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Communications on Pure & Applied Analysis

March 2020 , Volume 19 , Issue 3

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Bifurcation and stability of a two-species diffusive Lotka-Volterra model
Li Ma and Shangjiang Guo
2020, 19(3): 1205-1232 doi: 10.3934/cpaa.2020056 +[Abstract](118) +[HTML](12) +[PDF](419.64KB)
Abstract:

This paper is devoted to a two-species Lotka-Volterra model with general functional response. The existence, local and global stability of boundary (including trivial and semi-trivial) steady-state solutions are analyzed by means of the signs of the associated principal eigenvalues. Moreover, the nonexistence and steady-state bifurcation of coexistence steady-state solutions at each of the boundary steady states are investigated. In particular, the coincidence of bifurcating coexistence steady-state solution branches is also described. It should be pointed out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov-Schmidt reduction, and bifurcation theory.

Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime
Moon-Jin Kang, Seung-Yeal Ha, Jeongho Kim and Woojoo Shim
2020, 19(3): 1233-1256 doi: 10.3934/cpaa.2020057 +[Abstract](51) +[HTML](19) +[PDF](394.75KB)
Abstract:

We present a hydrodynamic limit from the kinetic thermomechanical Cucker-Smale (TCS) model to the hydrodynamic Cucker-Smale (CS) model in a strong local alignment regime. For this, we first provide a global existence of weak solution, and flocking dynamics for classical solution to the kinetic TCS model with local alignment force. Then we consider one-parameter family of well-prepared initial data to the kinetic TCS model in which the temperature tends to common constant value determined by initial datum, as singular parameter \begin{document}$ \varepsilon $\end{document} tends to zero. In a strong local alignment regime, the limit model is the hydrodynamic CS model in [8]. To verify this hydrodynamic limit rigorously, we adopt the technique introduced in [5] which combines the relative entropy method together with the 2-Wasserstein metric.

Stochastic functional Hamiltonian system with singular coefficients
Xing Huang and Wujun Lv
2020, 19(3): 1257-1273 doi: 10.3934/cpaa.2020060 +[Abstract](67) +[HTML](17) +[PDF](365.47KB)
Abstract:

By Zvonkin type transforms, the existence and uniqueness of the strong solutions for a class of stochastic functional Hamiltonian systems are obtained, where the drift contains a Hölder-Dini continuous perturbation. Moreover, under some reasonable conditions, the non-explosion of the solution is proved. In addition, as applications, the Harnack and shift Harnack inequalities are derived by method of coupling by change of measure. These inequalities are new even in the case without delay and the shift Harnack inequality is also new even in the non-degenerate functional SDEs with singular drifts.

Convergence of lacunary SU(1, 1)-valued trigonometric products
Jelena Rupčić
2020, 19(3): 1275-1289 doi: 10.3934/cpaa.2020062 +[Abstract](43) +[HTML](14) +[PDF](338.0KB)
Abstract:

This note attempts to study lacunary trigonometric products with values in the matrix group \begin{document}$ \rm{SU}(1,1) $\end{document} in analogy with lacunary trigonometric series. The central questions are the characterization of their convergence in an appropriately defined \begin{document}$ \rm{L}^p $\end{document}-metric and the characterization of their convergence almost everywhere. These can be interpreted as nonlinear analogues of the classical results by Zygmund and Kolmogorov.

Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process
Xiaobin Sun and Jianliang Zhai
2020, 19(3): 1291-1319 doi: 10.3934/cpaa.2020063 +[Abstract](58) +[HTML](22) +[PDF](436.13KB)
Abstract:

In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by cylindrical \begin{document}$ \alpha $\end{document}-stable process with \begin{document}$ \alpha\in (1, 2) $\end{document}. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.

Admissibility and polynomial dichotomies for evolution families
Davor Dragičević
2020, 19(3): 1321-1336 doi: 10.3934/cpaa.2020064 +[Abstract](84) +[HTML](15) +[PDF](327.87KB)
Abstract:

For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a (strong) nonuniform polynomial dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of a strong nonuniform polynomial dichotomy under sufficiently small linear perturbations.

Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation
Meng Qu, Ping Li and Liu Yang
2020, 19(3): 1337-1349 doi: 10.3934/cpaa.2020065 +[Abstract](59) +[HTML](13) +[PDF](346.34KB)
Abstract:

In this paper, we consider equations involving the fully nonlinear fractional order operator with homogeneous Dirichlet condition:

where \begin{document}$ \Omega $\end{document} is a domain(bounded or unbounded) in \begin{document}$ \mathbb R^n $\end{document} which is convex in \begin{document}$ x_1- $\end{document}direction. By using some ideas of maximum principle, we prove that the solution is strictly increasing in \begin{document}$ x_1- $\end{document}direction in the left half of \begin{document}$ \Omega $\end{document}. Symmetry of solution is also proved. Meanwhile we obtain a Liouville type theorem on the half space \begin{document}$ \mathbb R^n_+ $\end{document}.

A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities
Gui-Dong Li, Yong-Yong Li, Xiao-Qi Liu and Chun-Lei Tang
2020, 19(3): 1351-1365 doi: 10.3934/cpaa.2020066 +[Abstract](59) +[HTML](17) +[PDF](385.72KB)
Abstract:

In this paper, we investigate the following Choquard equation

where \begin{document}$ N\geq 3, \lambda>0, \alpha\in (0, N) $\end{document}, \begin{document}$ V $\end{document} is an asymptotically periodic potential, \begin{document}$ I_\alpha $\end{document} is the Riesz potential, the nonlinearity term \begin{document}$ F(s) = \int_{0}^{s}f(t)dt $\end{document} and \begin{document}$ f $\end{document} is only locally defined in a neighborhood of \begin{document}$ u = 0 $\end{document} and satisfies the suitable conditions. By using the Nehari manifold and the Moser iteration, we prove the existence of positive solutions for the equation with sufficiently large \begin{document}$ \lambda $\end{document}.

Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold
César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka and Thales Maier Souza
2020, 19(3): 1367-1386 doi: 10.3934/cpaa.2020067 +[Abstract](48) +[HTML](19) +[PDF](424.8KB)
Abstract:

In this paper we study the asymptotic dynamics for semilinear defocusing Schrödinger equation subject to a damping locally distributed on a n-dimentional compact Riemannian manifold \begin{document}$ M^n $\end{document} without boundary. The proofs are based on a result of unique continuation property, in the construction of a function \begin{document}$ f $\end{document} whose Hessian is positive definite and \begin{document}$ \Delta f = C_0 $\end{document} in some region contained in \begin{document}$ M $\end{document} and about the smoothing effect due to Aloui adapted to the present context.

Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary
Yuebin Hao
2020, 19(3): 1387-1397 doi: 10.3934/cpaa.2020068 +[Abstract](57) +[HTML](16) +[PDF](296.17KB)
Abstract:

The interior transmission eigenvalue problem plays a basic role in the study of inverse scattering problems for an inhomogeneous medium. In this paper, we consider the electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with conductive boundary. Our main focus is to understand the associated eigenvalue problem, more specifically to prove the transmission eigenvalues form a discrete set and show that they exist by employing a variety of variational techniques under various assumptions on the index of refraction.

Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian
Changchun Liu and Pingping Li
2020, 19(3): 1399-1419 doi: 10.3934/cpaa.2020070 +[Abstract](127) +[HTML](57) +[PDF](374.95KB)
Abstract:

This paper deals with a chemotaxis-haptotaxis model with the slow \begin{document}$ p $\end{document}-Laplacian diffusion in three-dimensional smooth bounded domains. It is proved that for any \begin{document}$ p>2 $\end{document}, the chemotaxis-haptotaxis model problem admits a global bounded weak solution if \begin{document}$ \frac{\chi}{\mu} $\end{document} is appropriately small.

Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data
Xinhua Zhao and Zilai Li
2020, 19(3): 1421-1448 doi: 10.3934/cpaa.2020052 +[Abstract](53) +[HTML](33) +[PDF](399.05KB)
Abstract:

In this paper, we study the asymptotic behavior of global spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations for the viscous heat conducting ideal polytropic gas flow with large initial data in \begin{document}$ H^1 $\end{document}, when the heat conductivity coefficient depends on the temperature, practically, \begin{document}$ \kappa(\theta) = \tilde{\kappa}_1+\tilde{\kappa}_2\theta^q $\end{document} where constants \begin{document}$ \tilde{\kappa}_1>0 $\end{document}, \begin{document}$ \tilde{\kappa}_2>0 $\end{document} and \begin{document}$ q>0 $\end{document} (as to the case of \begin{document}$ \tilde{\kappa}_1 = 0 $\end{document}, please refer to the Appendix). In addition, the exponential decay rate of solutions toward to the constant state as time tends to infinity for the initial boundary value problem in bounded domain is obtained. The mass density and temperature are proved to be pointwise bounded from below and above, independent of time although strong nonlinearity in heat diffusion. The analysis is based on some delicate uniform energy estimates independent of time.

Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation
Gang Li, Fen Gu and Feida Jiang
2020, 19(3): 1449-1462 doi: 10.3934/cpaa.2020071 +[Abstract](58) +[HTML](13) +[PDF](336.83KB)
Abstract:

In this paper, we investigate positive viscosity solutions of a third degree homogeneous parabolic equation \begin{document}$ u^{2}u_{t} = \Delta_{\infty}u $\end{document}. We prove a comparison principle, existence and uniqueness of continuous positive viscosity solutions.

Fredholm theory for an elliptic differential operator defined on $ \mathbb{R}^n $ and acting on generalized Sobolev spaces
Melvin Faierman
2020, 19(3): 1463-1483 doi: 10.3934/cpaa.2020074 +[Abstract](53) +[HTML](14) +[PDF](399.1KB)
Abstract:

We consider a spectral problem for an elliptic differential operator debined on \begin{document}$ \mathbb{R}^n $\end{document} and acting on the generalized Sobolev space \begin{document}$ W^{0, \chi}_p(\mathbb{R}^n) $\end{document} for \begin{document}$ 1 < p < \infty $\end{document}. We note that similar problems, but with \begin{document}$ \mathbb{R}^n $\end{document} replaced by either a bounded region in \begin{document}$ \mathbb{R}^n $\end{document} or by a closed manifold have been the subject of investigation by various authors. Then in this paper we establish, under the assumption of parameter-ellipticity, results pertaining to the existence and uniqueness of solutions of the spectral problem. Furthrermore, by utilizing the aforementioned results, we obain results pertaining to the spectral properties of the Banach space operator induced by the spectral problem.

On $ L^p $ estimates for a simplified Ericksen-Leslie system
Jinrui Huang, Wenjun Wang and Huanyao Wen
2020, 19(3): 1485-1507 doi: 10.3934/cpaa.2020075 +[Abstract](45) +[HTML](23) +[PDF](404.39KB)
Abstract:

In this paper, we study Cauchy problem for a simplified Ericksen-Leslie system in three dimensions. With the initial data of small perturbation near a steady state in \begin{document}$ H^2 $\end{document} norm, we obtain the global well-posedness of strong solutions as well as the \begin{document}$ L^p(p\in[1, 6]) $\end{document} estimates. In addition, sharper decay rates for the density and the momentum are obtained.

Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations
G. Deugoué, J. K. Djoko, A. C. Fouape and A. Ndongmo Ngana
2020, 19(3): 1509-1535 doi: 10.3934/cpaa.2020076 +[Abstract](111) +[HTML](15) +[PDF](434.54KB)
Abstract:

In this paper, we provide a detailed investigation of the problem of existence and uniqueness of strong solutions of a three-dimensional system of globally modified magnetohydrodynamic equations which describe the motion of turbulent particles of fluids in a magnetic field. We use the flattening property to establish the existence of the global \begin{document}$ V $\end{document}-attractor and a limit argument to obtain the existence of bounded entire weak solutions of the three-dimensional magnetohydrodynamic equations with time independent forcing.

On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $
O. A. Veliev
2020, 19(3): 1537-1562 doi: 10.3934/cpaa.2020077 +[Abstract](62) +[HTML](17) +[PDF](394.42KB)
Abstract:

In this paper we investigate the non-self-adjoint operator\begin{document}$ \ H $\end{document} generated in \begin{document}$ L_{2}(-\infty, \infty) $\end{document} by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which \begin{document}$ H $\end{document} has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator \begin{document}$ H $\end{document} by investigating the essential spectral singularities.

Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent
Ting Guo, Xianhua Tang, Qi Zhang and Zu Gao
2020, 19(3): 1563-1579 doi: 10.3934/cpaa.2020078 +[Abstract](51) +[HTML](18) +[PDF](383.96KB)
Abstract:

This paper is dedicated to studying the Choquard equation

where \begin{document}$ N\geq4 $\end{document}, \begin{document}$ \alpha\in(0, N) $\end{document}, \begin{document}$ V\in\mathcal{C}(\mathbb{R}^{N}, \mathbb{R}) $\end{document} is sign-changing and periodic, \begin{document}$ I_{\alpha} $\end{document} is the Riesz potential, \begin{document}$ p = \frac{N+\alpha}{N-2} $\end{document} and \begin{document}$ g\in\mathcal{C}(\mathbb{R}, \mathbb{R}) $\end{document}. The equation is strongly indefinite, i.e., the operator \begin{document}$ -\Delta+V $\end{document} has infinite-dimensional negative and positive spaces. Moreover, the exponent \begin{document}$ p = \frac{N+\alpha}{N-2} $\end{document} is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Under some mild assumptions on \begin{document}$ g $\end{document}, we obtain the existence of nontrivial solutions for this equation.

Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping
Tuan Anh Dao and Hironori Michihisa
2020, 19(3): 1581-1608 doi: 10.3934/cpaa.2020079 +[Abstract](66) +[HTML](19) +[PDF](479.49KB)
Abstract:

In this article, we study semi-linear \begin{document}$ \sigma $\end{document}-evolution equations with double damping including frictional and visco-elastic damping for any \begin{document}$ \sigma\ge 1 $\end{document}. We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the \begin{document}$ L^p-L^q $\end{document} framework, with \begin{document}$ 1\le p\le q\le \infty $\end{document}, to the corresponding linear equations. By assuming additional \begin{document}$ L^{m} $\end{document} regularity on the initial data, with \begin{document}$ m\in [1, 2) $\end{document}, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of global obtained solutions as well to semi-linear equations. Moreover, we also determine the so-called critical exponent when \begin{document}$ \sigma $\end{document} is integers.

Travelling corners for spatially discrete reaction-diffusion systems
H. J. Hupkes and L. Morelli
2020, 19(3): 1609-1667 doi: 10.3934/cpaa.2020058 +[Abstract](45) +[HTML](14) +[PDF](659.64KB)
Abstract:

We consider reaction-diffusion equations on the planar square lattice that admit spectrally stable planar travelling wave solutions. We show that these solutions can be continued into a branch of travelling corners. As an example, we consider the monochromatic and bichromatic Nagumo lattice differential equation and show that both systems exhibit interior and exterior corners.

Our result is valid in the setting where the group velocity is zero. In this case, the equations for the corner can be written as a difference equation posed on an appropriate Hilbert space. Using a non-standard global center manifold reduction, we recover a two-component difference equation that describes the behaviour of solutions that bifurcate off the planar travelling wave. The main technical complication is the lack of regularity caused by the spatial discreteness, which prevents the symmetry group from being factored out in a standard fashion.

Homogenization of a locally periodic time-dependent domain
Morteza Fotouhi and Mohsen Yousefnezhad
2020, 19(3): 1669-1695 doi: 10.3934/cpaa.2020061 +[Abstract](62) +[HTML](21) +[PDF](509.7KB)
Abstract:

We consider the homogenization of a Robin boundary value problem in a locally periodic perforated domain which is also time-dependent. We aim at justifying the homogenization limit, that we derive through asymptotic expansion technique. More exactly, we obtain the so-called corrector homogenization estimate that specifies the convergence rate. The major challenge is that the media is not cylindrical and changes over time. We also show the existence and uniqueness of solutions of the microscopic problem.

Global higher integrability of weak solutions of porous medium systems
Kristian Moring, Christoph Scheven, Sebastian Schwarzacher and Thomas Singer
2020, 19(3): 1697-1745 doi: 10.3934/cpaa.2020069 +[Abstract](52) +[HTML](23) +[PDF](599.74KB)
Abstract:

We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given by

where \begin{document}$ m>1 $\end{document}. More precisely, we prove that under suitable assumptions the spatial gradient \begin{document}$ D(|u|^{m-1}u) $\end{document} of any weak solution is integrable to a larger power than the natural power \begin{document}$ 2 $\end{document}. Our analysis includes both the case of the lateral boundary and the initial boundary.

Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity
Rita Ferreira and Elvira Zappale
2020, 19(3): 1747-1793 doi: 10.3934/cpaa.2020072 +[Abstract](47) +[HTML](18) +[PDF](1040.82KB)
Abstract:

Here, we address a dimension-reduction problem in the context of nonlinear elasticity where the applied external surface forces induce bending-torsion moments. The underlying body is a multi-structure in \begin{document}$\mathbb{R}^3$\end{document} consisting of a thin tube-shaped domain placed upon a thin plate-shaped domain. The problem involves two small parameters, the radius of the cross-section of the tube-shaped domain and the thickness of the plate-shaped domain. We characterize the different limit models, including the limit junction condition, in the membrane-string regime according to the ratio between these two parameters as they converge to zero.

The mathieu differential equation and generalizations to infinite fractafolds
Shiping Cao, Anthony Coniglio, Xueyan Niu, Richard H. Rand and Robert S. Strichartz
2020, 19(3): 1795-1845 doi: 10.3934/cpaa.2020073 +[Abstract](48) +[HTML](17) +[PDF](6797.02KB)
Abstract:

One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by converting the equation into an infinite system of linear equations for the Fourier coefficients. We study the asymptotic behavior of these Fourier coefficients and discuss the ways in which to numerically approximate solutions. We present both theoretical and numerical results pertaining to the stability of the Mathieu differential equation and the properties of solutions. Further, based on the idea of using Fourier series, we provide a method in which the Mathieu differential equation can be generalized to be defined on the infinite Sierpinski gasket. We discuss the stability of solutions to this fractal differential equation and describe further results concerning properties and behavior of these solutions.

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