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Communications on Pure & Applied Analysis

January 2018 , Volume 17 , Issue 1

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Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system
Narcisse Batangouna and Morgan Pierre
2018, 17(1): 1-19 doi: 10.3934/cpaa.2018001 +[Abstract](418) +[HTML](37) +[PDF](464.4KB)

We consider a time semi-discretization of the Caginalp phase-field model based on an operator splitting method. For every time-step parameter \begin{document}$ \tau $\end{document}, we build an exponential attractor \begin{document}$ \mathcal{M}_\tau $\end{document} of the discrete-in-time dynamical system. We prove that \begin{document}$ \mathcal{M}_\tau $\end{document} converges to an exponential attractor $\mathcal{M}_0$ of the continuous-in-time dynamical system for the symmetric Hausdorff distance as \begin{document}$ \tau $\end{document} tends to $0$. We also provide an explicit estimate of this distance and we prove that the fractal dimension of \begin{document}$ \mathcal{M}_\tau $\end{document} is bounded by a constant independent of \begin{document}$ \tau $\end{document}.

Unilateral global interval bifurcation for Kirchhoff type problems and its applications
Wenguo Shen
2018, 17(1): 21-37 doi: 10.3934/cpaa.2018002 +[Abstract](184) +[HTML](23) +[PDF](449.8KB)

In this paper, we establish a unilateral global bifurcation result from interval for a class of Kirchhoff type problems with nondifferentiable nonlinearity. By applying the above result, we shall prove the existence of one-sign solutions for the following Kirchhoff type problems.

where Ω is a bounded domain in \begin{document}$ \mathbb{R}^{N} $\end{document} with a smooth boundary \begin{document}$ \partial $\end{document}Ω, \begin{document}$ M $\end{document} is a continuous function, r is a parameter, \begin{document}$ a(x) \in C(\overline \Omega ) $\end{document} is positive, \begin{document}$ u^{+} = \max\{u, 0\}, u^{-}= -\min\{u, 0\} $\end{document}, \begin{document}$ \alpha ,\beta \in C\left( {\overline \Omega } \right) $\end{document}; \begin{document}$ f \in C\left( {\mathbb{R},\mathbb{R}} \right) $\end{document}, \begin{document}$ sf(s)>0 $\end{document} for \begin{document}$ s \in {\mathbb{R}^ + }, $\end{document} and \begin{document}$ {f_0} \in \left( {0,\infty } \right) $\end{document} and \begin{document}$ {f_\infty } \in \left( {0,\infty } \right] $\end{document} or \begin{document}$ {f_0} \in \infty $\end{document} and f∈[0, ∞], where \begin{document}$ {f_0} = {\lim _{\left| s \right| \to 0}}f\left( s \right)/s,{f_\infty } = {\lim _{\left| s \right| \to + \infty }}f\left( s \right)/s $\end{document}. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition
VicenŢiu D. RǍdulescu and Somayeh Saiedinezhad
2018, 17(1): 39-52 doi: 10.3934/cpaa.2018003 +[Abstract](276) +[HTML](14) +[PDF](436.0KB)

We are concerned with the study of the following nonlinear eigenvalue problem with Robin boundary condition

The abstract setting involves Sobolev spaces with variable exponent. The main result of the present paper establishes a sufficient condition for the existence of an unbounded sequence of eigenvalues. Our arguments strongly rely on the Lusternik-Schnirelmann principle. Finally, we focus to the following particular case, which is a $p(x)$-Laplacian problem with several variable exponents:

Combining variational arguments, we establish several properties of the eigenvalues family of this nonhomogeneous Robin problem.

Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential
Hongxia Shi and Haibo Chen
2018, 17(1): 53-66 doi: 10.3934/cpaa.2018004 +[Abstract](245) +[HTML](17) +[PDF](410.9KB)

We investigate a class of generalized quasilinear Schrödinger equations

where \begin{document}$ g(u):\mathbb{R}\to\mathbb{R}^{+} $\end{document} is a nondecreasing function with respect to \begin{document}$ |u| $\end{document}, the potential \begin{document}$ V(x) $\end{document} and the primitive of the nonlinearity \begin{document}$ f(x,u) $\end{document} are allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of infinitely many nontrivial solutions. The proof is based on a change of variable as well as symmetric Mountain Pass Theorem.

Stochastic spatiotemporal diffusive predator-prey systems
Guanqi Liu and Yuwen Wang
2018, 17(1): 67-84 doi: 10.3934/cpaa.2018005 +[Abstract](174) +[HTML](12) +[PDF](165.3KB)

In this paper, a spatiotemporal diffusive predator-prey system with Holling type-Ⅲ is considered. By using a Lyapunov-like function, it is proved that the unique local solution of the system must be a a global one if the interaction intensity is small enough. A comparison theorem is used to show that the system can be extinction or stability in mean square under some additional conditions. Finally, an unique invariant measure for the system is obtained.

Liouville results for fully nonlinear integral elliptic equations in exterior domains
Hongxia Zhang and Ying Wang
2018, 17(1): 85-112 doi: 10.3934/cpaa.2018006 +[Abstract](152) +[HTML](17) +[PDF](470.6KB)

In this paper, we obtain Liouville type theorems both in the whole space and exterior domain in viscosity sense for fully nonlinear elliptic inequality involving nonlocal Pucci's operator. The nonlocal property of the operator, we only have a much weaker comparison principle, compared with the inequality with classical Pucci's operators, which give rise to the difficulties for the Hadamard type property in exterior domain.

Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent
Peng Chen and Xiaochun Liu
2018, 17(1): 113-125 doi: 10.3934/cpaa.2018007 +[Abstract](241) +[HTML](28) +[PDF](410.1KB)

This paper is concerned with the existence and multiplicity of solutions to the following Kirchhoff type elliptic equations with critical nonlinearity:

where $Ω\subset\mathbb{R}^3$ is a bounded smooth domain, $μ$ is a positive parameter and $f:Ω×\mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying some further conditions. Our approach is based on concentration-compactness principle and symmetry mountain pass theorem.

The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas
Jianjun Chen and Wancheng Sheng
2018, 17(1): 127-142 doi: 10.3934/cpaa.2018008 +[Abstract](175) +[HTML](20) +[PDF](444.5KB)

This paper is concerned with the Euler equations in the magnetogasdynamics for generalized Chaplygin gas. The global solutions to the Riemann problems of the Euler equations in the magnetogasdynamics for generalized Chaplygin gas are obtained constructively by using phase plane analysis method. The formation of delta shock wave is studied as magnetic field vanishes. The limit behaviors of the Riemann solutions as magnetic field vanishes are also obtained.

Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities
Bartosz Bieganowski and Jaros law Mederski
2018, 17(1): 143-161 doi: 10.3934/cpaa.2018009 +[Abstract](306) +[HTML](32) +[PDF](424.2KB)

We look for ground state solutions to the following nonlinear Schrödinger equation

where $V=V_{per}+V_{loc}∈ L^{∞}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $Γ∈ L^{∞}(\mathbb{R}^N)$ is periodic and $Γ(x)≥ 0$ for a.e. $x∈\mathbb{R}^N$ and $2≤q <2^*$. We assume that $\inf σ(-Δ+V)>0$, where $σ(-Δ+V)$ stands for the spectrum of $-Δ +V$ and $f$ has the subcritical growth but higher than $Γ(x)|u|^{q-2}u$, however the nonlinearity $f(x, u)-Γ(x)|u|^{q-2}u$ may change sign. Although a Nehari-type monotonicity condition for the nonlinearity is not satisfied, we investigate the existence of ground state solutions being minimizers on the Nehari manifold.

Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field
Alex H. Ardila
2018, 17(1): 163-175 doi: 10.3934/cpaa.2018010 +[Abstract](289) +[HTML](12) +[PDF](393.7KB)

In this paper we study the existence and orbital stability of ground states for logarithmic Schrödinger equation under a constant magnetic field. For this purpose we establish the well-posedness of the Cauchy Problem in a magnetic Sobolev space and an appropriate Orlicz space. Then we show the existence of ground state solutions via a constrained minimization on the Nehari manifold. We also show that the ground state is orbitally stable.

Time decay in dual-phase-lag thermoelasticity: Critical case
Zhuangyi Liu and Ramón Quintanilla
2018, 17(1): 177-190 doi: 10.3934/cpaa.2018011 +[Abstract](212) +[HTML](22) +[PDF](383.7KB)

This note is devoted to the study of the time decay of the one-dimensional dual-phase-lag thermoelasticity. In this theory two delay parameters \begin{document} $τ_q$ \end{document} and \begin{document} $τ_{θ}$ \end{document} are proposed. It is known that the system is exponentially stable if \begin{document} $τ_q<2 τ_{θ}$ \end{document} [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that \begin{document} $τ_q=2 τ_{θ}$ \end{document} as well the optimality of this decay rate. Second, we prove that the exponential stability remains true even if the inequality only holds in a proper sub-interval of the spatial domain, when \begin{document} $τ_{θ}$ \end{document} is spatially dependent.

Higher order eigenvalues for non-local Schrödinger operators
Niels Jacob and Feng-Yu Wang
2018, 17(1): 191-208 doi: 10.3934/cpaa.2018012 +[Abstract](228) +[HTML](20) +[PDF](475.2KB)

Two-sided estimates for higher order eigenvalues are presented for a class of non-local Schrödinger operators by using the jump rate and the growth of the potential. For instance, let \begin{document} $L$ \end{document} be the generator of a Lévy process with Lévy measure \begin{document} $ν(\text{d} z):= \rho(z)\text{d} z$ \end{document} such that \begin{document} $\rho(z)=\rho(-z)$ \end{document} and

for some constants \begin{document} $\kappa, c_1,c_2>0$ \end{document} and \begin{document} $\alpha_1,\alpha_2∈ (0,2),$ \end{document} and let \begin{document} $c_3|x|^{θ_1} ≤ V(x)≤ c_4|x|^{θ_2}$ \end{document} for some constants \begin{document} $θ_1,θ_2, c_3,c_4>0$ \end{document} and large \begin{document} $|x|$ \end{document}. Then the eigenvalues \begin{document} $\lambda_1≤ \lambda_2≤··· \lambda_n≤ ··· $ \end{document} of \begin{document} $-L+V$ \end{document} satisfies the following two-side estimate: there exists a constant \begin{document} $C>1$ \end{document} such that

When \begin{document} $\alpha_1$ \end{document} is variable, a better lower bound estimate is derived.

Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary
Shaodong Wang
2018, 17(1): 209-230 doi: 10.3934/cpaa.2018013 +[Abstract](157) +[HTML](18) +[PDF](464.9KB)

We consider the Yamabe-type problem

when \begin{document} $(M,g)$ \end{document} is the standard half sphere of dimensions \begin{document} $n≥ 3$ \end{document}. We establish existence results of positive blowing-up solutions with unbounded energy to this problem for all dimensions \begin{document} $n≥ 3$ \end{document}.

Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term
Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu and Dušan D. Repovš
2018, 17(1): 231-241 doi: 10.3934/cpaa.2018014 +[Abstract](184) +[HTML](24) +[PDF](363.1KB)

We consider a nonlinear Robin problem driven by the p-Laplacian plus an indefinite potential. The reaction term is of arbitrary growth and only conditions near zero are imposed. Using critical point theory together with suitable truncation and perturbation techniques and comparison principles, we show that the problem admits a sequence of distinct smooth nodal solutions converging to zero in \begin{document} $C^1(\overline{Ω})$ \end{document}.

Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay
Luciano Abadías, Carlos Lizama and Marina Murillo-Arcila
2018, 17(1): 243-265 doi: 10.3934/cpaa.2018015 +[Abstract](161) +[HTML](17) +[PDF](526.8KB)

We characterize the well-posedness of a third order in time equation with infinite delay in Hölder spaces, solely in terms of spectral properties concerning the data of the problem. Our analysis includes the case of the linearized Kuznetzov and Westerwelt equations. We show in case of the Laplacian operator the new and surprising fact that for the standard memory kernel \begin{document} $g(t)=\frac{t^{ν-1}}{Γ(ν)}e^{-at}$ \end{document} the third order problem is ill-posed whenever \begin{document} $0<ν ≤q 1$ \end{document} and \begin{document} $a$ \end{document} is inversely proportional to one of the terms of the given model.

On stability of functional differential equations with rapidly oscillating coefficients
Vitalii G. Kurbatov and Valentina I. Kuznetsova
2018, 17(1): 267-283 doi: 10.3934/cpaa.2018016 +[Abstract](184) +[HTML](24) +[PDF](556.5KB)

The paper deals with the functional differential equation

where the functions \begin{document} $y$ \end{document} and \begin{document} $f$ \end{document} take their values in a Hilbert space, \begin{document} $ω_k∈\mathbb{R}$ \end{document}, \begin{document} $μ_k$ \end{document} are bounded operator-valued measures concentrated on \begin{document} $[0, +∞)$ \end{document}, and \begin{document} $\sum_{k=1}^∞\Vertμ_k\Vert < ∞$ \end{document}. It is shown that the equation is stable provided the unperturbed equation \begin{document} $y'(t)+\int{{_0^{∞}}}μ_0(ds)\, y(t-s)=f(t)$ \end{document} is at least strictly passive (and consequently stable) and a special estimate holds; this estimate is certainly true if \begin{document} $|ω_k|$ \end{document} are sufficiently large.

Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions
Anna Kostianko and Sergey Zelik
2018, 17(1): 285-317 doi: 10.3934/cpaa.2018017 +[Abstract](206) +[HTML](16) +[PDF](536.0KB)

This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in [6] and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed with periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for this class of dissipative systems strongly depend on the choice of boundary conditions.

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