# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

June 2021 , Volume 20 , Issue 6

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2021, 20(6): 2139-2154 doi: 10.3934/cpaa.2021061 +[Abstract](220) +[HTML](115) +[PDF](388.09KB)
Abstract:

In this paper, we investigate the compactness theory of the complex Green operator on smooth, embedded, orientable CR manifolds of hypersurface type that satisfy the weak \begin{document}$Y(q)$\end{document} condition. The sufficient condition that we define is an adaption of the CR-\begin{document}$P_q$\end{document} property for weak \begin{document}$Y(q)$\end{document} manifolds and does not require that the CR manifold is the boundary of a domain.

We also provide several non-pseudoconvex examples (and a level \begin{document}$q$\end{document}) for which the complex Green operator is compact.

2021, 20(6): 2155-2185 doi: 10.3934/cpaa.2021062 +[Abstract](207) +[HTML](111) +[PDF](3665.16KB)
Abstract:

This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator permits to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated. Flat and non-flat domains are considered.

2021, 20(6): 2187-2209 doi: 10.3934/cpaa.2021063 +[Abstract](209) +[HTML](98) +[PDF](433.52KB)
Abstract:

We obtain explicit characterization of orbital and spectral stability of solitary wave solutions to the \begin{document}${\bf{U}}(1)$\end{document}-invariant 1D Klein–Gordon equation coupled to an anharmonic oscillator. We also give the complete analysis of the spectrum of the linearization at a solitary wave.

2021, 20(6): 2211-2236 doi: 10.3934/cpaa.2021064 +[Abstract](213) +[HTML](104) +[PDF](435.25KB)
Abstract:

This paper deals with the following quasilinear two-species chemotaxis system

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$\Omega\subset \mathbb{R}^{n}$\end{document} \begin{document}$(n\geq2)$\end{document}. The diffusivity and the density-dependent sensitivity are given by \begin{document}$D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i}$\end{document} and \begin{document}$S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1}$\end{document} for all \begin{document}$s\geq0$\end{document}, respectively, where \begin{document}$C_{d_{i}},C_{s_{i}}>0$\end{document} and \begin{document}$\alpha_i,\beta_{i} \in \mathbb{R}$\end{document}; the logistic source and the signal productions are given by \begin{document}$f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}}$\end{document} and \begin{document}$g_{i}(s)\leq s^{\gamma_{i}}$\end{document} for all \begin{document}$s\geq0$\end{document} respectively, where \begin{document}$r_{i} \in \mathbb{R}$\end{document}, \begin{document}$\mu_{i},\gamma_{i} > 0$\end{document} and \begin{document}$k_{i} > 1$\end{document} \begin{document}$(i = 1,2)$\end{document}. It is proved that this system possesses a global bounded smooth solution under some specific conditions with or without the logistic functions \begin{document}$f_{i}(s)$\end{document}, which partially improves the results in [25]. Moreover, in case \begin{document}$r_{i}>0$\end{document}, if \begin{document}$\mu_{i}$\end{document} are sufficiently large, it is shown that the global bounded solution exponentially converges to \begin{document}$((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{1}}{k_{1}-1}} + (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{2}}{k_{2}-1}})$\end{document} as \begin{document}$t\rightarrow \infty$\end{document}.

2021, 20(6): 2237-2256 doi: 10.3934/cpaa.2021065 +[Abstract](237) +[HTML](95) +[PDF](376.96KB)
Abstract:

This work concerns with the following Choquard equation

where \begin{document}$\Omega\subseteq \mathbb{R}^{N}$\end{document} is an exterior domain with smooth boundary. We prove that the equation has at least one positive solution by variational and toplogical methods. Moreover, we establish a nonlocal version of global compactness result in unbounded domain.

2021, 20(6): 2257-2277 doi: 10.3934/cpaa.2021067 +[Abstract](246) +[HTML](107) +[PDF](442.93KB)
Abstract:

In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation

subject to Dirichlet boundary condition on \begin{document}$\partial \Omega$\end{document}, where \begin{document}$\Omega$\end{document} is a bounded smooth domain in \begin{document}${\mathbb{R}}^N$\end{document}, \begin{document}$N\geq 3$\end{document}, the function \begin{document}$\omega$\end{document} is a time-dependent damping, \begin{document}$\mu$\end{document} is a time-dependent squared speed of propagation, and \begin{document}$f$\end{document} is a nonlinear functional. Under structural assumptions on \begin{document}$\omega$\end{document} and \begin{document}$\mu$\end{document} we establish the existence of time-dependent attractor for the fractional models in the sense of Carvalho, Langa, Robinson [6], and Di Plinio, Duane, Temam [10].

2021, 20(6): 2279-2290 doi: 10.3934/cpaa.2021069 +[Abstract](198) +[HTML](127) +[PDF](347.17KB)
Abstract:

In this manuscript, we study the theory of conformal relativistic viscous hydrodynamics introduced in [4], which provided a causal and stable first-order theory of relativistic fluids with viscosity. Local existence and uniqueness of solutions to its equations of motion have been previously established in Gevrey spaces. Here, we improve this result by proving local existence and uniqueness of solutions in Sobolev spaces.

2021, 20(6): 2291-2312 doi: 10.3934/cpaa.2021077 +[Abstract](134) +[HTML](52) +[PDF](411.88KB)
Abstract:

In this paper, we investigate the existence and asymptotic behavior of least energy sign-changing solutions for the following Schrödinger-Poisson system

where \begin{document}$\lambda>0$\end{document} is a parameter. Under some suitable conditions on \begin{document}$f$\end{document} and \begin{document}$V$\end{document}, we get a least energy sign-changing solution \begin{document}$u_\lambda$\end{document} via variational method and its energy is strictly larger than twice that of least energy solutions. Moreover, the asymptotic behavior of \begin{document}$u_\lambda$\end{document} as \begin{document}$\lambda\rightarrow 0^+$\end{document} is also analyzed.

2021, 20(6): 2313-2322 doi: 10.3934/cpaa.2021078 +[Abstract](106) +[HTML](45) +[PDF](336.94KB)
Abstract:

Let \begin{document}$1<q<p$\end{document} and \begin{document}$a\in C(\overline{\Omega})$\end{document} be sign-changing, where \begin{document}$\Omega$\end{document} is a bounded and smooth domain of \begin{document}$\mathbb{R}^{N}$\end{document}. We show that the functional

has exactly one nonnegative minimizer \begin{document}$U_{q}$\end{document} (in \begin{document}$W_{0}^{1,p}(\Omega)$\end{document} or \begin{document}$W^{1,p}(\Omega)$\end{document}). In addition, we prove that \begin{document}$U_{q}$\end{document} is the only possible positive solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if \begin{document}$q$\end{document} is close enough to \begin{document}$p$\end{document} then \begin{document}$U_{q}$\end{document} is positive, which also guarantees that minimizers of \begin{document}$I_{q}$\end{document} do not change sign. Several of these results are new even for \begin{document}$p = 2$\end{document}.

2021, 20(6): 2323-2340 doi: 10.3934/cpaa.2021081 +[Abstract](109) +[HTML](55) +[PDF](363.88KB)
Abstract:

The image in \begin{document}$C^{\infty}$\end{document} for a class of complex vector fields, containing the Mizohata operator, was characterized.

2021, 20(6): 2341-2359 doi: 10.3934/cpaa.2021082 +[Abstract](84) +[HTML](46) +[PDF](975.67KB)
Abstract:

In this paper, an optimal control problem for a concrete dam system is considered. First, a mathematical model on the optimal osmotic control for basis of concrete dams is built up, and an optimal line-wise control of the system governed by the hybrid problem for elliptic partial differential equations is investigated. Then, the regularity of the generalized solution to the adjoint state equations, and the existence and uniqueness of the \begin{document}${\rm L^2}$\end{document}-solution for state equations are discussed and examined. Subsequently, the existence and uniqueness of the optimal control for the system, and a necessary and sufficient conditions for a control to be optimal and the optimality system are claimed and derived. Finally, the applications of the penalty shifting method with calculation of the optimal control of the system are studied, and the convergence of the method on an appropriate Hilbert space is claimed and proved.

2021, 20(6): 2361-2377 doi: 10.3934/cpaa.2021085 +[Abstract](106) +[HTML](63) +[PDF](351.87KB)
Abstract:

In this paper, we are concerned with the equations that are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix on compact K\begin{document}$\ddot{a}$\end{document}hler manifolds. Under the assumption of the cone condition, we obtain a priori estimates for the class of complex quotient equations. Then using the method of continuity, we prove an existence result.

2021, 20(6): 2379-2398 doi: 10.3934/cpaa.2021086 +[Abstract](126) +[HTML](44) +[PDF](418.44KB)
Abstract:

When hard inclusions are frequently spaced very closely, the electric field, which is the gradient of the solution to the perfect conductivity equation, may be arbitrarily large as the distance between two inclusions goes to zero. In this paper, our objectives are two-fold: first, we extend the asymptotic expansions of [26] to the higher dimensions greater than three by capturing the blow-up factors in all dimensions, which consist of some certain integrals of the solutions to the case when two inclusions are touching; second, our results answer the optimality of the blow-up rate for any \begin{document}$m,n\geq2$\end{document}, where \begin{document}$m$\end{document} and \begin{document}$n$\end{document} are the parameters of convexity and dimension, respectively, which is only partially solved in [29].

2021, 20(6): 2399-2419 doi: 10.3934/cpaa.2021087 +[Abstract](104) +[HTML](59) +[PDF](380.86KB)
Abstract:

By Malliavin calculus for Wiener-Poisson functionals and Lions derivative for probability measures, existence and smoothness of density functions for distribution dependent SDEs with Lévy noises are derived.

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