# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

September 2019 , Volume 18 , Issue 5

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2019, 18(5): 2199-2215 doi: 10.3934/cpaa.2019099 +[Abstract](2641) +[HTML](225) +[PDF](376.28KB)
Abstract:

In this paper we study existence of ground state solution to the following problem

where \begin{document}$(-\Delta)^{\alpha}$\end{document} is the fractional Laplacian, \begin{document}$\alpha\in (0,1)$\end{document}. We treat both cases \begin{document}$N\geq2$\end{document} and \begin{document}$N = 1$\end{document} with \begin{document}$\alpha = 1/2$\end{document}. The function \begin{document}$g$\end{document} is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case \begin{document}$N\geq2$\end{document}, exponential if \begin{document}$N = 1$\end{document}.

2019, 18(5): 2217-2242 doi: 10.3934/cpaa.2019100 +[Abstract](2266) +[HTML](230) +[PDF](416.6KB)
Abstract:

Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1-dimensional three-component systems of nonlinear Schrödinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.

2019, 18(5): 2243-2264 doi: 10.3934/cpaa.2019101 +[Abstract](2118) +[HTML](264) +[PDF](472.24KB)
Abstract:

We study in this article a stochastic version of a 2D simplified Ericksen-Leslie systems, which model the dynamic of nematic liquid crystals under the influence of stochastic external forces. We prove the existence and uniqueness of strong solution. The proof relies on a new formulation of the model proposed in [19] as well as a Galerkin approximation

2019, 18(5): 2265-2282 doi: 10.3934/cpaa.2019102 +[Abstract](2355) +[HTML](220) +[PDF](336.54KB)
Abstract:

For \begin{document}$-n<p<0,$\end{document} \begin{document}$0<q$\end{document} and

the existence of minimizers of

under

with given \begin{document}$m$\end{document} and \begin{document}$M$\end{document}, is proved in [3]. Moreover, except for translation, uniqueness and radial symmetry of the minimizer is proved for \begin{document}$-n<p<0$\end{document} and \begin{document}$q = 2$\end{document}. Here in the present paper, we show that, except for translation, uniqueness and radial symmetry of the minimizer hold for \begin{document}$-n<p<0$\end{document} and \begin{document}$2\leq q \leq 4.$\end{document} Applications are given.

2019, 18(5): 2283-2298 doi: 10.3934/cpaa.2019103 +[Abstract](2422) +[HTML](291) +[PDF](338.53KB)
Abstract:

This paper is concerned with the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Thanks to the strong coupling at the boundary, it is very difficult to obtain the uniqueness of an energy solution for problem (1)-(3) even in two dimension. To overcome this difficulty, inspired by the idea of Sell's radical approach (see [49]) to the global attractor of the three dimensional Navier-Stokes equations, we prove the closedness of the set \begin{document}$W$\end{document} of all global energy solutions for problem (1)-(3) equipped with some metric such that the \begin{document}$\omega$\end{document}-limit set of any bounded subset in \begin{document}$W$\end{document} still stay in \begin{document}$W,$\end{document} which is crucial to prove the existence of a global attractor for problem (1)-(3). In addition, we prove the existence of an absorbing set in \begin{document}$W$\end{document} and the uniform compactness of the semigroup \begin{document}$S_t$\end{document} for problem (1)-(3), which entails the existence of a global attractor in \begin{document}$W$\end{document} for problem (1)-(3).

2019, 18(5): 2299-2324 doi: 10.3934/cpaa.2019104 +[Abstract](2477) +[HTML](227) +[PDF](441.36KB)
Abstract:

In this paper, we study the modified Schr\begin{document}$\ddot{\mbox{o}}$\end{document}dinger-Poisson system:

where \begin{document}$V,K,g$\end{document} are asymptotically periodic functions of \begin{document}$x$\end{document}. Based on variational methods and the dual approach, we prove the existence of ground state solution by using the Nehari manifold method, the Mountain Pass theorem and the concentration-compactness principle.

2019, 18(5): 2325-2347 doi: 10.3934/cpaa.2019105 +[Abstract](2559) +[HTML](253) +[PDF](390.1KB)
Abstract:

By using bifurcation theory, we investigate the local asymptotical stability of non-negative steady states for a coupled dynamic system of ordinary differential equations and partial differential equations. The system models the interaction of pelagic algae, benthic algae and one essential nutrient in an oligotrophic shallow aquatic ecosystem with ample supply of light. The asymptotic profile of positive steady states when the diffusion coefficients are sufficiently small or large are also obtained.

2019, 18(5): 2349-2376 doi: 10.3934/cpaa.2019106 +[Abstract](2482) +[HTML](245) +[PDF](420.36KB)
Abstract:

We consider the long-time behavior for stochastic 2D nematic liquid crystals flows with the velocity field perturbed by an additive noise. The presence of the noises destroys the basic balance law of the nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow even in the weak solution space under non-periodic boundary conditions. To overcome the difficulty we use a new technique some kind of logarithmic energy estimates to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. Considering the existence of random attractor, the common method is to derive uniform a priori estimates in functional space which is more regular than the solution space. We can follow the common method to prove the existence of random attractor in the weak solution space. However, if we consider the existence of random attractor in the strong solution space, it is very difficult and very complicated for such highly non-linear stochastic system with no basic balance law and non-periodic boundary conditions. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to obtain the existence of the random attractor in the strong solution space, which implies the support of the invariant measure lies in a more regular space. As far as we know, it is the first article to attack the long-time behavior of stochastic nematic liquid crystals.

2019, 18(5): 2377-2395 doi: 10.3934/cpaa.2019107 +[Abstract](2051) +[HTML](234) +[PDF](424.11KB)
Abstract:

Let \begin{document}$A$\end{document} be a real \begin{document}$n\times n$\end{document} matrix with all its eigenvalues \begin{document}$\lambda$\end{document} satisfy \begin{document}$|\lambda|>1$\end{document}. Let \begin{document}$\varphi: \mathbb{R}^n\times[0, \, \infty)\to[0, \, \infty)$\end{document} be an anisotropic Musielak-Orlicz function, i.e., \begin{document}$\varphi(x, \cdot)$\end{document} is an Orlicz function uniformly in \begin{document}$x\in{\mathbb{R}^n}$\end{document} and \begin{document}$\varphi(\cdot, \, t)$\end{document} is an anisotropic Muckenhoupt \begin{document}$\mathcal {A}_\infty({\mathbb{R}^n})$\end{document} weight uniformly in \begin{document}$t\in(0, \, \infty)$\end{document}. In this article, the authors introduce the anisotropic weak Musielak-Orlicz Hardy space \begin{document}$WH^{\varphi}_A(\mathbb{R}^n)$\end{document} via the grand maximal function and establish its molecular characterization which are anisotropic extensions of Liang, Yang and Jiang (Math. Nachr. 289: 634-677, 2016). As an application, the boundedness of anisotropic Calderón-Zygmund operators from \begin{document}$H_A^\varphi(\mathbb{R}^n)$\end{document} to \begin{document}$WH_A^\varphi(\mathbb{R}^n)$\end{document} in the critical case is presented.

2019, 18(5): 2397-2408 doi: 10.3934/cpaa.2019108 +[Abstract](2216) +[HTML](213) +[PDF](346.85KB)
Abstract:

In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points \begin{document}$I_0$\end{document} is open and that the blow-up curve is of class \begin{document}$C^{1, \mu_0}$\end{document} and the phase \begin{document}$\theta$\end{document} is \begin{document}$C^{\mu_0}$\end{document} on this set. In order to prove this result, we introduce a Liouville Theorem for that equation. Our results hold also for the case of solutions with values in \begin{document}$\mathbb{R}^m$\end{document} with \begin{document}$m\ge 3$\end{document}, with the same proof.

Yun Lan and
2019, 18(5): 2409-2431 doi: 10.3934/cpaa.2019109 +[Abstract](2610) +[HTML](239) +[PDF](406.19KB)
Abstract:

This paper is concerned with the asymptotic behavior of solutions for non-autonomous stochastic fractional complex Ginzburg-Landau equations driven by multiplicative noise with \begin{document}$\alpha\in(0, 1)$\end{document}. We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions, which is slightly different from the deterministic fractional case with \begin{document}$\alpha\in(\frac{1}{2}, 1)$\end{document} and the real fractional case with \begin{document}$\alpha\in(0, 1)$\end{document}. Consequently, we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain. At last, we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero.

2019, 18(5): 2433-2455 doi: 10.3934/cpaa.2019110 +[Abstract](2225) +[HTML](204) +[PDF](407.56KB)
Abstract:

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term

where \begin{document}$z = (u, v)\in\mathbb{R}^{2}$\end{document}. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground state solutions, such as compactness of set of ground state solutions and exponential decay of ground state solutions.

2019, 18(5): 2457-2472 doi: 10.3934/cpaa.2019111 +[Abstract](2291) +[HTML](204) +[PDF](331.91KB)
Abstract:

In this paper, we study the asymptotic stability of traveling wave fronts to the Allen-Cahn equation with a fractional Laplacian. The main tools that we used are super- and subsolutions and squeezing methods.

2019, 18(5): 2473-2489 doi: 10.3934/cpaa.2019112 +[Abstract](1875) +[HTML](148) +[PDF](403.55KB)
Abstract:

We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in \begin{document}$L^2$\end{document}-based Sobolev spaces. We also obtain global well-posedness in \begin{document}$H^{\frac12+} \times L^2$\end{document}, which is sharp (up to endpoints) in the class of \begin{document}$L^2$\end{document}-based Sobolev spaces.

2019, 18(5): 2491-2509 doi: 10.3934/cpaa.2019113 +[Abstract](2636) +[HTML](273) +[PDF](488.97KB)
Abstract:

In this paper we establish the Freidlin-Wentzell's large deviation principle for stochastic 3D Leray-\begin{document}$\alpha$\end{document} model with general fractional dissipation and small multiplicative noise. This model is the stochastic 3D Navier-Stokes equations regularized through a smoothing kernel of order \begin{document}$\theta_1$\end{document} in the nonlinear term and a \begin{document}$\theta_2$\end{document}-fractional Laplacian. The main result generalizes the corresponding LDP result of the classical stochastic 3D Leray-\begin{document}$\alpha$\end{document} model (\begin{document}$\theta_1 = 1$\end{document}, \begin{document}$\theta_2 = 1$\end{document}), and it is also applicable to the stochastic 3D hyperviscous Navier-Stokes equations (\begin{document}$\theta_1 = 0$\end{document}, \begin{document}$\theta_2\geq\frac{5}{4}$\end{document}) and stochastic 3D critical Leray-\begin{document}$\alpha$\end{document} model (\begin{document}$\theta_1 = \frac{1}{4}$\end{document}, \begin{document}$\theta_2 = 1$\end{document}).

2019, 18(5): 2511-2528 doi: 10.3934/cpaa.2019114 +[Abstract](2208) +[HTML](197) +[PDF](350.23KB)
Abstract:

In this paper, we study the stationary problem of a predator-prey cross-diffusion system with a protection zone for the prey. We first apply the bifurcation theory to establish the existence of positive stationary solutions. Furthermore, as the cross-diffusion coefficient goes to infinity, the limiting behavior of positive stationary solutions is discussed. These results implies that the large cross-diffusion has beneficial effects on the coexistence of two species. Finally, we analyze the limiting behavior of positive stationary solutions as the intrinsic growth rate of the predator species goes to infinity.

2019, 18(5): 2529-2574 doi: 10.3934/cpaa.2019115 +[Abstract](1911) +[HTML](183) +[PDF](479.45KB)
Abstract:

In this paper, we establish the global existence, uniqueness and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model in \begin{document}$H^i\times H^i\times H^i\times H^{i+1}\;(i = 1,2,4)$\end{document}.

2019, 18(5): 2575-2605 doi: 10.3934/cpaa.2019116 +[Abstract](2191) +[HTML](218) +[PDF](559.14KB)
Abstract:

We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.

2019, 18(5): 2607-2661 doi: 10.3934/cpaa.2019117 +[Abstract](2274) +[HTML](186) +[PDF](621.02KB)
Abstract:

This paper is a continuation of authors' previous work [6]. We extend the argument [6] to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the \begin{document}$X^{s,b}$\end{document} nonlinear estimates for \begin{document}$b < \frac12$\end{document}, which is almost optimal compared to the standard \begin{document}$X^{s,b}$\end{document} nonlinear estimates for \begin{document}$b > \frac12$\end{document} [8,17]. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.

2019, 18(5): 2663-2677 doi: 10.3934/cpaa.2019118 +[Abstract](2031) +[HTML](168) +[PDF](340.08KB)
Abstract:

In this paper, we prove some qualitative properties of stationary solutions of the NLS on the Hyperbolic space. First, we prove a variational characterization of the ground state and give a complete characterization of the spectrum of the linearized operator around the ground state. Then we prove some rigidity theorems and necessary conditions for the existence of solutions in weighted spaces. Finally, we add a slowly varying potential to the homogeneous equation and prove the existence of non-trivial solutions concentrating on the critical points of a reduced functional. The results are the natural counterparts of the corresponding theorems on the Euclidean space. We produce also the natural virial identity on the Hyperbolic space for the complete evolution, which however requires the introduction of a weighted energy, which is not conserved and so does not lead directly to finite time blow-up as in the Euclidean case.

2019, 18(5): 2679-2691 doi: 10.3934/cpaa.2019119 +[Abstract](2062) +[HTML](174) +[PDF](360.91KB)
Abstract:

The classical Faber-Krahn inequality states that, among all domains with given measure, the ball has the smallest first Dirichlet eigenvalue of the Laplacian. Another inequality related to the first eigenvalue of the Laplacian has been proved by Lieb in 1983 and it relates the first Dirichlet eigenvalues of the Laplacian of two different domains with the first Dirichlet eigenvalue of the intersection of translations of them. In this paper we prove the analogue of Faber-Krahn and Lieb inequalities for the composite membrane problem.

2019, 18(5): 2693-2715 doi: 10.3934/cpaa.2019120 +[Abstract](2789) +[HTML](219) +[PDF](477.94KB)
Abstract:

This paper is concerned with the critical quasilinear Schrödinger systems in \begin{document}${\Bbb R}^N:$\end{document}

where \begin{document}$\lambda>0$\end{document} is a parameter, \begin{document}$p>2, q>2, \alpha>2, \beta>2,$\end{document} \begin{document}$2\cdot(2^*-1) < p+q<2\cdot2^*$\end{document} and \begin{document}$\alpha+ \beta = 2\cdot2^*.$\end{document} By using variational method, we prove the existence of positive ground state solutions which localize near the set \begin{document}$\Omega = int \left\{a^{-1}(0)\right\}\cap int \left\{b^{-1}(0)\right\}$\end{document} for \begin{document}$\lambda$\end{document} large enough.

2019, 18(5): 2717-2733 doi: 10.3934/cpaa.2019121 +[Abstract](2545) +[HTML](222) +[PDF](455.9KB)
Abstract:

We present the singular Hardy-Trudinger-Moser inequality and the existence of their extremal functions on the unit disc \begin{document}$B$\end{document} in \begin{document}$\mathbb{R}^2$\end{document}. As our first main result, we show that for any \begin{document}$0<t<2$\end{document} and \begin{document}$u \in C_0^\infty({B})$\end{document} satisfying

there exists a constant \begin{document}$C_{0}>0$\end{document} such that the following inequality holds

Furthermore, by the method of blow-up analysis, we establish the existence of extremal functions in a suitable function space. Our results extend those in Wang and Ye [36] from the non-singular case \begin{document}$t = 0$\end{document} to the singular case for \begin{document}$0<t<2$\end{document}.

2019, 18(5): 2735-2755 doi: 10.3934/cpaa.2019122 +[Abstract](2459) +[HTML](169) +[PDF](510.23KB)
Abstract:

We investigate the defocusing inhomogeneous nonlinear Schrödinger equation

with \begin{document}$0<b<1$\end{document} and \begin{document}$\alpha = 2\pi(2-b)$\end{document}. First we show the decay of global solutions by assuming that the initial data \begin{document}$u_0$\end{document} belongs to the weighted space \begin{document}$\Sigma(\mathbb{R}^2) = \{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\}$\end{document}. Then we combine the local theory with the decay estimate to obtain scattering in \begin{document}$\Sigma$\end{document} when the Hamiltonian is below the value \begin{document}$\frac{2}{(1+b)(2-b)}$\end{document}.

2019, 18(5): 2757-2764 doi: 10.3934/cpaa.2019123 +[Abstract](1877) +[HTML](205) +[PDF](367.08KB)
Abstract:

In this paper we study the following conformally invariant poly-harmonic equation

with \begin{document}$m = 2,3$\end{document}. We prove the existence of positive smooth radial solutions with prescribed volume \begin{document}$\int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx$\end{document}. We show that the set of all possible values of the volume is a bounded interval \begin{document}$(0,\Lambda^*]$\end{document} for \begin{document}$m = 2$\end{document}, and it is \begin{document}$(0,\infty)$\end{document} for \begin{document}$m = 3$\end{document}. This is in sharp contrast to \begin{document}$m = 1$\end{document} case in which the volume \begin{document}$\int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx$\end{document} is a fixed value.

2019, 18(5): 2765-2787 doi: 10.3934/cpaa.2019124 +[Abstract](2595) +[HTML](175) +[PDF](507.76KB)
Abstract:

A time-fractional Fokker–Planck initial-boundary value problem is considered, with differential operator \begin{document}$u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u -{\bf{F}}\partial_t^{1-\alpha}u)$\end{document}, where \begin{document}$0<\alpha <1$\end{document}. The forcing function \begin{document}${\bf{F}} = {\bf{F}}(t,x)$\end{document}, which is more difficult to analyse than the case \begin{document}${\bf{F}} = {\bf{F}}(x)$\end{document} investigated previously by other authors. The spatial domain \begin{document}$\Omega \subset\mathbb{R}^d$\end{document}, where \begin{document}$d\ge 1$\end{document}, has a smooth boundary. Existence, uniqueness and regularity of a mild solution \begin{document}$u$\end{document} is proved under the hypothesis that the initial data \begin{document}$u_0$\end{document} lies in \begin{document}$L^2(\Omega)$\end{document}. For \begin{document}$1/2<\alpha<1$\end{document} and \begin{document}$u_0\in H^2(\Omega)\cap H_0^1(\Omega)$\end{document}, it is shown that \begin{document}$u$\end{document} becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived—these are known to be needed in numerical analyses of this problem.

2019, 18(5): 2789-2818 doi: 10.3934/cpaa.2019125 +[Abstract](2129) +[HTML](179) +[PDF](551.24KB)
Abstract:

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.

2019, 18(5): 2819-2833 doi: 10.3934/cpaa.2019126 +[Abstract](1951) +[HTML](202) +[PDF](375.22KB)
Abstract:

We consider an elliptic system of equations in a punctured bounded domain. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperplane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction. As a consequence, we prove symmetry results also for a related polyharmonic problem of any order with Navier boundary conditions.

2019, 18(5): 2835-2854 doi: 10.3934/cpaa.2019127 +[Abstract](2130) +[HTML](199) +[PDF](449.54KB)
Abstract:

We are concerned with the pointwise estimates of solutions to the scalar conservation law with a nonlocal dissipative term for arbitrary large initial data. Based on the Green's function method, time-frequency decomposition method as well as the classical energy estimates, pointwise estimates and the optimal decay rates are established in this paper. We emphasize that the decay rate is independent of the index s in the nonlocal dissipative term. This phenomenon is also coincident with the fact that the decay rate is determined by the low frequency part of the solution no matter the initial data is small or large.

2019, 18(5): 2855-2878 doi: 10.3934/cpaa.2019128 +[Abstract](2486) +[HTML](176) +[PDF](493.95KB)
Abstract:

In this paper we consider the homoclinic orbits for a class of second order Hamiltonian systems of the form

where \begin{document}$\lambda>0$\end{document} is a parameter, \begin{document}$\frac{|\nabla W(t,x)|}{|x|}$\end{document} asymptotically tends to a constant as \begin{document}$|x|\rightarrow\infty$\end{document} and \begin{document}$|t|\rightarrow\infty$\end{document}. Via the variational method, two new theorems are proved.

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