# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

January 2019 , Volume 18 , Issue 1

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2019, 18(1): 1-13 doi: 10.3934/cpaa.2019001 +[Abstract](2803) +[HTML](153) +[PDF](332.85KB)
Abstract:

In this paper, for a nematic liquid crystal system, we address the space-time decay properties of strong solutions in the whole space \begin{document}$\mathbb{R}^3$\end{document}. Based on a parabolic interpolation inequality, bootstrap argument and some weighted estimates, we obtain the higher order derivative estimates for such system.

2019, 18(1): 15-32 doi: 10.3934/cpaa.2019002 +[Abstract](2623) +[HTML](142) +[PDF](358.5KB)
Abstract:

In this article is perfomed a global stability analysis of an infection load-structured epidemic model using tools of dynamical systems theory. An explicit Duhamel formulation of the semiflow allows us to prove the existence of a compact attractor for the trajectories of the system. Then, according to the sharp threshold \begin{document}$\mathcal R_0$\end{document}, the basic reproduction number of the disease, we make explicit the basins of attractions of the equilibria of the system and prove their global stability with respect to these basins, the attractivness property being obtained using infinite dimensional Lyapunov functions.

2019, 18(1): 33-50 doi: 10.3934/cpaa.2019003 +[Abstract](2341) +[HTML](109) +[PDF](419.04KB)
Abstract:

The main contribution of the N-barrier maximum principle is that it provides rather generic a priori upper and lower bounds for the linear combination of the components of a vector-valued solution. We show that the N-barrier maximum principle (NBMP, C.-C. Chen and L.-C. Hung (2016)) remains true for \begin{document}$n$\end{document} \begin{document}$(n>2)$\end{document} species. In addition, a stronger lower bound in NBMP is given by employing an improved tangent line method. As an application of NBMP, we establish a nonexistence result for traveling wave solutions to the four species Lotka-Volterra system.

2019, 18(1): 51-64 doi: 10.3934/cpaa.2019004 +[Abstract](2561) +[HTML](115) +[PDF](367.13KB)
Abstract:

We consider the following quasilinear Schrödinger equation

where \begin{document}$N≥ 1$\end{document}, \begin{document}$0 < q(x)≤ \lim_{|x|\to∞}q(x)$\end{document}, \begin{document}$g∈ C(\mathbb{R}^+, \mathbb{R})$\end{document} and \begin{document}$g(u)/u^3 \to 1$\end{document}, as \begin{document}$u \to ∞.$\end{document} We establish the existence of a positive solution to this problem by using the method developed in Szulkin and Weth [27,28].

2019, 18(1): 65-81 doi: 10.3934/cpaa.2019005 +[Abstract](2320) +[HTML](119) +[PDF](408.92KB)
Abstract:

This paper is concerned with constraint minimizers of an \begin{document}$L^2-$\end{document}critical minimization problem (1) in \begin{document}$\mathbb{R}^N$\end{document} (\begin{document}$N≥ 1$\end{document}) under an \begin{document}$L^2-$\end{document}subcritical perturbation. We prove that the problem admits minimizers with mass \begin{document}$ρ^\frac{N}{2}$\end{document} if and only if \begin{document}$0≤ρ < ρ^*: = \|Q\|^{\frac{4}{N}}_2$\end{document} for \begin{document}$b≥0$\end{document} and \begin{document}$0 < ρ ≤ρ^*$\end{document} for \begin{document}$b < 0$\end{document}, where the constant \begin{document}$b$\end{document} comes from the coefficient of the perturbation term, and \begin{document}$Q$\end{document} is the unique positive radically symmetric solution of \begin{document}$Δ u(x)-u(x)+u^{1+\frac{4}{N}}(x) = 0$\end{document} in \begin{document}$\mathbb{R}^N$\end{document}. Furthermore, we analyze rigorously the concentration behavior of minimizers as \begin{document}$ρ \nearrow ρ^*$\end{document} for the case where \begin{document}$b>0$\end{document}, which shows that the concentration rates are determined by the subcritical perturbation, instead of the local profiles of the potential \begin{document}$V(x)$\end{document}.

2019, 18(1): 83-106 doi: 10.3934/cpaa.2019006 +[Abstract](2454) +[HTML](105) +[PDF](394.64KB)
Abstract:

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by the Φ-Laplacian operator. One of the solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev space framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.

2019, 18(1): 107-128 doi: 10.3934/cpaa.2019007 +[Abstract](2422) +[HTML](110) +[PDF](397.03KB)
Abstract:

We consider a function $U$ satisfying a degenerate elliptic equation on $\mathbb{R}_ + ^{N + 1}: = (0, +∞)×{\mathbb{R}^N}$ with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain $\Omega\subset{\mathbb{R}^N}$ of class $C^{1, 1}$, whereas the Dirichlet data is on the exterior of $\Omega$. We prove Hölder regularity estimates of $\frac{U}{d_\Omega^s}$, where $d_\Omega$ is a distance function defined as $d_\Omega(z): = \text{dist}(z, {\mathbb{R}^N}\setminus\Omega)$, for $z∈\overline{\mathbb{R}_ + ^{N + 1}}$. The degenerate elliptic equation arises from the Caffarelli-Silvestre extension of the Dirichlet problem for the fractional Laplacian. Our proof relies on compactness and blow-up analysis arguments.

2019, 18(1): 129-158 doi: 10.3934/cpaa.2019008 +[Abstract](2466) +[HTML](113) +[PDF](459.92KB)
Abstract:

In the present paper, we consider the following Kirchhoff type problem

where \begin{document}$a$ \end{document} is a positive constant, \begin{document}$λ$ \end{document} is a positive parameter, \begin{document}$V∈ L^{\frac{N}{2}}(\mathbb{R}^N)$ \end{document} is a given nonnegative function and \begin{document}$2^*$ \end{document} is the critical exponent. The existence of bounded state solutions for Kirchhoff type problem with critical exponents in the whole \begin{document}$\mathbb R^N$ \end{document} (\begin{document}$N≥5$ \end{document}) has never been considered so far. We obtain sufficient conditions on the existence of bounded state solutions in high dimension \begin{document}$N≥4$ \end{document}, and especially it is the fist time to consider the case when \begin{document}$N≥5$ \end{document} in the literature.

2019, 18(1): 159-180 doi: 10.3934/cpaa.2019009 +[Abstract](2934) +[HTML](146) +[PDF](367.7KB)
Abstract:

In this paper, we consider a viscoelastic plate equation with a logarithmic nonlinearity. Using the Galaerkin method and the multiplier method, we establish the existence of solutions and prove an explicit and general decay rate result. This result extends and improves many results in the literature such as Gorka [19], Hiramatsu et al. [27] and Han and Wang [26].

2019, 18(1): 181-193 doi: 10.3934/cpaa.2019010 +[Abstract](2468) +[HTML](133) +[PDF](342.2KB)
Abstract:

An optimal condition is given for the existence of positive solutions of nonlinear Kirchhoff PDE with strong singularities. A byproduct is that $-2$ is no longer the critical position for the existence of positive solutions of PDE's with singular potentials and negative powers of the form: \begin{document} $- |x{|^\alpha }\Delta u = {u^{{\rm{ - }}\gamma }}$ \end{document} in \begin{document} $Ω$ \end{document}, \begin{document} $u = 0$ \end{document} on \begin{document} $\partial \Omega$ \end{document}, where \begin{document} $\Omega$ \end{document} is a bounded domain of \begin{document} ${\mathbb{R}}^{N}$ \end{document} containing 0, with \begin{document} $N \ge 3$ \end{document}, \begin{document} $\alpha \in \left( {0, N} \right)$ \end{document} and \begin{document} $- \gamma \in \left( { - 3, - 1} \right)$ \end{document}.

2019, 18(1): 195-225 doi: 10.3934/cpaa.2019011 +[Abstract](2446) +[HTML](97) +[PDF](458.38KB)
Abstract:

Two-phase flow of two Newtonian incompressible viscous fluids with a soluble surfactant and different densities of the fluids can be modeled within the diffuse interface approach. We consider a Navier-Stokes/Cahn-Hilliard type system coupled to non-linear diffusion equations that describe the diffusion of the surfactant in the bulk phases as well as along the diffuse interface. Moreover, the surfactant concentration influences the free energy and therefore the surface tension of the diffuse interface. For this system existence of weak solutions globally in time for general initial data is proved. To this end a two-step approximation is used that consists of a regularization of the time continuous system in the first and a time-discretization in the second step.

2019, 18(1): 227-236 doi: 10.3934/cpaa.2019012 +[Abstract](2362) +[HTML](108) +[PDF](317.59KB)
Abstract:

In this paper, we prove the existence of the positive and negative solutions to p-Laplacian eigenvalue problems with supercritical exponent. This extends previous results on the problems with subcritical and critical exponents.

2019, 18(1): 237-253 doi: 10.3934/cpaa.2019013 +[Abstract](2547) +[HTML](102) +[PDF](376.82KB)
Abstract:

In this paper, we study the following critical system with fractional Laplacian:

By using the Nehari manifold, under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.

2019, 18(1): 255-284 doi: 10.3934/cpaa.2019014 +[Abstract](2433) +[HTML](129) +[PDF](454.0KB)
Abstract:

In this work, we are concerned with a class of parabolic-elliptic chemotaxis systems with the prototype given by

with nonnegative initial condition for \begin{document}$u$\end{document} and homogeneous Neumann boundary conditions in a smooth bounded domain \begin{document}$Ω\subset \mathbb{R}^n(n≥ 2)$\end{document}, where \begin{document}$χ, b, κ>0$\end{document}, \begin{document}$a∈ \mathbb{R}$\end{document} and \begin{document}$θ>1$\end{document}.

First, using different ideas from [9,11], we re-obtain the boundedness and global existence for the corresponding initial-boundary value problem under, either

or

Next, carrying out bifurcation from "old multiplicity", we show that the corresponding stationary system exhibits pattern formation for an unbounded range of chemosensitivity \begin{document}$χ$\end{document} and the emerging patterns converge weakly in \begin{document}$L^θ(Ω)$\end{document} to some constants as \begin{document}$χ \to ∞$\end{document}. This provides more details and also fills up a gap left in Kuto et al. [13] for the particular case that \begin{document}$θ = 2$\end{document} and \begin{document}$κ = 1$\end{document}. Finally, for \begin{document}$θ = κ+1$\end{document}, the global stabilities of the equilibria \begin{document}$((a/b)^{\frac{1}{κ}}, a/b)$\end{document} and \begin{document}$(0,0)$\end{document} are comprehensively studied and explicit convergence rates are computed out, which exhibits chemotaxis effects and logistic damping on long time dynamics of solutions. These stabilization results indicate that no pattern formation arises for small \begin{document}$χ$\end{document} or large damping rate \begin{document}$b$\end{document}; on the other hand, they cover and extend He and Zheng's [6,Theorems 1 and 2] for logistic source and linear secretion (\begin{document}$θ = 2$\end{document} and \begin{document}$κ = 1$\end{document}) (where convergence rate estimates were shown) to generalized logistic source and nonlinear secretion.

2019, 18(1): 285-300 doi: 10.3934/cpaa.2019015 +[Abstract](3435) +[HTML](107) +[PDF](383.2KB)
Abstract:

In this paper, we investigate the following a class of Choquard equation

where \begin{document}$N≥ 3,~α∈ (0,N),~I_α$\end{document} is the Riesz potential and \begin{document}$F(s) = ∈t_{0}^{s}f(t)dt$\end{document}. If \begin{document}$f$\end{document} satisfies almost necessary the upper critical growth conditions in the spirit of Berestycki and Lions, we obtain the existence of positive radial ground state solution by using the Pohožaev manifold and the compactness lemma of Strauss.

2019, 18(1): 301-322 doi: 10.3934/cpaa.2019016 +[Abstract](2309) +[HTML](108) +[PDF](399.39KB)
Abstract:

Let \begin{document}$N ≥ 3$\end{document} and \begin{document}$Ω \subset \mathbb{R}^N$\end{document} be a \begin{document}$C^2$\end{document} bounded domain. We study the existence of positive solution \begin{document}$u ∈ H^1(Ω)$\end{document} of

where \begin{document}$τ = 1$\end{document} or \begin{document}$-1$\end{document}, \begin{document}$0 < s <2$\end{document}, \begin{document}$2^*(s) = \frac{2(N-s)}{N-2}$\end{document} and \begin{document}$x_1, x_2 ∈ \overline{Ω}$\end{document} with \begin{document}$x_1 ≠ x_2$\end{document}. First, we show the existence of positive solutions to the equation provided the positive \begin{document}$λ$\end{document} is small enough. In case that one of the singularities locates on the boundary and the mean curvature of the boundary at this singularity is positive, the existence of positive solutions is obtained for any \begin{document}$λ > 0$\end{document} and some \begin{document}$s$\end{document} depending on \begin{document}$τ$\end{document} and \begin{document}$N$\end{document}. Furthermore, we extend the existence theory of solutions to the equations for the case of the multiple singularities.

2019, 18(1): 323-340 doi: 10.3934/cpaa.2019017 +[Abstract](2083) +[HTML](121) +[PDF](367.59KB)
Abstract:

In this paper we study a class of degenerate second-order elliptic differential operators, often referred to as Fleming-Viot type operators, in the framework of function spaces defined on the \begin{document}$d$\end{document}-dimensional hypercube \begin{document}$Q_d$\end{document} of \begin{document}$\mathbf{R}^d$\end{document}, \begin{document}$d ≥1$\end{document}.

By making mainly use of techniques arising from approximation theory, we show that their closures generate positive semigroups both in the space of all continuous functions and in weighted \begin{document}$L^{p}$\end{document}-spaces.

In addition, we show that the semigroups are approximated by iterates of certain polynomial type positive linear operators, which we introduce and study in this paper and which generalize the Bernstein-Durrmeyer operators with Jacobi weights on \begin{document}$[0, 1]$\end{document}.

As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we establish some of their regularity properties along with their asymptotic behaviours.

2019, 18(1): 341-360 doi: 10.3934/cpaa.2019018 +[Abstract](2719) +[HTML](122) +[PDF](385.17KB)
Abstract:

In this paper, some new results on the the regularity of Kolmogorov equations associated to the infinite dimensional OU-process are obtained. As an application, the average \begin{document}$L^2$\end{document}-error on \begin{document}$[0, T]$\end{document} of exponential integrator scheme for a range of semi-linear stochastic partial differential equations is derived, where the drift term is assumed to be Hölder continuous with respect to the Sobolev norm \begin{document}$\|·\|_{β}$\end{document} for some appropriate \begin{document}$β>0$\end{document}. In addition, under a stronger condition on the drift, the strong convergence estimate is obtained, which covers the result of the SDEs with Hölder continuous drift.

2019, 18(1): 361-396 doi: 10.3934/cpaa.2019019 +[Abstract](2891) +[HTML](136) +[PDF](502.44KB)
Abstract:

The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habitats with nonlocal dispersal. It is assumed that the trivial solution \begin{document}${\bf u} = {\bf 0}$\end{document} of such a system is unstable and the system has a stable space-time periodic positive solution \begin{document}${\bf u^*}(t,x)$\end{document}. We first show that in any direction \begin{document}$ξ∈ \mathbb{S}^{N-1}$\end{document}, such a system has a finite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed \begin{document}$c^{*}(ξ)$\end{document} in the direction of \begin{document}$ξ$\end{document}. Next, we show that for any \begin{document}$c>c^{*}(ξ)$\end{document}, there are space-time periodic traveling wave solutions of the form \begin{document}${\bf{u}}(t,x) = {\bf{Φ}}(x-ctξ,t,ctξ)$\end{document} connecting \begin{document}${\bf u^*}$\end{document} and \begin{document}${\bf 0}$\end{document}, and propagating in the direction of \begin{document}$ξ$\end{document} with speed \begin{document}$c$\end{document}, where \begin{document}$Φ(x,t,y)$\end{document} is periodic in \begin{document}$t$\end{document} and \begin{document}$y$\end{document}, and there is no such solution for \begin{document}$c<c^{*}(ξ)$\end{document}. We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.

2019, 18(1): 397-424 doi: 10.3934/cpaa.2019020 +[Abstract](2086) +[HTML](98) +[PDF](410.59KB)
Abstract:

In this paper we construct the spectral expansion for the differential operator generated in \begin{document}$L_{2}(-∞, ∞)$\end{document} by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.

2019, 18(1): 425-434 doi: 10.3934/cpaa.2019021 +[Abstract](2218) +[HTML](108) +[PDF](318.06KB)
Abstract:

Consider the second order self-adjoint discrete Hamiltonian system

where \begin{document}$p(n), L(n)$\end{document} and \begin{document}$W(n, x)$\end{document} are \begin{document}$N$\end{document}-periodic on \begin{document}$n$\end{document}, and \begin{document}zhongwenzy$\end{document}lies in a gap of the spectrum \begin{document}$σ(\mathcal{A})$\end{document}of the operator \begin{document}$\mathcal{A}$\end{document}, which is bounded self-adjoint in \begin{document}$l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$\end{document} defined by \begin{document}$(\mathcal{A}u)(n) = \triangle [p(n)\triangle u(n-1)]-L(n)u(n)$\end{document}. We obtain a sufficient condition on the existence of nontrivial homoclinic orbits for the above system under a much weaker condition than \begin{document}$\lim_{|x|\to ∞}\frac{W(n, x)}{|x|^2} = ∞$\end{document} uniformly in \begin{document}$ n∈ \mathbb{Z}$\end{document}, which has been a common condition used in the existing literature. We also give three examples to illustrate our result. 2019, 18(1): 435-453 doi: 10.3934/cpaa.2019022 +[Abstract](2101) +[HTML](108) +[PDF](392.14KB) Abstract: In this manuscript, we provide a point-wise estimate for the 3-commutators involving fractional powers of the sub-Laplacian on Carnot groups of homogeneous dimension \begin{document}$Q$\end{document}. This can be seen as a fractional Leibniz rule in the sub-elliptic setting. As a corollary of the point-wise estimate, we provide an \begin{document}$(L^{p}, L^{q})\to L^{r}$\end{document} estimate for the commutator, provided that \begin{document}$\frac{1}{r} = \frac{1}{p}+\frac{1}{q}-\frac{α}{Q}$\end{document} for \begin{document}$α ∈ (0, Q)$\end{document}. 2019, 18(1): 455-478 doi: 10.3934/cpaa.2019023 +[Abstract](2508) +[HTML](125) +[PDF](368.2KB) Abstract: This paper deals with the exact controllability for a class of fractional evolution systems in a Banach space. First, we introduce a new concept of exact controllability and give notion of the mild solutions of the considered evolutional systems via resolvent operators. Second, by utilizing the semigroup theory, the fixed point strategy and Kuratowski's measure of noncompactness, the exact controllability of the evolutional systems is investigated without Lipschitz continuity and growth conditions imposed on nonlinear functions. The results are established under the hypothesis that the resolvent operator is differentiable and analytic, respectively, instead of supposing that the semigroup is compact. An example is provided to illustrate the proposed results. 2019, 18(1): 479-492 doi: 10.3934/cpaa.2019024 +[Abstract](2130) +[HTML](132) +[PDF](371.04KB) Abstract: For any positive decreasing to zero sequence \begin{document}$a_n$\end{document} such that \begin{document}$\sum a_n$\end{document} diverges we consider the related series \begin{document}$\sum k_na_n$\end{document} and \begin{document}$\sum j_na_n.$\end{document} Here, \begin{document}$k_n$\end{document} and \begin{document}$j_n$\end{document} are real sequences such that \begin{document}$k_n∈\{0,1\}$\end{document} and \begin{document}$j_n∈\{-1,1\}.$\end{document} We study their convergence and characterize it in terms of the density of 1's in the sequences \begin{document}$k_n$\end{document} and \begin{document}$j_n.$\end{document} We extend our results to series \begin{document}$\sum m_na_n,$\end{document} with \begin{document}$m_n∈\{-1,0,1\}$\end{document} and apply them to study some associated random series. 2019, 18(1): 493-517 doi: 10.3934/cpaa.2019025 +[Abstract](3161) +[HTML](127) +[PDF](413.66KB) Abstract: In this paper, we study the following quasilinear Schrödinger equation where \begin{document}$ N>4, 2^* = \frac{2N}{N-2}, V: \mathbb{R}^N \to \mathbb{R}$\end{document} satisfies suitable assumptions. Unlike \begin{document}$ g∈ \mathcal{C}^1(\mathbb{R},\mathbb{R})$\end{document}, we only need to assume that \begin{document}$ g∈ \mathcal{C}(\mathbb{R},\mathbb{R})$\end{document}. By using a change of variable, we obtain the existence of ground state solutions with general critical growth. Our results extend some known results. 2019, 18(1): 519-538 doi: 10.3934/cpaa.2019026 +[Abstract](2423) +[HTML](114) +[PDF](422.76KB) Abstract: In this paper, we are concerned with fractional Choquard equation where \begin{document}$\epsilon>0$\end{document} is a parameter, \begin{document}$0<α<1$\end{document}, \begin{document}$0<μ<3$\end{document}, \begin{document}$2_{μ,α}^* = \frac{6-μ}{3-2α}$\end{document} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator, \begin{document}$f$\end{document} is a continuous subcritical term, and \begin{document}$F$\end{document} is the primitive function of \begin{document}$f$\end{document}. By virtue of the method of Nehari manifold and Ljusternik-Schnirelmann category theory, we prove that the equation has a ground state for \begin{document}$\epsilon$\end{document} small enough and investigate the relation between the number of solutions and the topology of the set where \begin{document}$V$\end{document} attains its global minimum for small \begin{document}$\epsilon\$ \end{document}. We also obtain sufficient conditions for the nonexistence of ground states.

2019, 18(1): 539-558 doi: 10.3934/cpaa.2019027 +[Abstract](2666) +[HTML](132) +[PDF](470.81KB)
Abstract:

The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from [18]. At the end of the paper, a numerical example satisfying the theory is provided.

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