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Communications on Pure & Applied Analysis

2014 , Volume 13 , Issue 2

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Nontrivial solutions for Kirchhoff type equations via Morse theory
Jijiang Sun and Shiwang Ma
2014, 13(2): 483-494 doi: 10.3934/cpaa.2014.13.483 +[Abstract](424) +[PDF](382.4KB)
Abstract:
In this paper, the existence of nontrivial solutions is obtained for a class of Kirchhoff type problems with Dirichlet boundary conditions by computing the critical groups and Morse theory.
Schrödinger-like operators and the eikonal equation
Jaime Cruz-Sampedro
2014, 13(2): 495-510 doi: 10.3934/cpaa.2014.13.495 +[Abstract](398) +[PDF](449.9KB)
Abstract:
Let $V$ be a real-valued function of class $C^5$ on $\mathbb{R}^n$, $n \geq 2$, and suppose that $\partial^\alpha V(x)=O(|x|^{-|\alpha|})$, as $|x| \to \infty$, for $|\alpha| \leq 5$. For $\lambda > 0$ we set $W_\lambda(x) = 1-(V(x)/\lambda)$ and consider the Schrödinger-like operator $\mathcal{H}_\lambda=W_\lambda^{-{1/2}} H_0 W_\lambda^{-{1/2}}$ acting on $L^2(\mathbb{R}^n)$, where $H_0=-\Delta$ is the classical laplacian on $\mathbb{R}^n$. Using properties of the maximal solution to the eikonal equation $|\nabla S_\lambda|^2=W_\lambda$, for $\lambda$ sufficiently large we establish the behavior of $(\mathcal{H}_\lambda-z^2)^{-1}$ as Im $z\downarrow 0$ in the framework of Besov Spaces $B(\mathbb{R}^n)$. For $k\in \mathbb{R}\setminus\{0\}$ and $f\in B(\mathbb{R}^n)$ we find the unique solution to $-\Delta u-k^2 W_\lambda u = f $ on $\mathbb{R}^n$ that satisfies a certain radiation condition. These results can be applied to the study of the scattering theory of the Schrödinger operator $H=-\Delta+V$.
Liouville type theorem to an integral system in the half-space
Weiwei Zhao, Jinge Yang and Sining Zheng
2014, 13(2): 511-525 doi: 10.3934/cpaa.2014.13.511 +[Abstract](484) +[PDF](420.8KB)
Abstract:
In this paper, by using the moving plane method in integral forms, we establish a Liouville type theorem for a coupled integral system with Navier boundary values in the half-space. Furthermore, we prove that the Liouville type theorem is valid for the related differential system as well under an additional assumption by showing the equivalence between the involved differential and integral systems.
Well-posedness for the supercritical gKdV equation
Nils Strunk
2014, 13(2): 527-542 doi: 10.3934/cpaa.2014.13.527 +[Abstract](459) +[PDF](453.7KB)
Abstract:
In this paper we consider the supercritical generalized Korteweg-de~Vries equation $\partial_t\psi + \partial_{x x x}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5 \leq p \in R$. We prove a local well-posedness result in the homogeneous Besov space $\dot B_\infty^{s_p,2}(R)$, where $s_p=\frac12-\frac{2}{p-1}$ is the scaling critical index. In particular local well-posedness in the smaller inhomogeneous Sobolev space $H^{s_p}(R)$ can be proved similarly. As a byproduct a global well-posedness result for small initial data is also obtained.
Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation
Lihua Min and Xiaoping Yang
2014, 13(2): 543-566 doi: 10.3934/cpaa.2014.13.543 +[Abstract](389) +[PDF](481.0KB)
Abstract:
We consider a fourth order degenerate equation describing thin films over an inclined plane in this paper. A new approximating problem is introduced in order to obtain the local energy estimate of the solution. Based on combined use of local entropy estimate, local energy estimate and the suitable extensions of Stampacchia's Lemma to systems, we obtain the finite speed of propagation property of strong solutions, which has been known for the case of strong slippage $ n<2, $ in the case of weak slippage $ 2 \leq n < 3. $ The long time behavior of positive classical solutions is also discussed. We apply the entropy dissipation method to quantify the explicit rate of convergence in the $ L^\infty $ norm of the solution, and this improves and extends the previous results.
Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent
Xudong Shang, Jihui Zhang and Yang Yang
2014, 13(2): 567-584 doi: 10.3934/cpaa.2014.13.567 +[Abstract](542) +[PDF](446.0KB)
Abstract:
In this paper, we study the following problem \begin{eqnarray} (-\Delta)^{\frac{\alpha}{2}}u = K(x)|u|^{2_{\alpha}^{*}-2}u + f(x) \quad in \ \Omega,\\ u=0 \quad on \ \partial \Omega, \end{eqnarray} where $\Omega\subset R^N$ is a smooth bounded domain, $0< \alpha < 2$, $N>\alpha$, $ 2_{\alpha}^{*}= \frac{2N}{N-\alpha}$, $f\in H^{-\frac{\alpha}{2}}(\Omega)$ and $K(x)\in L^\infty(\Omega)$. Under appropriate assumptions on $K$ and $f$, we prove that this problem has at least two positive solutions. When $\alpha = 1$, we also establish a nonexistence result for a positive solution in a class of linear positive-type domains are more general than star-shaped ones.
Regularity criterion for 3D Navier-Stokes equations in Besov spaces
Daoyuan Fang and Chenyin Qian
2014, 13(2): 585-603 doi: 10.3934/cpaa.2014.13.585 +[Abstract](448) +[PDF](446.8KB)
Abstract:
Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $\partial_3u_3$.
Well-posedness of abstract distributed-order fractional diffusion equations
Junxiong Jia, Jigen Peng and Kexue Li
2014, 13(2): 605-621 doi: 10.3934/cpaa.2014.13.605 +[Abstract](538) +[PDF](424.7KB)
Abstract:
In this paper, based on distributed-order fractional diffusion equation we propose the distributed-order fractional abstract Cauchy problem (DFACP) and study the well-posedness of DFACP. Using functional calculus technique, we prove that the general distributed-order fractional operator generates a bounded analytic $\alpha$-times resolvent operator family or a $C_0$-semigroup under some suitable conditions. In addition, we reveal the relation between two $\alpha$-times resolvent families generated by the sectorial operator $A$ and the special distributed-order fractional operator, $p_1A^{\beta_1}+ p_2A^{\beta_2}+\ldots +p_nA^{\beta_n}$, respectively.
Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials
Ziheng Zhang and Rong Yuan
2014, 13(2): 623-634 doi: 10.3934/cpaa.2014.13.623 +[Abstract](444) +[PDF](399.1KB)
Abstract:
In this paper we investigate the existence of infinitely many homoclinic solutions for the following damped vibration problems \begin{eqnarray} \ddot q+A \dot q-L(t)q+W_q(t,q)=0, \end{eqnarray} where $A$ is an antisymmetric constant matrix, $L\in C(R,R^{n^2})$ is a symmetric and positive definite matrix for all $t\in R$, $W\in C^1(R\times R^n,R)$. The novelty of this paper is that, for the case that $W$ is subquadratic at infinity, we establish two new criteria to guarantee the existence of infinitely many homoclinic solutions for (DS) via the genus properties in critical point theory. Recent results in the literature are generalized and significantly improved.
A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities
Aníbal Rodríguez-Bernal and Alejandro Vidal-López
2014, 13(2): 635-644 doi: 10.3934/cpaa.2014.13.635 +[Abstract](318) +[PDF](355.3KB)
Abstract:
We show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in $L^q(\Omega)$ for each $1\leq q < \infty$. In particular, we do not assume restriction on the growth of the nonlinearites required by the standar local existence theory.
Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials
Daxiong Piao and Xiang Sun
2014, 13(2): 645-655 doi: 10.3934/cpaa.2014.13.645 +[Abstract](326) +[PDF](403.2KB)
Abstract:
In this paper, we consider the boundedness of solutions for a class of impact oscillators with time dependent polynomial potentials, \begin{eqnarray} \ddot{x}+x^{2n+1}+\sum_{i=0}^{2n}p_{i}(t)x^{i}=0, \quad for\ x(t)> 0,\\ x(t)\geq 0,\\ \dot{x}(t_{0}^{+})=-\dot{x}(t_{0}^{-}), \quad if\ x(t_{0})=0, \end{eqnarray} where $n\in N^+$, $p_i(t+1)=p_i(t)$ and $p_i(t)\in C^5(R/Z).$
The Fueter primitive of biaxially monogenic functions
Fabrizio Colombo, Irene Sabadini and Frank Sommen
2014, 13(2): 657-672 doi: 10.3934/cpaa.2014.13.657 +[Abstract](267) +[PDF](410.6KB)
Abstract:
In the recent papers by F. Colombo, I. Sabadini, F. Sommen, "The inverse Fueter mapping theorem", Commun. Pure Appl. Anal., 10 (2011), 1165--1181, and "The inverse Fueter mapping theorem in integral form using spherical monogenics", Israel J. Math., 194 (2013), 485--505, the authors have started a systematic study of the inverse Fueter mapping theorem. In this paper we show that the inversion theorem holds for the case of biaxially monogenic functions. Here there are several additional difficulties with respect to the cases already treated. However, we are still able to prove an integral version of the inverse Fueter mapping theorem. The kernels appearing in the integral representation formula have an explicit representation that can be computed depending on the dimension of the Euclidean space in which the problem is considered.
Local well-posedness for the nonlinear Dirac equation in two space dimensions
Hartmut Pecher
2014, 13(2): 673-685 doi: 10.3934/cpaa.2014.13.673 +[Abstract](387) +[PDF](371.1KB)
Abstract:
The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in $H^s$ for $ s > 1/2$. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.
Topological conjugacies and behavior at infinity
Luis Barreira and Claudia Valls
2014, 13(2): 687-701 doi: 10.3934/cpaa.2014.13.687 +[Abstract](316) +[PDF](368.3KB)
Abstract:
We obtain a version of the Grobman--Hartman theorem in Banach spaces for perturbations of a nonuniform exponential contraction, both for discrete and continuous time. More precisely, we consider the general case of an exponential contraction with an arbitrary nonuniform part and obtained from a nonautonomous dynamics, and we establish the existence of Hölder continuous conjugacies between an exponential contraction and any sufficiently small perturbation. As a nontrivial application, we describe the asymptotic behavior of the topological conjugacies in terms of the perturbations: namely, we show that for perturbations growing in a certain controlled manner the conjugacies approach zero at infinity and that when the perturbations decay exponentially at infinity the conjugacies have the same exponential behavior.
Formal equivalence between normal forms of reversible and hamiltonian dynamical systems
Ricardo Miranda Martins
2014, 13(2): 703-713 doi: 10.3934/cpaa.2014.13.703 +[Abstract](450) +[PDF](342.1KB)
Abstract:
We show the existence of formal equivalences between $2n$-dimensional reversible and Hamiltonian vector fields. The main tool we employ is the normal form theory.
Lifespan theorem and gap lemma for the globally constrained Willmore flow
Yannan Liu and Linfen Cao
2014, 13(2): 715-728 doi: 10.3934/cpaa.2014.13.715 +[Abstract](342) +[PDF](347.1KB)
Abstract:
We study a fourth-order flow, which can be seen as a globally constrained Willmore flow. We obtain a lower bound on the lifespan of the smooth solution, which depends on the concentration of curvature for the initial surface and the constrained term. We also give a gap lemma for this flow, which is an important lemma in the study of the blowup analysis.
Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations
Luisa Arlotti, Bertrand Lods and Mustapha Mokhtar-Kharroubi
2014, 13(2): 729-771 doi: 10.3934/cpaa.2014.13.729 +[Abstract](402) +[PDF](611.8KB)
Abstract:
We provide a honesty theory of substochastic evolution families in real abstract state space, extending to an non-autonomous setting the result obtained for $C_0$-semigroups in our recent contribution [On perturbed substochastic semigroups in abstract state spaces, Z. Anal. Anwend. 30, 457--495, 2011]. The link with the honesty theory of perturbed substochastic semigroups is established. Application to non-autonomous linear Boltzmann equation is provided.
Positive solutions to involving Wolff potentials
Huan-Zhen Chen and Zhongxue Lü
2014, 13(2): 773-788 doi: 10.3934/cpaa.2014.13.773 +[Abstract](396) +[PDF](418.6KB)
Abstract:
In this paper, we consider the weighted integral system involving Wolff potentials in $R^{n}$: \begin{eqnarray} u(x) = R_1(x)W_{\beta, \gamma}(\frac{u^pv^q(y)}{|y|^\sigma})(x), \\ v(x) = R_2(x)W_{\beta,\gamma}(\frac{v^pu^q(y)}{|y|^\sigma})(x). \end{eqnarray} where $0< R(x) \leq C$, $1 < \gamma \leq 2$, $0\leq \sigma < \beta \gamma$, $n-\beta\gamma > \sigma(\gamma-1)$, $\gamma^{*}-1=\frac{n\gamma}{n-\beta\gamma+\sigma}-1\geq 1$. Due to the weight $\frac{1}{|y|^\sigma}$, we need more complicated analytical techniques to handle the properties of the solutions. First, we use the method of regularity lifting to obtain the integrability for the solutions of this Wolff type integral equation. Next, we use the modifying and refining method of moving planes established by Chen and Li to prove the radial symmetry for the positive solutions of related integral equation. Based on these results, we obtain the decay rates of the solutions of (0.1) with $R_1(x)\equiv R_2(x)\equiv 1$ near infinity. We generalize the results in the related references.
A strongly singular parabolic problem on an unbounded domain
G. P. Trachanas and Nikolaos B. Zographopoulos
2014, 13(2): 789-809 doi: 10.3934/cpaa.2014.13.789 +[Abstract](328) +[PDF](448.2KB)
Abstract:
We study the well-posedness and describe the asymptotic behavior of solutions of a strongly singular equation for the Cauchy problem on $R^N$. The strong singularity is exactly the critical case of the Caffarelli-Kohn-Nirenberg inequality. Moreover, we show the stabilization towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This equation is closely related to a heat equation with inverse-square potential, posed on $R^N$. In this case we have the appearance of the Hardy singularity energy.
Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$
Paolo Caldiroli
2014, 13(2): 811-821 doi: 10.3934/cpaa.2014.13.811 +[Abstract](411) +[PDF](379.5KB)
Abstract:
We prove existence of extremal functions for some Rellich-Sobolev type inequalities involving the $L^2$ norm of the Laplacian as a leading term and the $L^2$ norm of the gradient, weighted with a Hardy potential. Moreover we exhibit a breaking symmetry phenomenon when the nonlinearity has a growth close to the critical one and the singular potential increases in strength.
A BKM's criterion of smooth solution to the incompressible viscoelastic flow
Hua Qiu and Shaomei Fang
2014, 13(2): 823-833 doi: 10.3934/cpaa.2014.13.823 +[Abstract](394) +[PDF](385.9KB)
Abstract:
In this paper, we study the regularity criterion of smooth solution to the Oldroyd model in $R^n(n=2,3)$. We obtain a Beale-Kato-Majda-type criterion in terms of vorticity in two and three space dimensions, namely, the solution $(u(t,x),F(t,x))$ does not develop singularity until $t=T$ provided that $\nabla \times u \in L^1(0,T;\dot{B}_{\infty,\infty}^0(R^n))$ in the case $n=2,3$.
Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux
Tong Li and Hui Yin
2014, 13(2): 835-858 doi: 10.3934/cpaa.2014.13.835 +[Abstract](374) +[PDF](482.9KB)
Abstract:
This paper is concerned with the initial-boundary value problem for the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $R_+$ \begin{eqnarray} u_t-u_{txx}-u_{xx}+f(u)_{x}=0,\quad t>0, x\in R_+,\\ u(0,x)=u_0(x)\to u_+, \quad as \ \ x\to +\infty,\\ u(t,0)=u_b. \end{eqnarray} Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$, $u_+\not=u_b$ are two given constant states and the nonlinear function $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b
The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation
Irena Pawłow and Wojciech M. Zajączkowski
2014, 13(2): 859-880 doi: 10.3934/cpaa.2014.13.859 +[Abstract](409) +[PDF](429.8KB)
Abstract:
We consider again the sixth order Cahn-Hilliard type equation with a nonlinear diffusion, addressed in our previous paper in Commun. Pure Appl. Anal. 10 (2011), 1823--1847. Such PDE arises as a model of oil-water-surfactant mixtures. Applying the approach based on the Bäcklund transformation and the Leray-Schauder fixed point theorem we generalize the existence result of the above mentioned paper by imposing weaker assumptions on the data. Here we prove the global unique solvability of the problem in the Sobolev space $H^{6,1}(\Omega\times(0,T))$ under the assumption that the initial datum is in $H^3(\Omega)$ whereas previously $H^6(\Omega)$-regularity was required. Moreover, we admit a broarder class of nonlinear terms in the free energy potential.
Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type
Long Hu, Tatsien Li and Bopeng Rao
2014, 13(2): 881-901 doi: 10.3934/cpaa.2014.13.881 +[Abstract](424) +[PDF](447.1KB)
Abstract:
Several kinds of exact synchronizations are introduced for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type and these synchronizations can be realized by means of some boundary controls.
Hodge-de Rham theory on fractal graphs and fractals
S. Aaron, Z. Conn, Robert S. Strichartz and H. Yu
2014, 13(2): 903-928 doi: 10.3934/cpaa.2014.13.903 +[Abstract](379) +[PDF](505.3KB)
Abstract:
We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and 3-dimensional Sierpinski gasket (SG$^3$), but the method is expected to be effective for many PCF fractals, and also infinitely ramified fractals such as the Sierpinski carpet (SC). Our approach is to construct k-forms and de Rham differential operators $d$ and $\delta$ for a sequence of graphs approximating the fractal, and then pass to the limit with suitable renormalization, in imitation of Kigami's approach on constructing Laplacians on functions. One of our results is that our Laplacian on 0-forms is equal to Kigami's Laplacian on functions. We give explicit construction of harmonic 1-forms for our examples. We also prove that the measures on line segments provided by 1-forms are not absolutely continuous with respect to Lebesgue measures.
Quasilinear retarded differential equations with functional dependence on piecewise constant argument
Marat Akhmet
2014, 13(2): 929-947 doi: 10.3934/cpaa.2014.13.929 +[Abstract](477) +[PDF](417.7KB)
Abstract:
We introduce a new class of differential equations, retarded differential equations with functional dependence on piecewise constant argument, $RFDEPCA$ and focus on quasilinear systems. Formulation of the initial value problem, bounded solutions, periodic and almost periodic solutions, their stability are under investigation. Illustrating examples are provided.
Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space
Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern and Y. Kabeya
2014, 13(2): 949-960 doi: 10.3934/cpaa.2014.13.949 +[Abstract](488) +[PDF](370.9KB)
Abstract:
In this article, we consider the following semilinear elliptic equation on the hyperbolic space \begin{eqnarray} \Delta_{H^n} u-\lambda u+|u|^{p-1}u=0\quad on\quad H^n\setminus \{Q\} \end{eqnarray} where $\Delta_{H^n}$ is the Laplace-Beltrami operator on the hyperbolic space \begin{eqnarray} H^n=\{(x_1,\cdots, x_n,x_{n+1})|x_1^2+\cdots+x_n^2-x_{n+1}^2=-1\}, \end{eqnarray} $n>10,\ p>1, \lambda>0, $ and $Q=(0,\cdots,0,1)$. We provide the existence and uniqueness of a singular positive ``radial'' solution of the above equation for big $p$ (greater than the Joseph-Lundgren exponent, which appears if $n > 10$) as well as its asymptotic behavior.
Periodic solutions of the Brillouin electron beam focusing equation
Maurizio Garrione and Manuel Zamora
2014, 13(2): 961-975 doi: 10.3934/cpaa.2014.13.961 +[Abstract](378) +[PDF](437.6KB)
Abstract:
Quite unexpectedly with respect to the numerical and analytical results found in literature, we establish a new range for the real parameter $b$ for which the existence of $2\pi-$periodic solutions of the Brillouin focusing beam equation \begin{eqnarray} \ddot{x}+b(1+\cos t)x=\frac{1}{x} \end{eqnarray} is guaranteed. This is possible thanks to suitable nonresonance conditions acting on the rotation number of the solutions in the phase plane.

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