# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

November 2019 , Volume 18 , Issue 6

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2019, 18(6): 2879-2903 doi: 10.3934/cpaa.2019129 +[Abstract](218) +[HTML](74) +[PDF](427.21KB)
Abstract:

We prove a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solution pair (u,P ) to the Dirichlet problem of stationary Stokes system. It is mainly assumed that the leading coefficients are merely measurable in one spatial variable and have sufficiently small bounded mean oscillation (BMO) seminorm in the other variables, the boundary of underlying domain is Reifenberg flat, and the variable exponents p(x) satisfy the so-called log-Hölder continuity.

2019, 18(6): 2905-2921 doi: 10.3934/cpaa.2019130 +[Abstract](217) +[HTML](93) +[PDF](346.22KB)
Abstract:

This paper studies a porous elasticity system with past history

By introducing a new variable, we establish an explicit and a general decay of energy for the case of equal-speed wave propagation as well as for the nonequalspeed case. To establish our results, we mainly adopt the method developed by Guesmia, Messaoudi and Soufyane [Electron. J. Differ. Equa. 2012(2012), 1-45] and some properties of convex functions developed by Alabau-Boussouira and Cannarsa [C. R. Acad. Sci. Paris Ser. I, 347(2009), 867-872], Lasiecka and Tataru [Differ. Inte. Equa., 6(1993), 507-533]. In addition we remove the assumption that b is positive constant in [J. Math. Anal. Appl., 469(2019), 457-471] and hence improve the result.

2019, 18(6): 2923-2960 doi: 10.3934/cpaa.2019131 +[Abstract](185) +[HTML](87) +[PDF](466.89KB)
Abstract:

In this paper we deal with uniqueness of unbounded solutions to the following problem

where \begin{document}$Q_T = (0, T)\times \Omega$\end{document} is the parabolic cylinder, \begin{document}$\Omega$\end{document} is an open subset of \begin{document}$\mathbb{R}^N$\end{document} , \begin{document}$N\ge2$\end{document} , \begin{document}$1 < p < N$\end{document} , and the right hand side \begin{document}$\displaystyle H(t, x, \xi):(0, T)\times\Omega \times \mathbb{R}^N\to \mathbb{R}$\end{document} exhibits a superlinear growth with respect to the gradient term.

2019, 18(6): 2961-2981 doi: 10.3934/cpaa.2019132 +[Abstract](63) +[HTML](23) +[PDF](388.34KB)
Abstract:

This paper is concerned with the Camassa-Holm-KP equation, which is a model for shallow water waves. By using the analysis of the phase space, we obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the Camassa-Holm-KP equation without delay. Furthermore we show the existence of solitary wave solutions for the equation with a special local delay convolution kernel by combining the geometric singular perturbation theory and invariant manifold theory. In addition, we discuss the existence of solitary wave solutions for the Camassa-Holm-KP equation with strength \begin{document}$1$\end{document} of nonlinearity, and prove the monotonicity of the wave speed by analyzing the ratio of the Abelian integral.

2019, 18(6): 2983-2999 doi: 10.3934/cpaa.2019133 +[Abstract](204) +[HTML](73) +[PDF](355.45KB)
Abstract:

This paper is concerned with asymptotic spreading for a time-periodic predator-prey system where both species synchronously invade a new habitat. Under two different conditions, we show the bounds of spreading speeds of the predator and the prey, which is proved by the theory of asymptotic spreading of scalar equations, comparison principle and generalized eigenvalue. We show either the predator or the prey has a spreading speed that is determined by the linearized equation at the trivial steady state while the spreading speed of the other also depends on the interspecific nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative effect on the spreading of the prey while the prey may play a positive role on the spreading of the predator.

2019, 18(6): 3001-3009 doi: 10.3934/cpaa.2019134 +[Abstract](156) +[HTML](78) +[PDF](304.68KB)
Abstract:

In this paper, we show a new regularity result on the transport density \begin{document}$\sigma$\end{document} in the classical Monge-Kantorovich optimal mass transport problem between two measures, \begin{document}$\mu$\end{document} and \begin{document}$\nu$\end{document}, having some summable densities, \begin{document}$f^+$\end{document} and \begin{document}$f^-$\end{document}. More precisely, we prove that the transport density \begin{document}$\sigma$\end{document} belongs to \begin{document}$L^{p,q}(\Omega)$\end{document} as soon as \begin{document}$f^+,\,f^- \in L^{p,q}(\Omega)$\end{document}.

2019, 18(6): 3011-3034 doi: 10.3934/cpaa.2019135 +[Abstract](141) +[HTML](72) +[PDF](402.88KB)
Abstract:

We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [3] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.

2019, 18(6): 3035-3057 doi: 10.3934/cpaa.2019136 +[Abstract](158) +[HTML](94) +[PDF](401.37KB)
Abstract:

In this paper, by using Fourier splitting method and the properties of decay character \begin{document}$r^*$\end{document}, we consider the decay rate on higher order derivative of solutions to 3D incompressible electron inertial Hall-MHD system in Sobolev space \begin{document}$H^s(\mathbb{R}^3)\times H^{s+1}(\mathbb{R}^3)$\end{document} for \begin{document}$s\in\mathbb{N}^+$\end{document}. Moreover, based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also address the space-time decay properties of strong solutions in \begin{document}$\mathbb{R}^3$\end{document}.

2019, 18(6): 3059-3088 doi: 10.3934/cpaa.2019137 +[Abstract](170) +[HTML](82) +[PDF](514.93KB)
Abstract:

The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary-Domain Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.

2019, 18(6): 3089-3101 doi: 10.3934/cpaa.2019138 +[Abstract](176) +[HTML](78) +[PDF](350.62KB)
Abstract:

In this work we study the existence of positive solutions for the following class of coupled elliptic systems involving nonlinear Schrödinger equations

where \begin{document}$N\geq 3$\end{document} and the nonlinearities \begin{document}$f_{1}$\end{document} and \begin{document}$f_{2}$\end{document} are asymptotically linear at infinity. The potentials \begin{document}$V_{1}(x)$\end{document} and \begin{document}$V_{2}(x)$\end{document} are continuous functions which are bounded from below and above. The function \begin{document}$\lambda(x)$\end{document} is continuous and gives us a linear coupling due the terms \begin{document}$\lambda(x)u$\end{document} and \begin{document}$\lambda(x)v$\end{document}. Here we employ some variational arguments jointly with a Pohozaev identity.

2019, 18(6): 3103-3120 doi: 10.3934/cpaa.2019139 +[Abstract](152) +[HTML](104) +[PDF](480.41KB)
Abstract:

In this paper, we investigate bi-parameter modulation spaces on the product of two Euclidean spaces \begin{document}$\mathbb{R}^{n}$\end{document} and \begin{document}$\mathbb{R}^{m}$\end{document} via uniform decompositions of each factor. A molecular decomposition of these bi-parameter spaces are given, which generalizes the related single-parameter result of Kobayashi and Sawano [33]. Furthermore, we prove the boundedness of a class of Fourier multipliers on bi-parameter modulation spaces, generalizing the results of Bényi et al. [2] and Feichtinger and Narimani [17].

2019, 18(6): 3121-3135 doi: 10.3934/cpaa.2019140 +[Abstract](191) +[HTML](73) +[PDF](363.48KB)
Abstract:

This work is concerned with the invariant measure of a stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain. Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of the system becomes more complicated. In order to get over the difficulties caused by the fractional Laplacian operator, the usual Hilbert space does not fit the system, we introduce an appropriate weighted space to study it. Meanwhile, we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble caused by the degenerate noise, the corresponding Malliavin operator is not invertible. We finally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.

2019, 18(6): 3137-3160 doi: 10.3934/cpaa.2019141 +[Abstract](177) +[HTML](119) +[PDF](416.27KB)
Abstract:

In this paper we study subquadratic elliptic systems in divergence form with VMO leading coefficients in \begin{document}$\mathbb{R}^{n}$\end{document}. We establish pointwise estimates for gradients of local weak solutions to the system by involving the sharp maximal operator. As a consequence, the nonlinear Calderón-Zygmund gradient estimates for \begin{document}$L^{q}$\end{document} and BMO norms are derived.

2019, 18(6): 3161-3179 doi: 10.3934/cpaa.2019142 +[Abstract](147) +[HTML](85) +[PDF](384.73KB)
Abstract:

This paper deals with the nonlinear phase field system

in a general domain \begin{document}$\Omega\subseteq\mathbb{R}^{d}$\end{document}. Here \begin{document}$d \in \mathbb{N}$\end{document}, \begin{document}$T>0$\end{document}, \begin{document}$\ell>0$\end{document}, \begin{document}$f$\end{document} is a source term, \begin{document}$\beta$\end{document} is a maximal monotone graph and \begin{document}$\pi$\end{document} is a Lipschitz continuous function. We note that in the above system the nonlinearity \begin{document}$\beta+\pi$\end{document} replaces the derivative of a potential of double well type. Thus it turns out that the system is a generalization of the Caginalp phase field model and it has been studied by many authors in the case that \begin{document}$\Omega$\end{document} is a bounded domain. However, for unbounded domains the analysis of the system seems to be at an early stage. In this paper we study the existence of solutions by employing a time discretization scheme and passing to the limit as the time step \begin{document}$h$\end{document} goes to \begin{document}$0$\end{document}. In the limit procedure we face with the difficulty that the embedding \begin{document}$H^1(\Omega) \hookrightarrow L^2(\Omega)$\end{document} is not compact in the case of unbounded domains. Moreover, we can prove an interesting error estimate of order \begin{document}$h^{1/2}$\end{document} for the difference between continuous and discrete solutions.

2019, 18(6): 3181-3200 doi: 10.3934/cpaa.2019143 +[Abstract](215) +[HTML](91) +[PDF](401.37KB)
Abstract:

In this paper, we study the following fractional Kirchhoff equation with critical nonlinearity

where \begin{document}$a,b>0$\end{document}, \begin{document}$\lambda>0$\end{document}, \begin{document}$(-\Delta )^s$\end{document} is the fractional Laplace operator with \begin{document}$s\in(\frac{3}{4},1)$\end{document} and \begin{document}$2_s^* = \frac{6}{3-2s}$\end{document}, \begin{document}$V,K$\end{document} and \begin{document}$g$\end{document} are asymptotically periodic in \begin{document}$x$\end{document}. The existence of a positive ground state solution is obtained by variational method.

2019, 18(6): 3201-3216 doi: 10.3934/cpaa.2019144 +[Abstract](156) +[HTML](73) +[PDF](362.46KB)
Abstract:

In this paper, we prove the existence of bounded positive solutions for a class of semilinear degenerate elliptic equations involving supercritical cone Sobolev exponents. We also obtain the existence of multiple solutions by the Ljusternik-Schnirelman theory.

2019, 18(6): 3217-3242 doi: 10.3934/cpaa.2019145 +[Abstract](166) +[HTML](78) +[PDF](429.34KB)
Abstract:

We are concerned with a class of nonlocal elliptic equations as follows:

where \begin{document}$N\geq 1,$\end{document} \begin{document}$\lambda>0$\end{document} is a parameter, \begin{document}$M(t) = am(t)+b$\end{document} with \begin{document}$a, b>0$\end{document} and \begin{document}$m\in C(\mathbb{R}^{+}, \mathbb{R}^{+})$\end{document}, \begin{document}$V\in C(\mathbb{R}^{N}, \mathbb{R}^{+})$\end{document} and \begin{document}$f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$\end{document} satisfying \begin{document}$\lim_{|u|\rightarrow \infty }f(x, u) /|u|^{k-1} = q(x)$\end{document} uniformly in \begin{document}$x\in \mathbb{R}^{N}$\end{document} for any \begin{document}$2<k<2^{\ast}$\end{document}(\begin{document}$2^{\ast} = \infty$\end{document} for \begin{document}$N = 1, 2$\end{document} and \begin{document}$2^{\ast} = 2N/(N-2)$\end{document} for \begin{document}$N\geq 3$\end{document}). Unlike most other papers on this problem, we are more interested in the effects of the functions \begin{document}$m$\end{document} and \begin{document}$q$\end{document} on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.

2019, 18(6): 3243-3265 doi: 10.3934/cpaa.2019146 +[Abstract](161) +[HTML](73) +[PDF](407.02KB)
Abstract:

In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is \begin{document}$u_{\nu} = -\phi(x)(1+|Du|^2)^\frac{1-q}{2}$\end{document} for any parameter \begin{document}$q>0$\end{document}. Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. In addition, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at infinity time. The originality of the paper lies in the range \begin{document}$0<q<1$\end{document}, since there are no any related results before. For parabolic case, we generalize the result of Ma-Wang-Wei [25] to any \begin{document}$q>0$\end{document}. And in elliptic case, we generalize the results in [32] to any \begin{document}$q\ge 0$\end{document} and to any bounded smooth domain.

2019, 18(6): 3267-3284 doi: 10.3934/cpaa.2019147 +[Abstract](180) +[HTML](90) +[PDF](421.45KB)
Abstract:

In this paper, we study the global bifurcation curves and the exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem

where \begin{document}$p, q\geq 0$\end{document}, \begin{document}$p\neq q$\end{document}, \begin{document}$\lambda >0$\end{document} is a bifurcation parameter and \begin{document}$L>0$\end{document} is an evolution parameter. We prove that the bifurcation curve is continuous and further classify its exact shape (either monotone increasing or \begin{document}$\subset$\end{document}-shaped by \begin{document}$p$\end{document} and \begin{document}$q$\end{document}). Moreover, we can achieve the exact multiplicity of positive solutions.

2019, 18(6): 3285-3316 doi: 10.3934/cpaa.2019148 +[Abstract](221) +[HTML](75) +[PDF](461.09KB)
Abstract:

We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the unified transform method). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.

Yanbo Hu and
2019, 18(6): 3317-3336 doi: 10.3934/cpaa.2019149 +[Abstract](160) +[HTML](73) +[PDF](384.07KB)
Abstract:

We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the ideas of characteristic decomposition and the bootstrap method, we show that the solution is uniformly \begin{document}${C^{1,\frac{1}{6}}}$\end{document} up to the degenerate sonic boundary and that the sonic curve is \begin{document}${C^{1,\frac{1}{6}}}$\end{document}.

2019, 18(6): 3337-3349 doi: 10.3934/cpaa.2019150 +[Abstract](184) +[HTML](88) +[PDF](373.99KB)
Abstract:

This paper mainly investigates a class of almost periodic Nicholson's blowflies model involving a nonlinear density-dependent mortality term and time-varying delays. Combining Lyapunov function method and differential inequality approach, some novel assertions are established to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recent published literatures. Particularly, an example and its numerical simulations are given to support the proposed approach.

2019, 18(6): 3351-3365 doi: 10.3934/cpaa.2019151 +[Abstract](241) +[HTML](84) +[PDF](1494.37KB)
Abstract:

In this paper we are concerned with the multiplicity of solutions near resonance for the following nonlinear equation:

associated with the Dirichlet boundary condition, where \begin{document}$f$\end{document} satisfies some appropriate conditions. We will treat this problem in the framework of dynamical systems. It will be shown that there exist a one-sided neighborhood \begin{document}$\Lambda_-$\end{document} of the eigenvalue \begin{document}$\mu_k$\end{document} of the Laplacian operator and a dense subset \begin{document}${\mathcal D}$\end{document} of \begin{document}$\mathbb{R}$\end{document} such that the equation has at least four distinct nontrivial solutions generically for \begin{document}$\lambda\in\Lambda_- \cap {\mathcal D}$\end{document}.

2019, 18(6): 3367-3386 doi: 10.3934/cpaa.2019152 +[Abstract](178) +[HTML](131) +[PDF](1539.41KB)
Abstract:

In this article, we consider a quasilinear hyperbolic system of partial differential equations governing the dynamics of a thin film of a perfectly soluble anti-surfactant liquid. We construct elementary waves of the corresponding Riemann problem and study their interactions. Further, we provide exact solution of the Riemann problem along with numerical examples. Finally, we show that the solution of the Riemann problem is stable under small perturbation of the initial data.

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