# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

March 2021 , Volume 20 , Issue 3

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2021, 20(3): 955-974 doi: 10.3934/cpaa.2021001 +[Abstract](1118) +[HTML](182) +[PDF](343.12KB)
Abstract:

We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions. Under reasonable hypotheses, we establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm.

2021, 20(3): 975-994 doi: 10.3934/cpaa.2021002 +[Abstract](386) +[HTML](173) +[PDF](403.63KB)
Abstract:

Some weighted inequalities for the maximal operator with respect to the discrete diffusion semigroups associated with exceptional Jacobi and Jacobi-Dunkl polynomials are given. This setup allows to extend the corresponding results obtained for discrete heat semigroup recently to richer class of differential-difference operators.

2021, 20(3): 995-1023 doi: 10.3934/cpaa.2021003 +[Abstract](500) +[HTML](164) +[PDF](384.38KB)
Abstract:

This paper is concerned with the Cauchy problem of the 1-D unipolar hydrodynamic model for semiconductor device, a system of Euler-Poisson equations with time-dependent damping effect \begin{document}$-J(1+t)^{-\lambda}$\end{document} for \begin{document}$-1<\lambda<1$\end{document}, where \begin{document}$J$\end{document} denotes the current density, and the damping effect is asymptotically vanishing as \begin{document}$t \to \infty$\end{document} for \begin{document}$\lambda>0$\end{document}, and asymptotically enhancing to infinity as \begin{document}$t \to \infty$\end{document} for \begin{document}$\lambda<0$\end{document}. When the initial perturbation around the constant states are sufficiently small, by means of the time-weighted energy method, we prove that the smooth solutions to the Cauchy problem exist uniquely and globally. Particularly, we also obtain the large-time behavior of the solutions.

2021, 20(3): 1025-1038 doi: 10.3934/cpaa.2021004 +[Abstract](446) +[HTML](174) +[PDF](356.75KB)
Abstract:

We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.

2021, 20(3): 1039-1058 doi: 10.3934/cpaa.2021005 +[Abstract](329) +[HTML](143) +[PDF](415.78KB)
Abstract:

We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions \begin{document}$u$\end{document} with initial data \begin{document}$u_0 \in H^s$\end{document} are known to be global if \begin{document}$s \ge 1$\end{document}. We prove that for any integer \begin{document}$s \ge 2$\end{document}, \begin{document}$\| u(t) \|_{H^s}$\end{document} grows at most polynomially in \begin{document}$t$\end{document} for large times \begin{document}$t$\end{document}. This result is related to wave turbulence and how a solution of (ZK) can move energy to high frequencies.

It is inspired by analoguous results by Staffilani [21] on the non linear Schrödinger and Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces and a careful study of the variation of the \begin{document}$H^s$\end{document} norm.

2021, 20(3): 1059-1076 doi: 10.3934/cpaa.2021006 +[Abstract](355) +[HTML](145) +[PDF](406.34KB)
Abstract:

The paper investigates the well-posedness and the existence of global attractor for a strongly damped wave equation on \begin{document}$\mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u_{t}-\Delta u+u_{t}+u+g(u) = f(x)$\end{document}. It shows that when the nonlinearity \begin{document}$g(u)$\end{document} is of supercritical growth \begin{document}$p$\end{document}, with \begin{document}$\frac{N+2}{N-2}\equiv p^*< p< p^{**} \equiv\frac{N+4}{(N-4)^+}$\end{document}, (i) the initial value problem of the equation is well-posed and its weak solution possesses additionally partial regularity as \begin{document}$t>0$\end{document}; (ii) the related solution semigroup has a global attractor in natural energy space. By using a new double truncation method on frequency space \begin{document}$\mathbb{R}^N$\end{document} rather than approximating physical space \begin{document}$\mathbb{R}^N$\end{document} by a sequence of balls \begin{document}$\Omega_R$\end{document} as usual, we break through the longstanding existed restriction on this topic for \begin{document}$p: 1\leqslant p\leqslant p^*$\end{document}.

2021, 20(3): 1077-1089 doi: 10.3934/cpaa.2021007 +[Abstract](332) +[HTML](142) +[PDF](301.51KB)
Abstract:

We deal with Abel equations \begin{document}$dy/dx = A(x) y^2 + B(x) y^3$\end{document}, where \begin{document}$A(x)$\end{document} and \begin{document}$B(x)$\end{document} are real polynomials. We prove that these Abel equations can have at most two rational limit cycles and we characterize when this happens. Moreover we provide examples of these Abel equations with two nontrivial rational limit cycles.

2021, 20(3): 1091-1102 doi: 10.3934/cpaa.2021008 +[Abstract](345) +[HTML](135) +[PDF](328.7KB)
Abstract:

In this paper we introduce the notion of scale pressure and measure theoretic scale pressure for amenable group actions. A variational principle for amenable group actions is presented. We also describe these quantities by pseudo-orbits. Moreover, we prove that if \begin{document}$G$\end{document} is a finitely generated countable discrete amenable group, then the scale pressure of \begin{document}$G$\end{document} coincides with the scale pressure of \begin{document}$G$\end{document} with respect to pseudo-orbits.

2021, 20(3): 1103-1133 doi: 10.3934/cpaa.2021009 +[Abstract](367) +[HTML](148) +[PDF](495.86KB)
Abstract:

Two direct systems of Boundary-Domain Integral Equations, BDIEs, associated with a mixed boundary value problem for the stationary compressible Stokes system with variable viscosity coefficient in an exterior domain of \begin{document}$\mathbb{R}^3$\end{document} are derived. This is done by employing the Stokes surface and volume potentials based on an appropriate parametrix (Levi function) in the third Green identities for the velocity and pressure. Mapping properties of the potentials in weighted Sobolev spaces are analysed. Finally, the equivalence between the BDIE systems and the BVP is shown and the isomorphism of operators defined by the BDIE systems is proved.

2021, 20(3): 1135-1170 doi: 10.3934/cpaa.2021010 +[Abstract](304) +[HTML](125) +[PDF](438.79KB)
Abstract:

In this paper, we consider a stochastic Allen-Cahn Navier-Stokes system in a bounded domain of \begin{document}$\mathbb{R}^d,$\end{document} \begin{document}$d = 2,3$\end{document}. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on splitting-up method as well as some compactness method.

2021, 20(3): 1171-1186 doi: 10.3934/cpaa.2021011 +[Abstract](300) +[HTML](135) +[PDF](346.02KB)
Abstract:

Let \begin{document}$(K,d)$\end{document} be a compact metric space, \begin{document}$\mathcal A$\end{document} be a commutative semisimple Banach algebra and \begin{document}$0<\alpha\leq 1$\end{document}. The overall purpose of the present paper is to demonstrate that all BSE concepts of \begin{document}${\rm Lip}_\alpha(K,\mathcal A)$\end{document} are inherited from \begin{document}$\mathcal A$\end{document} and vice versa. Recently, the authors proved in the case that \begin{document}$\mathcal A$\end{document} is unital, \begin{document}${\rm Lip}_\alpha(K,\mathcal A)$\end{document} is a BSE-algebra if and only if \begin{document}$\mathcal A$\end{document} is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra \begin{document}$\mathcal A$\end{document}. Furthermore, we investigate the BSE-norm property for \begin{document}${\rm Lip}_\alpha(K,\mathcal A)$\end{document} and prove that \begin{document}${\rm Lip}_\alpha(K,\mathcal A)$\end{document} belongs to the class of BSE-norm algebras if and only if \begin{document}$\mathcal A$\end{document} is owned by this class. Moreover, we prove that for any natural number \begin{document}$n$\end{document} with \begin{document}$n\geq 2$\end{document}, if all continuous bounded functions on \begin{document}$\Delta({\rm Lip}_\alpha(K,\mathcal A))$\end{document} are \begin{document}$n$\end{document}-BSE-functions, then \begin{document}$K$\end{document} is finite. As a result, we obtain that \begin{document}${\rm Lip}_{\alpha}(K,\mathcal A)$\end{document} is a BSE-algebra of type I if and only if \begin{document}$\mathcal A$\end{document} is a BSE-algebra of type I and \begin{document}$K$\end{document} is finite. Furthermore, in according to a result of Kaniuth and Ülger, which disapproves the BSE-property for \begin{document}${\rm lip}_{\alpha}K$\end{document}, we show that for any commutative semisimple Banach algebra \begin{document}$\mathcal A$\end{document}, \begin{document}${\rm lip}_{\alpha}(K,\mathcal A)$\end{document} fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra \begin{document}${\rm Lip}_\alpha X$\end{document}, for an arbitrary metric space (not necessarily compact) \begin{document}$(X,d)$\end{document} and \begin{document}$\alpha>0$\end{document}, when \begin{document}${\rm Lip}_\alpha X$\end{document} separates the points of \begin{document}$X$\end{document}. In particular, we show that \begin{document}${\rm Lip}_\alpha X$\end{document} is a BSE-algebra, as well as a BSE-norm algebra.

2021, 20(3): 1187-1198 doi: 10.3934/cpaa.2021012 +[Abstract](303) +[HTML](134) +[PDF](305.56KB)
Abstract:

In this paper, we obtain the interior gradient estimate of the Hessian quotient curvature equation in the hyperbolic space. The method depends on the maximum principle.

2021, 20(3): 1199-1211 doi: 10.3934/cpaa.2021016 +[Abstract](268) +[HTML](100) +[PDF](322.29KB)
Abstract:

We consider the mean curvature flow of a closed hypersurface in hyperbolic space. Under a suitable pinching assumption on the initial data, we prove a priori estimate on the principal curvatures which implies that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes the estimates obtained in the previous works of Huisken, Sinestrari and Nguyen on the mean curvature flow of hypersurfaces in Euclidean spaces and in the spheres.

2021, 20(3): 1213-1227 doi: 10.3934/cpaa.2021017 +[Abstract](290) +[HTML](93) +[PDF](332.24KB)
Abstract:

We prove existence and uniqueness of a positive solution for a class of quasilinear parabolic equations. We also show some maximum principles on the derivatives of the solution and study the asymptotic behavior of the solution near the maximal time of existence.

2021, 20(3): 1229-1240 doi: 10.3934/cpaa.2021018 +[Abstract](264) +[HTML](98) +[PDF](358.62KB)
Abstract:

In this paper, we start to investigate the global existence and uniqueness of weak solutions of the \begin{document}$n$\end{document}-dimensional (\begin{document}$n\geq3$\end{document}) hyper-dissipative Boussinesq system without thermal diffusivity in the periodic domain \begin{document}$\mathbb{T}^n$\end{document} with the initial data \begin{document}$u_0\in L^2(\mathbb{T}^n)$\end{document} and \begin{document}$\theta_0 \in L^2(\mathbb{T}^n)\cap L^{\frac{4n}{n+2}}(\mathbb{T}^n)$\end{document}. Then we focus on the vanishing thermal diffusion limit and obtain the convergent result in the sense of \begin{document}$L^2$\end{document}-norm. Ultimately, we also prove the global regularity of this system in the case of 3-dimension.

2021, 20(3): 1241-1261 doi: 10.3934/cpaa.2021019 +[Abstract](266) +[HTML](105) +[PDF](378.71KB)
Abstract:

In this paper, we consider the positive viscosity solutions for certain fully nonlinear uniformly elliptic equations in unbounded cylinder with zero boundary condition. After establishing an Aleksandrov-Bakelman-Pucci maximum principle, we classify all positive solutions as three categories in unbounded cylinder. Two special solution spaces (exponential growth at one end and exponential decay at the another) are one dimensional, independently, while solutions in the third solution space can be controlled by the solutions in the other two special solution spaces under some conditions, respectively.

2021, 20(3): 1263-1296 doi: 10.3934/cpaa.2021020 +[Abstract](314) +[HTML](102) +[PDF](431.42KB)
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In this paper, we prove the continuity of global attractors and the Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation if every equilibrium of the unperturbed equation is hyperbolic.

2021, 20(3): 1297-1317 doi: 10.3934/cpaa.2021021 +[Abstract](316) +[HTML](101) +[PDF](386.64KB)
Abstract:

We study the large-time asymptotic behavior of solutions toward the rarefaction wave of the compressible non-isentropic Navier-Stokes equations coupling with Maxwell equations under some small perturbations of initial data and also under the assumption that the dielectric constant is bounded. For that, the dissipative structure of this hyperbolic-parabolic system is studied to include the effect of the electromagnetic field into the viscous fluid and turns out to be more complicated than that in the simpler compressible Navier-Stokes system. The proof of the main result is based on the elementary \begin{document}$L^2$\end{document} energy methods.

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