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Communications on Pure & Applied Analysis

December 2020 , Volume 19 , Issue 12

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Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency
Wenmeng Geng and Kai Tao
2020, 19(12): 5305-5335 doi: 10.3934/cpaa.2020240 +[Abstract](327) +[HTML](101) +[PDF](487.82KB)
Abstract:

In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Rüssmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-Rüssmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

Approximations of stochastic 3D tamed Navier-Stokes equations
Xuhui Peng and Rangrang Zhang
2020, 19(12): 5337-5365 doi: 10.3934/cpaa.2020241 +[Abstract](268) +[HTML](89) +[PDF](418.85KB)
Abstract:

In this paper, we are concerned with 3D tamed Navier-Stokes equations with periodic boundary conditions, which can be viewed as an approximation of the classical 3D Navier-Stokes equations. We show that the strong solution of 3D tamed Navier-Stokes equations driven by Poisson random measure converges weakly to the strong solution of 3D tamed Navier-Stokes equations driven by Gaussian noise on the state space \begin{document}$ \mathcal{D}([0, T];\mathbb{H}^1) $\end{document}.

Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains
Lin Shi, Xuemin Wang and Dingshi Li
2020, 19(12): 5367-5386 doi: 10.3934/cpaa.2020242 +[Abstract](309) +[HTML](89) +[PDF](413.54KB)
Abstract:

This article is concerned with the limiting behavior of dynamics of a class of non-autonomous stochastic partial differential equations driven by colored noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on \begin{document}$ (n+1) $\end{document}-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the \begin{document}$ (n+1) $\end{document}-dimensional unbounded thin domains collapse onto the \begin{document}$ n $\end{document}-dimensional space \begin{document}$ \mathbb{R}^n $\end{document}. Here, the tail estimates are utilized to deal with the non-compactness of Sobolev embeddings on unbounded domains.

A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain
Ahmad Z. Fino and Wenhui Chen
2020, 19(12): 5387-5411 doi: 10.3934/cpaa.2020243 +[Abstract](341) +[HTML](99) +[PDF](388.62KB)
Abstract:

We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity \begin{document}$ |u|^p+|u_t|^q $\end{document} in an exterior domain, where \begin{document}$ p, q>1 $\end{document}. We prove global (in time) existence of small data solution with suitable higher regularity by using a weighted energy method, and assuming some conditions on powers of nonlinearity.

Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $
Mathew Gluck
2020, 19(12): 5413-5436 doi: 10.3934/cpaa.2020246 +[Abstract](206) +[HTML](48) +[PDF](422.22KB)
Abstract:

This work concerns the distributional solutions of a conformally invariant system of \begin{document}$ n^{\rm th} $\end{document}-order elliptic equations on \begin{document}$ \mathbb R^n $\end{document} having exponential type nonlinearity. The system in question is a natural generalization of the constant \begin{document}$ Q $\end{document}-curvature equation on \begin{document}$ \mathbb R^n $\end{document}. Under an \begin{document}$ L^1 $\end{document}-finiteness assumption and some assumptions on the coupling coefficients, an asymptotic estimate for solutions as \begin{document}$ \left|x\right|\to \infty $\end{document} is obtained. Under a growth constraint and further \begin{document}$ L^1 $\end{document}-norm assumptions the method of moving spheres is used to show that, up to an additive polynomial of low degree, each of the unknown functions is a standard bubble with common center and scale parameters.

Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation
Kihoon Seong
2020, 19(12): 5437-5473 doi: 10.3934/cpaa.2020247 +[Abstract](198) +[HTML](47) +[PDF](500.19KB)
Abstract:

We consider the low regularity behavior of the fourth order cubic nonlinear Schrödinger equation (4NLS)

In [29], the author showed that this equation is globally well-posed in \begin{document}$ H^s(\mathbb{R}), s\\\geq -\frac{1}{2} $\end{document} and mildly ill-posed in the sense that the solution map fails to be locally uniformly continuous for \begin{document}$ -\frac{15}{14}<s<-\frac{1}{2} $\end{document}. Therefore, \begin{document}$ s = -\frac{1}{2} $\end{document} is the lowest regularity that can be handled by the contraction argument. In spite of this mild ill-posedness result, we obtain an a priori bound below \begin{document}$ s<-1/2 $\end{document}. This an a priori estimate guarantees the existence of a weak solution for \begin{document}$ -3/4<s<-1/2 $\end{document}. Our method is inspired by Koch-Tataru [17]. We use the \begin{document}$ U^p $\end{document} and \begin{document}$ V^p $\end{document} based spaces adapted to frequency dependent time intervals on which the nonlinear evolution can still be described by linear dynamics.

Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry
Antoine Benoit
2020, 19(12): 5475-5486 doi: 10.3934/cpaa.2020248 +[Abstract](193) +[HTML](51) +[PDF](359.69KB)
Abstract:

In this article we investigate the possible losses of regularity of the solution for hyperbolic boundary value problems defined in the strip \begin{document}$ \mathbb{R}^{d-1}\times \left[0,1 \right] $\end{document}.

This question has already been widely studied in the half-space geometry in which a full characterization is almost completed (see [16,7,6]). In this setting it is known that several behaviours are possible, for example, a loss of a derivative on the boundary only or a loss of a derivative on the boundary combined with one or a half loss in the interior.

Crudely speaking the question addressed here is "can several boundaries make the situation becomes worse?".

Here we focus our attention to one special case of loss (namely the elliptic degeneracy of [16]) and we show that (in terms of losses of regularity) the situation is exactly the same as the one described in the half-space, meaning that the instability does not meet the geometry. This result has to be compared with the one of [2] in which the geometry has a real impact on the behaviour of the solution.

Parabolic equations involving Laguerre operators and weighted mixed-norm estimates
Huiying Fan and Tao Ma
2020, 19(12): 5487-5508 doi: 10.3934/cpaa.2020249 +[Abstract](207) +[HTML](52) +[PDF](385.34KB)
Abstract:

In this paper, we study evolution equation \begin{document}$ \partial_t u = -L_\alpha u+f $\end{document} and the corresponding Cauchy problem, where \begin{document}$ L_\alpha $\end{document} represents the Laguerre operator \begin{document}$ L_\alpha = \frac 12(-\frac{d^2}{dx^2}+x^2+\frac 1{x^2}(\alpha^2-\frac 14)) $\end{document}, for every \begin{document}$ \alpha\geq-\frac 12 $\end{document}. We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup \begin{document}$ \{ e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0} $\end{document}. In addition, we define the Poisson operator related to the fractional power \begin{document}$ (\partial_t+L_\alpha)^s $\end{document} and reveal weighted mixed-norm estimates for revelent maximal operators.

A truncated real interpolation method and characterizations of screened Sobolev spaces
Noah Stevenson and Ian Tice
2020, 19(12): 5509-5566 doi: 10.3934/cpaa.2020250 +[Abstract](232) +[HTML](59) +[PDF](683.17KB)
Abstract:

In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.

Mathematical analysis of bump to bucket problem
Min Chen, Olivier Goubet and Shenghao Li
2020, 19(12): 5567-5580 doi: 10.3934/cpaa.2020251 +[Abstract](204) +[HTML](62) +[PDF](331.43KB)
Abstract:

In this article, several systems of equations which model surface water waves generated by a sudden bottom deformation (bump) are studied. Because the effect of such deformation are often approximated by assuming the initial water surface has a deformation (bucket), this procedure is investigated and we prove rigorously that by using the correct bucket, the solutions of the regularized bump problems converge to the solution of the bucket problem.

Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth
Jun Zhou
2020, 19(12): 5581-5590 doi: 10.3934/cpaa.2020252 +[Abstract](227) +[HTML](52) +[PDF](313.33KB)
Abstract:

This paper deals with a fourth order parabolic PDE arising in the theory of epitaxial growth, which was studied in [4]. We estimated the lifespan under the blow-up conditions given in [4]. Moreover, we extend the blow-up conditions of [4] from subcritical initial energy to critical initial energy.

Solutions of nonlocal problem with critical exponent
Qingfang Wang and Hua Yang
2020, 19(12): 5591-5608 doi: 10.3934/cpaa.2020253 +[Abstract](231) +[HTML](55) +[PDF](384.29KB)
Abstract:

This paper deals with the system with linearly coupled of nonlocal problem with critical exponent,

Here \begin{document}$ \Omega $\end{document} is a smooth bounded domain in \begin{document}$ {\mathbb{R}}^N(N>4\alpha) $\end{document}, \begin{document}$ 0<\alpha<1 $\end{document}, \begin{document}$ \lambda_1,\lambda_2>-\lambda_1(\Omega) $\end{document} are constants, \begin{document}$ \lambda_1(\Omega) $\end{document} is the first eigenvalue of fractional Laplacian with Dirichlet boundary, \begin{document}$ 2_\alpha^* = \frac{2N}{N-2\alpha} $\end{document} is the Sobolev critical exponent and \begin{document}$ \beta\in {\mathbb{R}} $\end{document} is a coupling parameter. By variational method, we prove that this system has a positive ground state solution for some \begin{document}$ \beta>0 $\end{document}. Via a perturbation argument, by doing some delicate estimates for the nonlocal term, we overcome some difficulties and find a positive higher energy solution when \begin{document}$ |\beta| $\end{document} is small. Moreover, the asymptotic behaviors of the positive ground state and higher energy solutions as \begin{document}$ \beta\rightarrow 0 $\end{document} are analyzed.

On competition models under allee effect: Asymptotic behavior and traveling waves
Wei Feng, Michael Freeze and Xin Lu
2020, 19(12): 5609-5626 doi: 10.3934/cpaa.2020256 +[Abstract](248) +[HTML](48) +[PDF](941.09KB)
Abstract:

In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species (\begin{document}$ u $\end{document} and \begin{document}$ v $\end{document}). Under one-side Allee effect on \begin{document}$ u $\end{document}-species, the model demonstrates complexity on its coexistence and \begin{document}$ u $\end{document}-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant \begin{document}$ K $\end{document} is large relative to other biological parameters, the asymptotic stability of the \begin{document}$ v $\end{document}-dominance state \begin{document}$ (0,\:1) $\end{document} indicates the competitive exclusion of the \begin{document}$ u $\end{document}-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the \begin{document}$ u $\end{document}-dominance states), there exist traveling wave solutions flowing from the \begin{document}$ u $\end{document}-dominance states to the \begin{document}$ v $\end{document}-dominance state. The asymptotic rates of the traveling waves at \begin{document}$ \xi \rightarrow \mp \infty $\end{document} are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.

2019  Impact Factor: 1.105

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