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Communications on Pure & Applied Analysis

July 2019 , Volume 18 , Issue 4

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On the existence of solutions and causality for relativistic viscous conformal fluids
Marcelo M. Disconzi
2019, 18(4): 1567-1599 doi: 10.3934/cpaa.2019075 +[Abstract](839) +[HTML](164) +[PDF](464.53KB)
Abstract:

We consider a stress-energy tensor describing a pure radiation viscous fluid with conformal symmetry introduced in [3]. We show that the corresponding equations of motions are causal in Minkowski background and also when coupled to Einstein's equations, and solve the associated initial-value problem.

The interior gradient estimate for some nonlinear curvature equations
Liangjun Weng
2019, 18(4): 1601-1612 doi: 10.3934/cpaa.2019076 +[Abstract](927) +[HTML](161) +[PDF](296.92KB)
Abstract:

In this paper, we obtain the interior gradient estimate of some nonlinear equations which arise naturally from prescribed curvature problem of graphs in hyperbolic space. The method depends on the maximum principle.

Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems
Liang Kong, Tung Nguyen and Wenxian Shen
2019, 18(4): 1613-1636 doi: 10.3934/cpaa.2019077 +[Abstract](654) +[HTML](149) +[PDF](408.1KB)
Abstract:

This paper is concerned with the asymptotic dynamics of two species competition systems of the form

where \begin{document}$ (\mathcal{A}u)(t,x) = u_{xx}(t,x) $\end{document}, or \begin{document}$ (\mathcal{A}u)(t,x) = \int_{ {\mathbb{R}} }\kappa(y-x)u(t,y)dy-u(t,x) $\end{document} (\begin{document}$ \kappa(\cdot) $\end{document} is a smooth non-negative convolution kernel supported on an interval centered at the origin), \begin{document}$ a_i(t+T,x) = a_i(t,x) $\end{document}, \begin{document}$ b_i(t+T,x) = b_i(t,x) $\end{document}, \begin{document}$ c_i(t+T,x) = c_i(t,x) $\end{document}, and \begin{document}$ a_i $\end{document}, \begin{document}$ b_i $\end{document}, and \begin{document}$ c_i $\end{document} (\begin{document}$ i = 1,2 $\end{document}) are spatially homogeneous when \begin{document}$ |x|\gg 1 $\end{document}, that is, \begin{document}$ a_i(t,x) = a_i^0(t) $\end{document}, \begin{document}$ b_i(t,x) = b_i^0(t) $\end{document}, \begin{document}$ c_i(t,x) = c_i^0(t) $\end{document} for some \begin{document}$ a_i^0(t) $\end{document}, \begin{document}$ b_i^0(t) $\end{document}, \begin{document}$ c_i^0(t) $\end{document}, and \begin{document}$ |x|\gg 1 $\end{document}. Such a system can be viewed as a time periodic competition system subject to certain localized spatial variations. In particular, we study the effects of localized spatial variations on the uniform persistence and spreading speeds of the system. Among others, it is proved that any localized spatial variation does not affect the uniform persistence of the system, does not slow down the spreading speeds of the system, and under some linear determinant condition, does not speed up the spreading speeds. We also study a relevant problem, that is, the continuity of the spreading speeds of time periodic two species competition systems with respect to time periodic perturbations, and prove that the spread speeds of such systems are lower semicontinuous with respect to time periodic perturbations.

New general decay results in a finite-memory bresse system
Salim A. Messaoudi and Jamilu Hashim Hassan
2019, 18(4): 1637-1662 doi: 10.3934/cpaa.2019078 +[Abstract](730) +[HTML](163) +[PDF](392.82KB)
Abstract:

This paper is concerned with the following memory-type Bresse system

with homogeneous Dirichlet-Neumann-Neumann boundary conditions, where \begin{document}$ (x,t) \in (0,L) \times (0, \infty) $\end{document}, \begin{document}$ g $\end{document} is a positive strictly increasing function satisfying, for some nonnegative functions \begin{document}$ \xi $\end{document} and \begin{document}$ H $\end{document},

Under appropriate conditions on \begin{document}$ \xi $\end{document} and \begin{document}$ H $\end{document}, we prove, in cases of equal and non-equal speeds of wave propagation, some new decay results that generalize and improve the recent results in the literature.

Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $
Lun Guo, Wentao Huang and Huifang Jia
2019, 18(4): 1663-1693 doi: 10.3934/cpaa.2019079 +[Abstract](1081) +[HTML](158) +[PDF](475.74KB)
Abstract:

We consider the existence of positive solutions for the following fractional Schrödinger-Poisson system

where \begin{document}$ s \in (\frac{3}{4}, 1) $\end{document}, \begin{document}$ \varepsilon $\end{document} is a small and positive parameter, \begin{document}$ V $\end{document} and \begin{document}$ K $\end{document} are nonnegative potential functions. \begin{document}$ 2_{s}^{*} $\end{document} is the critical exponent with respect to fractional Sobolev embedding theorem. Under some suitable conditions on the nonlinearity \begin{document}$ f $\end{document} and potential functions \begin{document}$ V $\end{document} and \begin{document}$ K $\end{document}, we prove that for \begin{document}$ \varepsilon $\end{document} small, the system has a positive ground state solution concentrating around a concrete set related to \begin{document}$ V $\end{document} and \begin{document}$ K $\end{document}. This result generalizes the result for fractional Schrödinger-Poisson system with subcritical exponent by Yu et al. [39] to critical exponent. Moreover, when \begin{document}$ V $\end{document} attains its minimum and \begin{document}$ K $\end{document} attains its maximum, we also obtain multiple solutions by Ljusternik-Schnirelmann theory.

Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium
Pablo Amster, Mariel Paula Kuna and Gonzalo Robledo
2019, 18(4): 1695-1709 doi: 10.3934/cpaa.2019080 +[Abstract](799) +[HTML](128) +[PDF](363.52KB)
Abstract:

Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of \begin{document}$ T $\end{document}-periodic solutions lying inside a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^{N} $\end{document} is, generically, at least \begin{document}$ |\chi \pm 1|+1 $\end{document}, where \begin{document}$ \chi $\end{document} denotes the Euler characteristic of \begin{document}$ \Omega $\end{document}. Moreover, some connections between the associated fixed point operator and the Poincaré operator are explored.

Scattering results for Dirac Hartree-type equations with small initial data
Changhun Yang
2019, 18(4): 1711-1734 doi: 10.3934/cpaa.2019081 +[Abstract](839) +[HTML](139) +[PDF](412.87KB)
Abstract:

We consider the Dirac equations with cubic Hartree-type nonlinearity which are derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data global well-posedness and scattering results in the full scaling subcritical regularity regime. The strategy of the proof relies on the localized Strichartz estimates and bilinear estimates in \begin{document}$ V^2 $\end{document} spaces, together with the use of the null structure that the nonlinear term exhibits. This result is shown to be almost optimal in the sense that the iteration method based on Duhamel's formula fails over the supercritical range.

Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition
Kunio Hidano and Dongbing Zha
2019, 18(4): 1735-1767 doi: 10.3934/cpaa.2019082 +[Abstract](578) +[HTML](134) +[PDF](467.74KB)
Abstract:

We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scalar unknowns is also refined.

Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces
Koya Nishimura
2019, 18(4): 1769-1782 doi: 10.3934/cpaa.2019083 +[Abstract](943) +[HTML](134) +[PDF](348.94KB)
Abstract:

We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the \begin{document}$ L^r_v L^\infty_x $\end{document} spaces where \begin{document}$ r \in (1,\infty] $\end{document} by using the excess conservation laws and entropy inequality introduced in [5].

Existence and regularity of solutions for an evolution model of perfectly plastic plates
P. Gidoni, G. B. Maggiani and R. Scala
2019, 18(4): 1783-1826 doi: 10.3934/cpaa.2019084 +[Abstract](764) +[HTML](132) +[PDF](479.94KB)
Abstract:

We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived in [19] from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable.

Second order non-autonomous lattice systems and their uniform attractors
Ahmed Y. Abdallah and Rania T. Wannan
2019, 18(4): 1827-1846 doi: 10.3934/cpaa.2019085 +[Abstract](586) +[HTML](134) +[PDF](388.29KB)
Abstract:

The existence of the uniform global attractor for a second order non-autonomous lattice dynamical system (LDS) with almost periodic symbols has been carefully studied. Considering the nonlinear operators \begin{document}$ \left( f_{1i}\left( \overset{.}{u}_{j}\mid j\in I_{iq_{1}}\right) \right) _{i\in \mathbb{Z} ^{n}} $\end{document} and \begin{document}$ \left( f_{2i}\left( u_{j}\mid j\in I_{iq_{2}}\right) \right) _{i\in \mathbb{Z} ^{n}} $\end{document} of this LDS, up to our knowledge it is the first time to investigate the existence of uniform global attractors for such second order LDSs. In fact there are some previous studies for first order autonomous and non-autonomous LDSs with similar nonlinear parts, cf. [3, 24]. Moreover, the LDS under consideration covers a wide range of second order LDSs. In fact, for specific choices of the nonlinear functions \begin{document}$ f_{1i} $\end{document} and \begin{document}$ f_{2i} $\end{document} we get the autonomous and non-autonomous second order systems given by [1, 25, 26].

Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $
Silvia Frassu
2019, 18(4): 1847-1867 doi: 10.3934/cpaa.2019086 +[Abstract](837) +[HTML](131) +[PDF](392.09KB)
Abstract:

In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational methods and Morse theory. We present some results about regularity of solutions such as \begin{document}$ L^{\infty} $\end{document} bound and Hopf's lemma, for the latter we first consider a non negative nonlinearity and then a strictly negative one. Moreover, we prove that, for the corresponding functional, local minimizers with respect to a \begin{document}$ C^0 $\end{document}-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the \begin{document}$ X(\Omega) $\end{document}-topology.

Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms
M. L. Santos and Mirelson M. Freitas
2019, 18(4): 1869-1890 doi: 10.3934/cpaa.2019087 +[Abstract](1268) +[HTML](148) +[PDF](380.3KB)
Abstract:

This paper is concerned with long-time dynamics of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. We also establish the existence of a global attractor, and we study the fractal dimension and exponential attractors.

Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients
Wenming Hu and Huicheng Yin
2019, 18(4): 1891-1919 doi: 10.3934/cpaa.2019088 +[Abstract](596) +[HTML](158) +[PDF](453.0KB)
Abstract:

In this paper, we establish the local well-posedness of low regularity solutions to the general second order strictly hyperbolic equation of divergence form \begin{document}$ \partial _t(a_0 \partial _t u)+ \mathop \sum \limits_{j = 1}^n [ \partial _t(a_j \partial _j u)+ \partial _j(a_j \partial _t u)] -\mathop \sum \limits_{j,k = 1}^n \partial _j(a_{jk} \partial _k u) $\end{document} \begin{document}$ +b_0 \partial _t u+ \partial _t(c_0u)+ \mathop \sum \limits_{j = 1}^n [b_j \partial _ju+ \partial _j(c_ju)] +du = f $\end{document} in domain \begin{document}$ \Omega = (0, T_0)\times \mathbb R ^n $\end{document}, where the coefficients \begin{document}$ a_0, a_j, a_{jk}\in L^\infty( \Omega )\cap LL(\bar\Omega) $\end{document} \begin{document}$ (1\le j, k\le n) $\end{document}, \begin{document}$ b_0, c_0, b_j, c_j\in L^\infty( \Omega )\cap C^ \alpha (\bar\Omega) $\end{document} \begin{document}$ (1\le j\le n) $\end{document} for \begin{document}$ \alpha \in(\frac{1}{2},1) $\end{document}, \begin{document}$ d\in L^\infty(\Omega) $\end{document}, \begin{document}$ (u(0,x), Xu(0,x))\in (H^{1- \theta +\beta \log}, H^{- \theta +\beta \log}) $\end{document} with \begin{document}$ \theta\in (1- \alpha , \alpha ) $\end{document}, \begin{document}$ \beta\in\Bbb R $\end{document}, and \begin{document}$ Xu = a_0 \partial _tu+ \mathop \sum \limits_{j = 1}^n a_j \partial _ju $\end{document}. Compared with previous references, except a little more general initial data in the space \begin{document}$ (H^{1- \theta +\beta \log}, H^{- \theta +\beta \log}) $\end{document} (only \begin{document}$ \beta = 0 $\end{document} is considered as before), we improve both the lifespan of \begin{document}$ u $\end{document} up to the precise number \begin{document}$ T^* $\end{document} and the range of \begin{document}$ \theta $\end{document} to the left endpoint \begin{document}$ 1- \alpha $\end{document} under some suitable conditions.

Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy
Jaeyoung Byeon and Sang-hyuck Moon
2019, 18(4): 1921-1965 doi: 10.3934/cpaa.2019089 +[Abstract](628) +[HTML](143) +[PDF](523.48KB)
Abstract:

We consider the following singularly perturbed problem

Existence of a solution with a spike layer near a min-max critical point of the mean curvature on the boundary \begin{document}$ \partial \Omega $\end{document} is well known when a nondegeneracy for a limiting problem holds. In this paper, we use a variational method for the construction of such a solution which does not depend on the nondengeneracy for the limiting problem. By a purely variational approach, we construct the solution for an optimal class of nonlinearities \begin{document}$ f $\end{document} satisfying the Berestycki-Lions conditions.

$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data
Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto and Yuta Wakasugi
2019, 18(4): 1967-2008 doi: 10.3934/cpaa.2019090 +[Abstract](923) +[HTML](143) +[PDF](544.49KB)
Abstract:

We study the Cauchy problem of the damped wave equation

and give sharp \begin{document}$ L^p $\end{document}-\begin{document}$ L^q $\end{document} estimates of the solution for \begin{document}$ 1\le q \le p < \infty\ (p\neq 1) $\end{document} with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in \begin{document}$ (H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r) $\end{document} with \begin{document}$ r \in (1,2] $\end{document}, \begin{document}$ s\ge 0 $\end{document}, and \begin{document}$ \beta = (n-1)|\frac{1}{2}-\frac{1}{r}| $\end{document}, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power \begin{document}$ 1+\frac{2r}{n} $\end{document}, while it is known that the critical power \begin{document}$ 1+\frac{2}{n} $\end{document} belongs to the blow-up region when \begin{document}$ r = 1 $\end{document}. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument.

Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $
Claudianor O. Alves, Vincenzo Ambrosio and Teresa Isernia
2019, 18(4): 2009-2045 doi: 10.3934/cpaa.2019091 +[Abstract](871) +[HTML](210) +[PDF](511.84KB)
Abstract:

In this work we consider the following class of fractional \begin{document}$p \& q$\end{document} Laplacian problems

where \begin{document}$ \varepsilon >0 $\end{document} is a parameter, \begin{document}$ s\in (0, 1) $\end{document}, \begin{document}$ 1< p<q<\frac{N}{s} $\end{document}, \begin{document}$ (-\Delta)^{s}_{t} $\end{document}, with \begin{document}$ t\in \{p,q\} $\end{document}, is the fractional \begin{document}$ t $\end{document}-Laplacian operator, \begin{document}$ V: \mathbb{R} ^{N}\rightarrow \mathbb{R} $\end{document} is a continuous potential and \begin{document}$ f: \mathbb{R} \rightarrow \mathbb{R} $\end{document} is a \begin{document}$ \mathcal{C} ^{1} $\end{document}-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that \begin{document}$ \varepsilon $\end{document} is sufficiently small.

A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production
Youshan Tao and Michael Winkler
2019, 18(4): 2047-2067 doi: 10.3934/cpaa.2019092 +[Abstract](922) +[HTML](146) +[PDF](386.29KB)
Abstract:

We consider the chemotaxis-haptotaxis system

in a bounded convex domain \begin{document}$ \Omega\subset \mathbb{R} ^n $\end{document} with smooth boundary, where \begin{document}$ \chi, \xi, \mu $\end{document} and \begin{document}$ \eta $\end{document} are positive constants, and where \begin{document}$ f \in C^1([0,\infty)) $\end{document} is a given function fulfilling \begin{document}$ f(0) \ge 0 $\end{document} and

with some \begin{document}$ K_f >0 $\end{document} and \begin{document}$ \alpha>0 $\end{document}.

It is asserted that whenever

the Neumann boundary problem with suitably regular initial data possesses a unique global and bounded classical solution.

Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system
Zhaohai Ma, Rong Yuan, Yang Wang and Xin Wu
2019, 18(4): 2069-2092 doi: 10.3934/cpaa.2019093 +[Abstract](635) +[HTML](143) +[PDF](519.32KB)
Abstract:

In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in \begin{document}$ n $\end{document}–dimensional space. More precisely, we prove that all planar traveling waves with speed \begin{document}$ c>c^* $\end{document} are exponentially stable in \begin{document}$ L^{\infty}(\mathbb{R}^n ) $\end{document} in the form of \begin{document}$ t^{-\frac{n}{2\alpha }}{\rm{e}}^{-\varepsilon_{\tau} \sigma t} $\end{document} for some constants \begin{document}$ \sigma >0 $\end{document} and \begin{document}$ \varepsilon_{\tau} \in (0,1) $\end{document}, where \begin{document}$ \varepsilon_{\tau} = \varepsilon(\tau) $\end{document} is a decreasing function refer to the time delay \begin{document}$ \tau>0 $\end{document}. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed \begin{document}$ c = c^* $\end{document}, we show that they are algebraically stable in the form of \begin{document}$ t^{-\frac{n}{2\alpha}} $\end{document}. The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.

On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction
Jaime Angulo Pava and César A. Hernández Melo
2019, 18(4): 2093-2116 doi: 10.3934/cpaa.2019094 +[Abstract](926) +[HTML](138) +[PDF](468.08KB)
Abstract:

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schródinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the case of an attractive-attractive case and an focusing interaction we give an approach based in the extension theory of symmetric operators for determining the Morse index.

On a formula for sets of constant width in 2d
Bernd Kawohl and Guido Sweers
2019, 18(4): 2117-2131 doi: 10.3934/cpaa.2019095 +[Abstract](976) +[HTML](170) +[PDF](637.45KB)
Abstract:

A formula for smooth orbiforms originating from Euler can be adjusted to describe all sets of constant width in 2d. Moreover, the formula allows short proofs of some laborious approximation results for sets of constant width.

Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids
Zhenhua Guo, Wenchao Dong and Jinjing Liu
2019, 18(4): 2133-2161 doi: 10.3934/cpaa.2019096 +[Abstract](770) +[HTML](135) +[PDF](399.92KB)
Abstract:

In this paper, we study the large time behaviors of boundary layer solution of the inflow problem on the half space for a class of isentropic compressible non-Newtonian fluids. We establish the existence and uniqueness of the boundary layer solution to the non-Newtonian fluids. Especially, it is shown that such a boundary layer solution have a maximal interval of existence. Then we prove that if the strength of the boundary layer solution and the initial perturbation are suitably small, the unique global solution in time to the non-Newtonian fluids exists and asymptotically tends toward the boundary layer solution. The proof is given by the elementary energy method.

Word combinatorics for stochastic differential equations: Splitting integrators
A. Alamo and J. M. Sanz-Serna
2019, 18(4): 2163-2195 doi: 10.3934/cpaa.2019097 +[Abstract](591) +[HTML](151) +[PDF](465.98KB)
Abstract:

We present an analysis based on word combinatorics of splitting integrators for Ito or Stratonovich systems of stochastic differential equations. In particular we present a technique to write down systematically the expansion of the local error; this makes it possible to easily formulate the conditions that guarantee that a given integrator achieves a prescribed strong or weak order. This approach bypasses the need to use the Baker-Campbell-Hausdorff (BCH) formula and shows the existence of an order barrier of two for the attainable weak order. The paper also provides a succinct introduction to the combinatorics of words.

Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems
Nikos Katzourakis
2019, 18(4): 2197-2198 doi: 10.3934/cpaa.2019098 +[Abstract](853) +[HTML](145) +[PDF](221.02KB)
Abstract:

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