# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure and Applied Analysis

October 2022 , Volume 21 , Issue 10

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Yan Liu and
2022, 21(10): 3247-3261 doi: 10.3934/cpaa.2022098 +[Abstract](281) +[HTML](79) +[PDF](435.17KB)
Abstract:

Consider the following second-order perturbed Hamiltonian systems

where \begin{document}$F(t,u)=-K(t,u)+W(t,u)$\end{document}, \begin{document}$K$\end{document}, \begin{document}$W$\end{document} are measurable and \begin{document}$T-$\end{document}periodic in \begin{document}$t$\end{document} for all \begin{document}$u\in\bf{R}^N$\end{document}, continuously differentiable in \begin{document}$u$\end{document} for a.e. \begin{document}$t\in[0,T]$\end{document} and even in \begin{document}$u$\end{document}, \begin{document}$G\in C^1\left(\bf{R}\times\bf{R}^N,\bf{R}\right)$\end{document} is also \begin{document}$T-$\end{document}periodic in \begin{document}$t$\end{document}, but \begin{document}$G$\end{document} maybe has no parity in \begin{document}$u$\end{document}. Assume that \begin{document}$W$\end{document} is local superquadratic and \begin{document}$G$\end{document} is subquadratic at the infinity and \begin{document}$K$\end{document} satisfies "pinched" condition, the existence of infinitely many weak periodic solutions for above perturbed systems is obtained via the Bolle's perturbation method, which generalizes and improves some previous results.

2022, 21(10): 3263-3282 doi: 10.3934/cpaa.2022099 +[Abstract](272) +[HTML](106) +[PDF](484.64KB)
Abstract:

In this paper, we investigate the threshold results for a nonlocal and time-delayed reaction-diffusion system involving the spatial heterogeneity and the seasonality. Due to the complexity of the model, we rigorously analyze the well-posedness of the model. The basic reproduction number \begin{document}$\Re_0$\end{document} is characterized with the next generation operator method. We show that the disease-free \begin{document}$\omega$\end{document}-periodic solution is globally attractive when \begin{document}$\Re_0 < 1$\end{document}; while the system is uniformly persistent and a positive \begin{document}$\omega$\end{document}-periodic solution exists when \begin{document}$\Re_0 > 1$\end{document}. In a special case that the parameters are all independent of the spatial heterogeneity and the seasonality, the global attractivity of the constant equilibria of the model is investigated by the technique of Lyapunov functionals.

2022, 21(10): 3283-3307 doi: 10.3934/cpaa.2022100 +[Abstract](235) +[HTML](71) +[PDF](599.1KB)
Abstract:

We prove new \begin{document}$L^p$\end{document}\begin{document}$L^q$\end{document} estimates for averages over dilates of the circle with respect to fractal measures, which unify different types of maximal estimates for the circular average. Our results are consequences of \begin{document}$L^p$\end{document}\begin{document}$L^q$\end{document} smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages.

2022, 21(10): 3309-3334 doi: 10.3934/cpaa.2022101 +[Abstract](293) +[HTML](77) +[PDF](478.81KB)
Abstract:

We consider the Cauchy problem for the system of quadratic derivative nonlinear Schrödinger equations, which was introduced by Colin and Colin (2004). In the previous paper, the authors (2021) determined the almost optimal Sobolev regularity to be well-posed in \begin{document}$H^s ( \mathbb{R}^d)$\end{document} as long as we use the iteration argument. In this paper, we consider the well-posedness under the conditions where the flow map fails to be twice differentiable. To prove the well-posedness, we construct a modified energy and apply the energy method.

2022, 21(10): 3335-3351 doi: 10.3934/cpaa.2022102 +[Abstract](224) +[HTML](164) +[PDF](351.88KB)
Abstract:

A semiflow is eventually expansive if there is a prefixed radius up to which two orbits eventually coincide. We prove the following result: Every injective eventually expansive semiflow of a compact metric space consists of finitely many closed orbits and, in the noncompact case, the semiflow cannot have global attractors. The suspension of a continuous map is eventually expansive only if the map is positively expansive. The suspension of every positively expansive map is an eventually expansive semiflow. The topological entropy of an eventually expansive semiflow is bounded from below by the growth rate of the periodic orbits. We present some related examples.

2022, 21(10): 3353-3369 doi: 10.3934/cpaa.2022103 +[Abstract](201) +[HTML](70) +[PDF](374.68KB)
Abstract:

We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an affirmative answer for the case \begin{document}$k = n$\end{document} when the asymptotic boundary \begin{document}$\Gamma$\end{document} bounds a uniformly convex domain, and for \begin{document}$k < n$\end{document} when \begin{document}$\Gamma$\end{document} bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work [12, 13], and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.

2022, 21(10): 3371-3387 doi: 10.3934/cpaa.2022104 +[Abstract](221) +[HTML](63) +[PDF](388.79KB)
Abstract:

We consider a class of time-fractional phase field models including the Allen-Cahn and Cahn-Hilliard equations. We establish several weighted positivity results for functionals driven by the Caputo time-fractional derivative. Several novel criterions are examined for showing the positive-definiteness of the associated kernel functions. We deduce strict energy-dissipation for a number of non-local energy functionals, thereby proving fractional energy dissipation laws.

2022, 21(10): 3389-3405 doi: 10.3934/cpaa.2022105 +[Abstract](308) +[HTML](59) +[PDF](406.42KB)
Abstract:

We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.

2022, 21(10): 3407-3420 doi: 10.3934/cpaa.2022106 +[Abstract](305) +[HTML](65) +[PDF](368.75KB)
Abstract:

We use Perron method to prove the existence of ancient solutions of the exterior problem for parabolic k-Hessian equations \begin{document}$-u_tS_k(D^2u) = 1$\end{document} with prescribed asymptotic behavior at infinity.

2022, 21(10): 3421-3424 doi: 10.3934/cpaa.2022107 +[Abstract](221) +[HTML](64) +[PDF](275.22KB)
Abstract:

We provide a proof via direct energy estimates of the optimal exponential decay rate of the semigroup generated by the weakly damped wave equation.

2022, 21(10): 3425-3439 doi: 10.3934/cpaa.2022108 +[Abstract](220) +[HTML](133) +[PDF](399.72KB)
Abstract:

We devote to studying the condition of a holomorphic function \begin{document}$g$\end{document} in the complex unit ball \begin{document}$\mathbb{B}_n$\end{document} so that the Volterra operator \begin{document}$T_g:A_\alpha^2\to A_\alpha^2$\end{document} belongs to the Schatten \begin{document}$p$\end{document}-class. Assuming \begin{document}$n\ge2$\end{document} and \begin{document}$\alpha>-3$\end{document}, we get the following conclusions

1. For \begin{document}$0<p\le n$\end{document}, \begin{document}$T_g\in \mathcal{S}_p(A^2_\alpha)$\end{document} if and only if \begin{document}$g$\end{document} is a constant.

2. For \begin{document}$n<p<\infty$\end{document} and \begin{document}$p(\alpha+1)+4n>0$\end{document}, \begin{document}$T_g\in \mathcal{S}_p(A^2_\alpha)$\end{document} if and only if

where \begin{document}$t>\max\{\frac np-\frac{n+1+\alpha}2, \frac{n-1}2\}$\end{document} and and \begin{document}$\mathrm{d} \tau(w) = (1-|w|^2)^{-n-1}{ \mathrm{d} v(w)}$\end{document} is the Möbius invariant measure in \begin{document}$\mathbb{B}_n$\end{document}. Here \begin{document}$\mathrm{d} v$\end{document} is the normalized Lebesgue measure on \begin{document}$\mathbb{B}_n$\end{document} so that \begin{document}$v( \mathbb{B}_n) = 1$\end{document} and \begin{document}$\mathrm{d} v_{\alpha+2}(z) = c_{\alpha+2}(1-|z|^2)^{\alpha+2} \mathrm{d} v (z)$\end{document} with a normalized constant \begin{document}$c_{\alpha+2}$\end{document} so that \begin{document}$v_{\alpha+2}( \mathbb{B}_n) = 1$\end{document}.

2022, 21(10): 3441-3462 doi: 10.3934/cpaa.2022109 +[Abstract](275) +[HTML](51) +[PDF](419.73KB)
Abstract:

In this paper, we establish the nonlinear stability and large time behavior of hydrostatic equilibrium in a uniform magnetic field for the Boussinesq system with magnetohydrodynamics convection in the whole space \begin{document}$\mathbb{R}^{2}$\end{document} with mixed partial dissipation, motivated by Lai, Wu, Zhong [18] and Lin, Ji, Wu and Yan [22]. Due to the lack of horizontal dissipation and vertical dissipation in the second component of velocity, the natural energy is not easy to be closed, which is overcome by introducing an additional functional of the horizontal derivative of the second component of velocity. This shows that the magnetic field and the temperature have a stabilizing effect on the fluid. Large time behavior and linear decay rate of the solution are also obtained.

2022, 21(10): 3463-3478 doi: 10.3934/cpaa.2022110 +[Abstract](184) +[HTML](65) +[PDF](383.36KB)
Abstract:

The aim of this paper is to determine when polynomial planar systems can be reduced to Abel equations. The study is based on earlier works by Christopher, Devlin, Lloyd, and Pearson. The novelty of the approach is that the properties of the functions involved in the change of variables are determined. In particular, a constructive method of obtaining planar systems that can be reduced to Abel equations is provided. With the results obtained, it is possible to obtain new families of planar systems that can be reduced to Abel equations.

2022, 21(10): 3479-3498 doi: 10.3934/cpaa.2022111 +[Abstract](179) +[HTML](148) +[PDF](466.02KB)
Abstract:

A Freidlin-Wentzell-type large deviation principle is established for \begin{document}$2D$\end{document} stochastic Navier-Stokes equations on time-dependent domains driven by Brownian motion, which captures situations where the regions of the fluid change with time.

2022, 21(10): 3499-3527 doi: 10.3934/cpaa.2022112 +[Abstract](265) +[HTML](60) +[PDF](510.86KB)
Abstract:

The theory of abstract Friedrichs operators, introduced by Ern, Guermond and Caplain (2007), proved to be a successful setting for studying positive symmetric systems of first order partial differential equations (Fried-richs, 1958), nowadays better known as Friedrichs systems. Recently, Antonić, Michelangeli and Erceg (2017) presented a purely operator-theoretic description of abstract Friedrichs operators, allowing for application of the universal operator extension theory (Grubb, 1968). In this paper we make a further theoretical step by developing a decomposition of the graph space (maximal domain) as a direct sum of the minimal domain and the kernels of corresponding adjoints. We then study one-dimensional scalar (classical) Friedrichs operators with variable coefficients and present a complete classification of admissible boundary conditions.

2022, 21(10): 3529-3558 doi: 10.3934/cpaa.2022113 +[Abstract](179) +[HTML](49) +[PDF](549.44KB)
Abstract:

In this paper we consider a stochastic variant of the generalized Camassa-Holm equation. We first establish some local results, including local existence, uniqueness and a blow-up criterion characterizing the possible blow-up of the solutions, for the corresponding stochastic partial differential equation. Then we study the effect of noise. For the relatively small noise, in terms of the dependence on initial data, we construct an example to show that the SPDE is weakly instable in the sense that either the exiting time is not strongly stable, or the dependence on initial data is not uniformly continuous. Finally, for the large noise, we prove that singularities can be prevented, i.e., large noise has regularization effect.

2021 Impact Factor: 1.273
5 Year Impact Factor: 1.282
2021 CiteScore: 2.2