the main results are skipped and difficult to follow. In the present paper, we adopt a different approach and give a short and easy-followed detailed proof for their main results.
2015, 14(2): 527-548
doi: 10.3934/cpaa.2015.14.527
+[Abstract](2066)
+[PDF](481.9KB)
Abstract:
In this article we consider the following integral equation involving Bessel potentials on a half space $\mathbb{R}^n_+ $: \begin{eqnarray} u(x)=\int_{ \mathbb{R}^n_+ }\{g_\alpha(x-y)-g_\alpha(\bar x-y)\} u^\beta(y) dy,\;\;x\in \mathbb{R}^n_+, \end{eqnarray} where $\alpha>0$, $\beta>1$, $\bar x$ is the reflection of $x$ about $x_n=0$, and $g_\alpha(x)$ denotes the Bessel kernel. We first enhance the regularity of positive solutions for the integral equation by regularity-lifting-method, which has been extensively used by many authors. Then, employing the method of moving planes in integral forms, we demonstrate that there is no positive solution for the integral equation.
In this article we consider the following integral equation involving Bessel potentials on a half space $\mathbb{R}^n_+ $: \begin{eqnarray} u(x)=\int_{ \mathbb{R}^n_+ }\{g_\alpha(x-y)-g_\alpha(\bar x-y)\} u^\beta(y) dy,\;\;x\in \mathbb{R}^n_+, \end{eqnarray} where $\alpha>0$, $\beta>1$, $\bar x$ is the reflection of $x$ about $x_n=0$, and $g_\alpha(x)$ denotes the Bessel kernel. We first enhance the regularity of positive solutions for the integral equation by regularity-lifting-method, which has been extensively used by many authors. Then, employing the method of moving planes in integral forms, we demonstrate that there is no positive solution for the integral equation.
2015, 14(2): 549-564
doi: 10.3934/cpaa.2015.14.549
+[Abstract](2006)
+[PDF](420.8KB)
Abstract:
In this paper, we investigate the singularities formation for the relativistic Euler and Euler-Poisson equations with repulsive force, in spherical symmetry. We will show the non-trivial regular solution $(p^{\frac{\gamma-1}{2\gamma}}\in C^1$) with compact support in $[0,R]$, under a certain condition on initial data \begin{eqnarray} H_0:=\int_0^R rv_0dr>0, \end{eqnarray} will blow up, where $0 < R < \min\{\frac{8}{45},\frac{8(\gamma-1)}{5\gamma+23}\}$ is a constant. Since every term in the relativistic Euler-Poisson equations corresponding to $\rho, v$ in non-relativistic case [34] has a relativistic factor $\frac{1}{\sqrt{1-v^2/c^2}}$, we will separate variables and estimate some integration items instead of direct using integration method as [34].
In this paper, we investigate the singularities formation for the relativistic Euler and Euler-Poisson equations with repulsive force, in spherical symmetry. We will show the non-trivial regular solution $(p^{\frac{\gamma-1}{2\gamma}}\in C^1$) with compact support in $[0,R]$, under a certain condition on initial data \begin{eqnarray} H_0:=\int_0^R rv_0dr>0, \end{eqnarray} will blow up, where $0 < R < \min\{\frac{8}{45},\frac{8(\gamma-1)}{5\gamma+23}\}$ is a constant. Since every term in the relativistic Euler-Poisson equations corresponding to $\rho, v$ in non-relativistic case [34] has a relativistic factor $\frac{1}{\sqrt{1-v^2/c^2}}$, we will separate variables and estimate some integration items instead of direct using integration method as [34].
2015, 14(2): 565-576
doi: 10.3934/cpaa.2015.14.565
+[Abstract](1816)
+[PDF](386.0KB)
Abstract:
In this paper we establish a Liouville type theorem for positive solutions of a class of system of integral equations. Firstly, we show the local regularity lifting result with the help of the Hardy-Littlewood-Sobolev inequality. Then by the method of moving planes in integral forms, we obtain a Liouville type theorem for this system.
In this paper we establish a Liouville type theorem for positive solutions of a class of system of integral equations. Firstly, we show the local regularity lifting result with the help of the Hardy-Littlewood-Sobolev inequality. Then by the method of moving planes in integral forms, we obtain a Liouville type theorem for this system.
2015, 14(2): 577-584
doi: 10.3934/cpaa.2015.14.577
+[Abstract](1771)
+[PDF](360.2KB)
Abstract:
We prove that the energy over balls of entire, nonconstant bounded solutions to the vector Allen-Cahn equation grows faster than $(\ln R)^k R^{n-2}$, for any $k>0$, as the radius $R$ of the $n$-dimensional ball tends to infinity. This improves the growth rate of order $R^{n-2}$ if $n\geq 3$ and $\ln R$ if $n=2$ that follows from the general weak monotonicity formula. Moreover, our estimate may be considered as an approximation to the corresponding rate of order $R^{n-1}$ that is known to hold in the scalar case.
We prove that the energy over balls of entire, nonconstant bounded solutions to the vector Allen-Cahn equation grows faster than $(\ln R)^k R^{n-2}$, for any $k>0$, as the radius $R$ of the $n$-dimensional ball tends to infinity. This improves the growth rate of order $R^{n-2}$ if $n\geq 3$ and $\ln R$ if $n=2$ that follows from the general weak monotonicity formula. Moreover, our estimate may be considered as an approximation to the corresponding rate of order $R^{n-1}$ that is known to hold in the scalar case.
Note on global
regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero
diffusivity
2015, 14(2): 585-595
doi: 10.3934/cpaa.2015.14.585
+[Abstract](1603)
+[PDF](352.5KB)
Abstract:
In this paper we consider the Cauchy problem of a three-dimensional incompressible magnetohydrodynamic-$\alpha$ (MHD-$\alpha$) model with fractional velocity and zero magnetic diffusivity. We prove the global existence results of classical solutions for the model in the endpoint case with arbitrarily large initial data in Sobolev spaces.
In this paper we consider the Cauchy problem of a three-dimensional incompressible magnetohydrodynamic-$\alpha$ (MHD-$\alpha$) model with fractional velocity and zero magnetic diffusivity. We prove the global existence results of classical solutions for the model in the endpoint case with arbitrarily large initial data in Sobolev spaces.
2015, 14(2): 597-607
doi: 10.3934/cpaa.2015.14.597
+[Abstract](1612)
+[PDF](357.4KB)
Abstract:
Fix a function $W(x_1,\ldots,x_d) = \sum_{k=1}^d W_k(x_k)$ where each $W_k: R \to R$ is a right continuous with left limits and strictly increasing function, and consider the $W$-laplacian given by $\Delta_W = \sum_{i=1}^d \partial_{x_i}\partial_{W_i}$, which is a generalization of the laplacian operator. In this work we introduce the $W$-Sobolev spaces of higher order, thus extending the notion of $W$-Sobolev spaces introduced in Simas and Valentim (2011) [7]. We then provide a characterization of these spaces in terms of a suitable Fourier series, and conclude the paper with some results on elliptic regularity of the problem $\lambda u - \Delta_Wu = f,$ for $\lambda\geq 0$.
Fix a function $W(x_1,\ldots,x_d) = \sum_{k=1}^d W_k(x_k)$ where each $W_k: R \to R$ is a right continuous with left limits and strictly increasing function, and consider the $W$-laplacian given by $\Delta_W = \sum_{i=1}^d \partial_{x_i}\partial_{W_i}$, which is a generalization of the laplacian operator. In this work we introduce the $W$-Sobolev spaces of higher order, thus extending the notion of $W$-Sobolev spaces introduced in Simas and Valentim (2011) [7]. We then provide a characterization of these spaces in terms of a suitable Fourier series, and conclude the paper with some results on elliptic regularity of the problem $\lambda u - \Delta_Wu = f,$ for $\lambda\geq 0$.
2015, 14(2): 609-622
doi: 10.3934/cpaa.2015.14.609
+[Abstract](1841)
+[PDF](420.9KB)
Abstract:
Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided only the horizontal components of the initial data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in [1,\infty)$.
Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided only the horizontal components of the initial data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in [1,\infty)$.
2015, 14(2): 623-626
doi: 10.3934/cpaa.2015.14.623
+[Abstract](1622)
+[PDF](287.2KB)
Abstract:
We establish the strong unique continuation property for solutions to (1.1) where $F$ satisfies the structural assumptions A)-D). This extends a recent result of Armstrong and Silvestre (see [3]) where $F$ was assumed to be independent of $x$. We also establish an analogous unique continuation result at the boundary along the lines of [1] when the domain is $C^{3, \alpha}$.
We establish the strong unique continuation property for solutions to (1.1) where $F$ satisfies the structural assumptions A)-D). This extends a recent result of Armstrong and Silvestre (see [3]) where $F$ was assumed to be independent of $x$. We also establish an analogous unique continuation result at the boundary along the lines of [1] when the domain is $C^{3, \alpha}$.
2015, 14(2): 627-636
doi: 10.3934/cpaa.2015.14.627
+[Abstract](1619)
+[PDF](333.0KB)
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In this paper, the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the pseudo-differential operator is obtained.
In this paper, the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the pseudo-differential operator is obtained.
2015, 14(2): 637-655
doi: 10.3934/cpaa.2015.14.637
+[Abstract](1768)
+[PDF](446.8KB)
Abstract:
In this paper, we prove some logarithmically improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain.
In this paper, we prove some logarithmically improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain.
2015, 14(2): 657-676
doi: 10.3934/cpaa.2015.14.657
+[Abstract](2194)
+[PDF](521.7KB)
Abstract:
This paper is devoted to the study of an age-structured population system with Riker type birth function. Two time lag factors is considered for the model. One lag lies in the birth process and the another is in the birth function. We investigate some dynamical properties of the equation by using integrated semigroup theory, through which we obtain some conditions of asymptotical stability and Hopf bifurcation occurring at positive steady state for the system. The obtained results show how the two delays affect these dynamical properties.
This paper is devoted to the study of an age-structured population system with Riker type birth function. Two time lag factors is considered for the model. One lag lies in the birth process and the another is in the birth function. We investigate some dynamical properties of the equation by using integrated semigroup theory, through which we obtain some conditions of asymptotical stability and Hopf bifurcation occurring at positive steady state for the system. The obtained results show how the two delays affect these dynamical properties.
2015, 14(2): 677-693
doi: 10.3934/cpaa.2015.14.677
+[Abstract](2353)
+[PDF](428.4KB)
Abstract:
We study a class of nonlinear fourth order differential equations which arise as models of suspension bridges. When it comes to power-like nonlinearities, it is known that solutions may blow up in finite time, if the initial data satisfy some positivity assumption. We extend this result to more general nonlinearities allowing exponential growth and to a wider class of initial data. We also give some hints on how to prevent blow-up.
We study a class of nonlinear fourth order differential equations which arise as models of suspension bridges. When it comes to power-like nonlinearities, it is known that solutions may blow up in finite time, if the initial data satisfy some positivity assumption. We extend this result to more general nonlinearities allowing exponential growth and to a wider class of initial data. We also give some hints on how to prevent blow-up.
2015, 14(2): 695-716
doi: 10.3934/cpaa.2015.14.695
+[Abstract](1651)
+[PDF](464.6KB)
Abstract:
we prove the existence of a global attractor to dissipative Klein-Gordon-Schrödinger (KGS) system with cubic nonlinearities in $H^1({\mathbb R}^2)\times H^1({\mathbb R}^2)\times L^2({\mathbb R}^2)$ and more particularly that this attractor is in fact a compact set of $H^2({\mathbb R}^2)\times H^2({\mathbb R}^2)\times H^1({\mathbb R}^2)$.
we prove the existence of a global attractor to dissipative Klein-Gordon-Schrödinger (KGS) system with cubic nonlinearities in $H^1({\mathbb R}^2)\times H^1({\mathbb R}^2)\times L^2({\mathbb R}^2)$ and more particularly that this attractor is in fact a compact set of $H^2({\mathbb R}^2)\times H^2({\mathbb R}^2)\times H^1({\mathbb R}^2)$.
2015, 14(2): 717-736
doi: 10.3934/cpaa.2015.14.717
+[Abstract](1866)
+[PDF](578.7KB)
Abstract:
Our aim in this paper is to prove the instability of multi-spot patterns in a shadow system, which is obtained as a limiting system of a reaction-diffusion model as one of the diffusion coefficients goes to infinity. Instead of investigating each eigenfunction for a linearized operator, we characterize the eigenspace spanned by unstable eigenfunctions.
Our aim in this paper is to prove the instability of multi-spot patterns in a shadow system, which is obtained as a limiting system of a reaction-diffusion model as one of the diffusion coefficients goes to infinity. Instead of investigating each eigenfunction for a linearized operator, we characterize the eigenspace spanned by unstable eigenfunctions.
2015, 14(2): 737-742
doi: 10.3934/cpaa.2015.14.737
+[Abstract](1642)
+[PDF](248.9KB)
Abstract:
The stated theorems in [1] remain completely unchanged. However, the proof of Proposition 2.1 has to be modified, because in several places Cor. 1.1 was used for $\beta_- < \frac{1}{4}$, which is not admissible. Instead we use that the nonlinearity satisfies two null conditions, namely $\langle \beta \psi,\psi \rangle$ on one hand and the factor $\beta \psi$ produces a second null condition by duality on the other hand. The latter property was not used before and gives an additional regularizing factor which allows to use Cor. 1.1 correctly. Here and in the following we use the numbering and notation of [1].
The stated theorems in [1] remain completely unchanged. However, the proof of Proposition 2.1 has to be modified, because in several places Cor. 1.1 was used for $\beta_- < \frac{1}{4}$, which is not admissible. Instead we use that the nonlinearity satisfies two null conditions, namely $\langle \beta \psi,\psi \rangle$ on one hand and the factor $\beta \psi$ produces a second null condition by duality on the other hand. The latter property was not used before and gives an additional regularizing factor which allows to use Cor. 1.1 correctly. Here and in the following we use the numbering and notation of [1].
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