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Communications on Pure & Applied Analysis

June 2020 , Volume 19 , Issue 6

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Classification of singular sets of solutions to elliptic equations
Haiyun Deng, Hairong Liu and Long Tian
2020, 19(6): 2949-2964 doi: 10.3934/cpaa.2020129 +[Abstract](320) +[HTML](108) +[PDF](475.84KB)

In this paper, we mainly investigate the classification of singular sets of solutions to elliptic equations. Firstly, we define the \begin{document}$ j $\end{document}-symmetric singular set \begin{document}$ S^j(u) $\end{document} of solution \begin{document}$ u $\end{document}, and show that the Hausdorff dimension of the \begin{document}$ j $\end{document}-symmetric singular set \begin{document}$ S^j(u) $\end{document} is not more than \begin{document}$ j $\end{document}. Then we prove the generalized \begin{document}$ \varepsilon $\end{document}-regularity lemma for \begin{document}$ j $\end{document}-symmetric homogeneous harmonic polynomial \begin{document}$ P $\end{document} with origin \begin{document}$ 0 $\end{document} as the isolated critical point in \begin{document}$ \mathbb{R}^{n-j} $\end{document}, and by the generalized \begin{document}$ \varepsilon $\end{document}-regularity lemma, we show the Hausdorff measure estimate of the \begin{document}$ j $\end{document}-symmetric singular set \begin{document}$ S^j(u) $\end{document}. Moreover, we study the geometric structure of interior singular points of solutions \begin{document}$ u $\end{document} in a planar bounded domain.

Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity
Yuta Ishii
2020, 19(6): 2965-3031 doi: 10.3934/cpaa.2020130 +[Abstract](243) +[HTML](82) +[PDF](1587.4KB)

In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity:

where \begin{document}$ d,c,D>0 $\end{document} are given constants, \begin{document}$ \varepsilon >0 $\end{document} is sufficiently small, and \begin{document}$ g(x) $\end{document} is a given positive function. Let \begin{document}$ N \ge 1 $\end{document} be an arbitrary natural number. We assume that \begin{document}$ g(x) $\end{document} is a periodic and symmetric function, namely \begin{document}$ g(x) = g(-x) $\end{document} and \begin{document}$ g(x) = g(x+2N^{-1}) $\end{document}. We study the stability of \begin{document}$ N $\end{document}-peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity \begin{document}$ g(x) $\end{document} above on their stability. For the standard Schnakenberg model, namely the case of \begin{document}$ g(x) = 1 $\end{document}, with \begin{document}$ d = 0 $\end{document}, the stability of \begin{document}$ N $\end{document}-peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of \begin{document}$ N $\end{document}-peak solution. In particular, we investigate how \begin{document}$ N $\end{document}-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case \begin{document}$ g(x) = 1 $\end{document}.

Real-variable characterizations of new anisotropic mixed-norm Hardy spaces
Long Huang, Jun Liu, Dachun Yang and Wen Yuan
2020, 19(6): 3033-3082 doi: 10.3934/cpaa.2020132 +[Abstract](189) +[HTML](72) +[PDF](856.46KB)

Let \begin{document}$ \vec{p}\in(0, \infty)^n $\end{document} and \begin{document}$ A $\end{document} be a general expansive matrix on \begin{document}$ \mathbb{R}^n $\end{document}. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces \begin{document}$ H_A^{\vec{p}}(\mathbb{R}^n) $\end{document} associated with \begin{document}$ A $\end{document} and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize \begin{document}$ H_A^{\vec{p}}(\mathbb{R}^n) $\end{document}, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood–Paley \begin{document}$ g $\end{document}-functions or \begin{document}$ g_{\lambda}^\ast $\end{document}-functions via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the mixed-norm Lebesgue space \begin{document}$ L^{\vec{p}}(\mathbb{R}^n) $\end{document}. In addition, the authors also obtain the duality between \begin{document}$ H_A^{\vec{p}}(\mathbb{R}^n) $\end{document} and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from \begin{document}$ H_A^{\vec{p}}(\mathbb{R}^n) $\end{document} into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional \begin{document}$ \delta $\end{document}-type and non-convolutional \begin{document}$ \beta $\end{document}-order Calderón–Zygmund operators from \begin{document}$ H_A^{\vec{p}}(\mathbb{R}^n) $\end{document} to itself [or to \begin{document}$ L^{\vec{p}}(\mathbb{R}^n) $\end{document}]. As a corollary, the boundedness of anisotropic convolutional \begin{document}$ \delta $\end{document}-type Calderón–Zygmund operators on the mixed-norm Lebesgue space \begin{document}$ L^{\vec{p}}(\mathbb{R}^n) $\end{document} with \begin{document}$ \vec{p}\in(1, \infty)^n $\end{document} is also presented.

A Lusin type result
Silvano Delladio
2020, 19(6): 3083-3091 doi: 10.3934/cpaa.2020133 +[Abstract](224) +[HTML](89) +[PDF](374.29KB)

By using the property known as Federer-Fleming conjecture (cf. [7, 3.1.17]), recently resolved by B. Bojarski, we prove the following Lusin type result:

Theorem. Let \begin{document}$ A\subset {\mathbb{R}}^n $\end{document} be a measurable set and let \begin{document}$ k $\end{document} be a nonnegative integer. Assume that to each \begin{document}$ x\in A $\end{document} corresponds a polynomial \begin{document}$ P_x: {\mathbb{R}}^n\to {\mathbb{R}} $\end{document} of degree less or equal to \begin{document}$ k+1 $\end{document} such that

holds for all \begin{document}$ \alpha\in {\mathbb{N}}^n $\end{document} such that \begin{document}$ \vert\alpha\vert\leq k $\end{document}, at a.e. \begin{document}$ a\in A $\end{document}. Then, for each \begin{document}$ \varepsilon >0 $\end{document}, there exists \begin{document}$ \varphi\in C^{k+1}( {\mathbb{R}}^n) $\end{document} such that

We will use such a theorem to provide a simple new proof of a well-known property of Sobolev functions.

Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity
Xiaoliang Li and Baiyu Liu
2020, 19(6): 3093-3112 doi: 10.3934/cpaa.2020134 +[Abstract](394) +[HTML](103) +[PDF](479.38KB)

This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time.

Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off
Lvqiao Liu and Hao Wang
2020, 19(6): 3113-3136 doi: 10.3934/cpaa.2020135 +[Abstract](187) +[HTML](72) +[PDF](488.72KB)

In this article we study the large-time behavior of perturbative classical solutions to the Fokker-Planck-Boltzmann equation for non-cutoff hard potentials. When the initial data is a small pertubation of an equilibrium state, global existence and temporal decay estimates of classical solutions are established.

Regular measurable backward compact random attractor for $ g $-Navier-Stokes equation
Fuzhi Li, Dongmei Xu and Jiali Yu
2020, 19(6): 3137-3157 doi: 10.3934/cpaa.2020136 +[Abstract](146) +[HTML](78) +[PDF](499.69KB)

In this paper, we study the backward compactness of random attractors, which describes the compactness of the union \begin{document}$ \cup_{s\leq\tau}\mathcal A(s,\omega) $\end{document} of random attractor sections over past times, \begin{document}$ \tau\in\mathbb R $\end{document}. In particular, we prove the backward compactness and the regularity of random attractors for stochastic \begin{document}$ g $\end{document}-Navier-Stokes equations under the condition that the force is backward tempered and backward limiting. The attraction universe in consideration is non-autonomous and consists of backward tempered sets.

Radial solutions for a class of Hénon type systems with partial interference with the spectrum
Eudes. M. Barboza, Olimpio H. Miyagaki, Fábio R. Pereira and Cláudia R. Santana
2020, 19(6): 3159-3187 doi: 10.3934/cpaa.2020137 +[Abstract](201) +[HTML](64) +[PDF](595.17KB)

We investigate the existence of radial solutions for a class of Hénon type systems with nonlinearities reaching the critical growth and interacting with the spectrum of the operator with the possibility of double resonance. The proof is made using variational methods, combining Brézis and Nirenberg arguments with Ni compactness result and Rabinowitz linking theorem.

Stability problems in nonautonomous linear differential equations in infinite dimensions
Hildebrando M. Rodrigues, J. Solà-Morales and G. K. Nakassima
2020, 19(6): 3189-3207 doi: 10.3934/cpaa.2020138 +[Abstract](190) +[HTML](64) +[PDF](1045.22KB)

In this paper we study the robustness of the stability in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel [3]. Based in Rodrigues [11] and in Kloeden & Rodrigues [10] we introduce a class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.

As counterparts, e show first in another example that it is possible to stabilize an unstable system by using a perturbation with a large period and a small mean value, and finally we give an example where we stabilize an unstable linear ODE with a small perturbation in infinite dimensions by using some ideas developed in Rodrigues & Solà-Morales [21] after an example due to Kakutani (see [13]).

Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl
Naoki Hamamoto and Futoshi Takahashi
2020, 19(6): 3209-3222 doi: 10.3934/cpaa.2020139 +[Abstract](173) +[HTML](93) +[PDF](414.07KB)

In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya [4] on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.

Ricci curvature of conformal deformation on compact 2-manifolds
Yoon-Tae Jung, Soo-Young Lee and Eun-Hee Choi
2020, 19(6): 3223-3231 doi: 10.3934/cpaa.2020140 +[Abstract](157) +[HTML](77) +[PDF](314.26KB)

In this paper, we consider Ricci curvature of conformal deformation on compact 2-manifolds. And we prove that, by the conformal deformation, the resulting manifold is an Einstein manifold.

Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow
Yaguang Wang and Shiyong Zhu
2020, 19(6): 3233-3244 doi: 10.3934/cpaa.2020141 +[Abstract](141) +[HTML](72) +[PDF](375.71KB)

In this paper, we study the formation of singularities in a finite time for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first order spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by using a Lyapunov functional approach. It is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.

Some global dynamics of a Lotka-Volterra competition-diffusion-advection system
Qi Wang
2020, 19(6): 3245-3255 doi: 10.3934/cpaa.2020142 +[Abstract](188) +[HTML](70) +[PDF](361.96KB)

This paper studies some population dynamics of a diffusive Lotka-Volterra competition advection model under no-flux boundary condition. We establish the main results that determine the stability of semi-trivial steady states.

Stable periodic solutions for Nazarenko's equation
Szandra Beretka and Gabriella Vas
2020, 19(6): 3257-3281 doi: 10.3934/cpaa.2020144 +[Abstract](161) +[HTML](73) +[PDF](507.28KB)

In 1976 Nazarenko proposed studying the delay differential equation

under the assumptions that \begin{document}$ p,q,r,\tau\in\left(0,\infty\right) $\end{document}, \begin{document}$ n\in\mathbb{N} = \left\{ 1,2,\ldots\right\} $\end{document} and \begin{document}$ q/p>r $\end{document}. We show that if \begin{document}$ \tau $\end{document} or \begin{document}$ n $\end{document} is large enough, then the positive periodic solution oscillating slowly about \begin{document}$ K = \left(q/p-r\right)^{1/n} $\end{document} is unique, and the corresponding periodic orbit is asymptotically stable. We also determine the asymptotic shape of the periodic solution as \begin{document}$ n\rightarrow\infty $\end{document}.

Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters
Yanqiong Lu and Ruyun Ma
2020, 19(6): 3283-3302 doi: 10.3934/cpaa.2020145 +[Abstract](154) +[HTML](76) +[PDF](416.23KB)

In this work, we apply the 'disconjugacy theory' and Elias's spectrum theory to study the disconjugacy \begin{document}$ u^{(4)} + \beta u''-\alpha u = 0 $\end{document} with two parameters \begin{document}$ \alpha,\beta\in\mathbb{R} $\end{document} and the spectrum structure of the linear operator \begin{document}$ u^{(4)} + \beta u''-\alpha u $\end{document} coupled with the clamped beam conditions \begin{document}$ u(0) = u'(0) = u(1) = u'(1) = 0 $\end{document}. As the application of our results, we obtain the global structure of nodal solutions of the corresponding nonlinear analogue based on the bifurcation theory.

Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces
Hartmut Pecher
2020, 19(6): 3303-3321 doi: 10.3934/cpaa.2020146 +[Abstract](165) +[HTML](77) +[PDF](476.87KB)

We prove a low regularity local well-posedness result for the Maxwell-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces \begin{document}$ \widehat{H}^{s,r} $\end{document}, where \begin{document}$ \|f\|_{\widehat{H}^{s,r}} = \|\langle \xi \rangle^s \widehat{f}(\xi)\|_{\widehat{L}^{r'}} $\end{document}, \begin{document}$ \frac{1}{r}+\frac{1}{r'} = 1 $\end{document}. The assumed regularity for the data is almost optimal with respect to scaling as \begin{document}$ r \to 1 $\end{document}. This closes the gap between what is known in the case \begin{document}$ r = 2 $\end{document}, namely \begin{document}$ s > \frac{3}{4} $\end{document}, and the critical value \begin{document}$ s_c = \frac{1}{2} $\end{document} with respect to scaling.

Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source
Siyao Zhu and Jinliang Wang
2020, 19(6): 3323-3340 doi: 10.3934/cpaa.2020147 +[Abstract](210) +[HTML](74) +[PDF](411.59KB)

Spatial heterogeneity and movement of population play an important role in disease spread and control in reality. This paper concerns with a spatial Susceptible-Infected-Susceptible epidemic model with spontaneous infection and logistic source, aiming to investigate the asymptotic profiles of the endemic steady state (whenever it exists) for large and small diffusion rates. We firstly establish uniform upper bound of solutions. By studying the local and global stability of the unique constant endemic equilibrium when spatial environment is homogeneous, we apply the well-known Leray-Schuauder degree index formula to confirm the existence of endemic steady state. Our theoretical results suggest that spontaneous infection and varying total population strongly enhance the persistence of disease spread in the sense that disease component of the endemic steady state will not approach zero whenever the large and small diffusion rates of the susceptible or infected population is used. This gives new insights and aspects for infectious disease modeling and control.

On the Sobolev embedding properties for compact matrix quantum groups of Kac type
Sang-Gyun Youn
2020, 19(6): 3341-3366 doi: 10.3934/cpaa.2020148 +[Abstract](179) +[HTML](87) +[PDF](600.57KB)

We study the optimal order of natural analogues of Sobolev embedding properties within the framework of compact matrix quantum groups of Kac type. One of the main results of this paper is that the optimal order is given by the polynomial growth order of dual discrete quantum groups in a broad class, which covers all connected compact Lie groups, duals of polynomially growing discrete groups, \begin{document}$ O_2^+ $\end{document} and \begin{document}$ S_4^+ $\end{document}. Outside the realm of co-amenable compact quantum groups, we prove that the optimal order is \begin{document}$ 3 $\end{document} for duals of free groups and free quantum groups \begin{document}$ O_N^+ $\end{document} and \begin{document}$ S_N^+ $\end{document}, and that Sobolev embedding properties can be generalized for all compact matrix quantum groups of Kac type whose duals have the rapid decay property. In addition, we generalize sharpened Hausdorff-Young inequalities, compute degrees of the rapid decay property for duals of \begin{document}$ O_N^+,S_N^+ $\end{document} and prove sharpness of Hardy-Littlewood inequalities on duals of free groups.

The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities
Shuai Zhang and Shaopeng Xu
2020, 19(6): 3367-3385 doi: 10.3934/cpaa.2020149 +[Abstract](170) +[HTML](85) +[PDF](548.93KB)

Chen and Zhang [7] consider the probabilistic Cauchy problem of the fourth order Schrödinger equation

where \begin{document}$ P_m $\end{document} is a homogeneous polynomial of degree \begin{document}$ m $\end{document}. The almost sure local well-posedness and small data global existence were obtained in \begin{document}$ H^s(\mathbb{R}^d) $\end{document} with the regularity threshold \begin{document}$ s_c-1/2 $\end{document} when \begin{document}$ d\geq3 $\end{document}, where \begin{document}$ s_c: = d/2-2/(m-1) $\end{document} is the scaling critical regularity. For the lower regularity threshold \begin{document}$ (d-1)s_c/d $\end{document} with \begin{document}$ m = 2 $\end{document} and \begin{document}$ s_c-\min\{1,d/4\} $\end{document} with \begin{document}$ m\geq3 $\end{document}, we get the corresponding well-posedness of the following fourth order nonlinear Schrödinger equation

on \begin{document}$ {\mathbb{R}}^d $\end{document} (\begin{document}$ d\geq2 $\end{document}) with random initial data.

Global solutions of shock reflection problem for the pressure gradient system
Hanchun Yang, Meimei Zhang and Qin Wang
2020, 19(6): 3387-3428 doi: 10.3934/cpaa.2020150 +[Abstract](173) +[HTML](91) +[PDF](752.24KB)

We are concerned with the shock reflection in gas dynamics for the pressure gradient system. Experimental and computational analysis has shown that two patterns of regular reflection may occur: supersonic and subsonic reflection. In this paper we establish the global existence of solutions for both configurations. The ideas and techniques developed here will be useful for the two-dimensional Riemann problems for hyperbolic conservation laws.

Existence results for quasilinear Schrödinger equations with a general nonlinearity
Haidong Liu and Leiga Zhao
2020, 19(6): 3429-3444 doi: 10.3934/cpaa.2020059 +[Abstract](208) +[HTML](95) +[PDF](446.03KB)

Consider the quasilinear Schrödinger equation

where \begin{document}$ N\geq 3 $\end{document}, \begin{document}$ V: {\mathbb{R}}^N\to{\mathbb{R}} $\end{document} and \begin{document}$ h: {\mathbb{R}}\to{\mathbb{R}} $\end{document} are functions. Under some general assumptions on \begin{document}$ V $\end{document} and \begin{document}$ h $\end{document}, we establish two existence results for problem (A) by using variational methods. The main novelty is that, unlike most other papers on this problem, we do not assume the nonlinear term to be 4-superlinear at infinity.

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