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Communications on Pure & Applied Analysis

March 2018 , Volume 17 , Issue 2

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Beltrami equations in the plane and Sobolev regularity
Martí Prats
2018, 17(2): 319-332 doi: 10.3934/cpaa.2018018 +[Abstract](738) +[HTML](161) +[PDF](413.34KB)
Abstract:

New results regarding the Sobolev regularity of the principal solution of the linear Beltrami equation \begin{document}$\bar{\partial} f = μ \partial f + ν \overline{\partial f}$\end{document} for discontinuous Beltrami coefficients \begin{document}$μ$\end{document} and \begin{document}$ν$\end{document} are obtained, using Kato-Ponce commutators, obtaining that \begin{document}$\overline \partial f$\end{document} belongs to a Sobolev space with the same smoothness as the coefficients but some loss in the integrability parameter. A conjecture on the cases where the limitations of the method do not work is raised.

Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities
Dongsheng Kang and Liangshun Xu
2018, 17(2): 333-346 doi: 10.3934/cpaa.2018019 +[Abstract](613) +[HTML](161) +[PDF](363.84KB)
Abstract:

In this paper, the minimizers of a Rellich-Sobolev constant are firstly investigated. Secondly, a system of biharmonic equations is investigated, which involves multiple Rellich-type terms and strongly coupled critical Rellich-Sobolev terms. The existence of nontrivial solutions to the system is established by variational arguments.

Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating
Yinxia Wang and Hengjun Zhao
2018, 17(2): 347-374 doi: 10.3934/cpaa.2018020 +[Abstract](598) +[HTML](133) +[PDF](502.86KB)
Abstract:

In this paper, we consider the initial value problem for the compressible viscoelastic flows with self-gravitating in \begin{document}$\mathbb{R}^n(n≥ 3)$\end{document}. Global existence and decay rates of classical solutions are established. The corresponding linear equations becomes two similar equations by using Hodge decomposition and then the solutions operator is derived. The proof is mainly based on the decay properties of the solutions operator and energy method. The decay properties of the solutions operator may be derived from the pointwise estimate of the solution operator to two linear wave equations.

Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction
Wonlyul Ko and Inkyung Ahn
2018, 17(2): 375-389 doi: 10.3934/cpaa.2018021 +[Abstract](742) +[HTML](230) +[PDF](409.22KB)
Abstract:

We consider a predator-prey system with a ratio-dependent functional response when a prey population is infected. First, we examine the global attractor and persistence properties of the time-dependent system. The existence of nonconstant positive steady-states are studied under Neumann boundary conditions in terms of the diffusion effect; namely, pattern formations, arising from diffusion-driven instability, are investigated. A comparison principle for the parabolic problem and the Leray-Schauder index theory are employed for analysis.

The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system
Chun Shen, Wancheng Sheng and Meina Sun
2018, 17(2): 391-411 doi: 10.3934/cpaa.2018022 +[Abstract](731) +[HTML](121) +[PDF](446.34KB)
Abstract:

The Riemann problem for the scaled Leroux system is considered. It is proven rigorously that the Riemann solutions for the scaled Leroux system converge to the corresponding ones for a non-strictly hyperbolic system of conservation laws when the perturbation parameter tends to zero. In addition, some interesting phenomena are displayed in the limiting process, such as the formation of delta shock wave and a rarefaction (or shock) wave degenerates to be a contact discontinuity.

The regularity of some vector-valued variational inequalities with gradient constraints
Mohammad Safdari
2018, 17(2): 413-428 doi: 10.3934/cpaa.2018023 +[Abstract](423) +[HTML](111) +[PDF](383.45KB)
Abstract:

We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar case and in the vector-valued case.

Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model
Caterina Calgaro, Meriem Ezzoug and Ezzeddine Zahrouni
2018, 17(2): 429-448 doi: 10.3934/cpaa.2018024 +[Abstract](539) +[HTML](123) +[PDF](492.2KB)
Abstract:

In this paper, we construct a fully discrete numerical scheme for approximating a two-dimensional multiphasic incompressible fluid model, also called the Kazhikhov-Smagulov model. We use a first-order time discretization and a splitting in time to allow us the construction of an hybrid scheme which combines a Finite Volume and a Finite Element method. Consequently, at each time step, one only needs to solve two decoupled problems, the first one for the density and the second one for the velocity and pressure. We will prove the stability of the scheme and the convergence towards the global in time weak solution of the model.

On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation
Yohei Fujishima
2018, 17(2): 449-475 doi: 10.3934/cpaa.2018025 +[Abstract](564) +[HTML](148) +[PDF](520.77KB)
Abstract:

This paper concerns the blow-up problem for a semilinear heat equation

where \begin{document}$\partial_t=\partial/\partial t$\end{document}, \begin{document}$p>1$\end{document}, \begin{document}$N≥ 1$\end{document}, \begin{document}$Ω\subset {\bf R}^N$\end{document}, \begin{document}$u_0$\end{document} is a bounded continuous function in \begin{document}$\overline{Ω}$\end{document}. For the case \begin{document}$u_0(x)=λ\varphi(x)$\end{document} for some function \begin{document}$\varphi$\end{document} and a sufficiently large \begin{document}$λ>0$\end{document}, it is known that the solution blows up only near the maximum points of \begin{document}$\varphi$\end{document} under suitable assumptions. Furthermore, if \begin{document}$\varphi$\end{document} has several maximum points, then the blow-up set for (P) is characterized by \begin{document}$Δ\varphi$\end{document} at its maximum points. However, for initial data \begin{document}$u_0(x)=λ\varphi(x)$\end{document}, it seems difficult to obtain further information on the blow-up set such that effect of higher order derivatives of initial data. In this paper, we consider another type large initial data \begin{document}$u_0(x)=λ+\varphi(x)$\end{document} and study the relationship between the blow-up set for (P) and higher order derivatives of initial data.

Nonexistence of nonconstant positive steady states of a diffusive predator-prey model
Shanshan Chen
2018, 17(2): 477-485 doi: 10.3934/cpaa.2018026 +[Abstract](619) +[HTML](104) +[PDF](326.58KB)
Abstract:

In this paper, we investigate a diffusive predator-prey model with a general predator functional response. We show that there exist no nonconstant positive steady states when the interaction between the predator and prey is strong. This result implies that the global bifurcating branches of steady state solutions are bounded loops for a predator-prey model with Holling type Ⅲ functional response.

Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation
Yuanyuan Ren, Yongsheng Li and Wei Yan
2018, 17(2): 487-504 doi: 10.3934/cpaa.2018027 +[Abstract](623) +[HTML](110) +[PDF](440.27KB)
Abstract:

In this paper, we investigate the Cauchy problem for the fourth order nonlinear Schrödinger equation

Zheng (Adv. Differential Equations, 16(2011), 467-486.) has proved that the problem is locally well-posed in \begin{document}$H^{s}(\mathbb{R})$\end{document} with \begin{document}$-\frac{7}{4} <s≤q 0.$\end{document} In this paper, we aim at extending Zheng's work to a lower regularity index. We prove that the equation is locally well-posed in \begin{document}$H^{s}(\mathbb{R})$\end{document} when \begin{document}$s≥q -2$\end{document} and ill-posed when \begin{document}$s < -2$\end{document} in the sense that the solution map is discontinuous for \begin{document}$s <-2$\end{document}. The key ingredient used in this paper is Besov-type space introduced by Bejenaru and Tao (Journal of Functional Analysis, 233(2006), 228-259.).

The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements
Chiun-Chuan Chen, Seung-Yeal Ha and Xiongtao Zhang
2018, 17(2): 505-538 doi: 10.3934/cpaa.2018028 +[Abstract](803) +[HTML](120) +[PDF](524.91KB)
Abstract:

We present a coupled kinetic-macroscopic equation describing the dynamic behaviors of Cucker-Smale(in short C-S) ensemble undergoing velocity jumps and chemotactic movements. The proposed coupled model consists of a kinetic C-S equation supplemented with a turning operator for the kinetic density of C-S particles, and a reaction-diffusion equation for the chemotactic density. We study a global existence of strong solutions for the proposed model, when initial data is sufficiently regular, compactly supported in velocity and has finite mass and energy. The turning operator can screw up the velocity alignment, and result in a dispersed state. However, under suitable structural assumptions on the turning kernel and ansatz for the reaction term, the effects of the turning operator can vanish asymptotically due to the diffusion of chemical substances. In this situation, velocity alignment can emerge algebraically slow. We also present parabolic and hyperbolic Keller-Segel models with alignment dissipation in two scaling limits.

Subsonic irrotational inviscid flow around certain bodies with two protruding corners
Volker Elling
2018, 17(2): 539-555 doi: 10.3934/cpaa.2018029 +[Abstract](474) +[HTML](142) +[PDF](566.14KB)
Abstract:

We prove non-existence of nontrivial uniformly subsonic inviscid irrotational flows around several classes of solid bodies with two protruding corners, in particular vertical and angled flat plates; horizontal plates are the only case where solutions exists. This fills the gap between classical results on bodies with a single protruding corner on one hand and recent work on bodies with three or more protruding corners.

Thus even with zero viscosity and slip boundary conditions solids can generate vorticity, in the sense of having at least one rotational but no irrotational solutions. Our observation complements the commonly accepted explanation of vorticity generation based on Prandtl's theory of viscous boundary layers.

On the existence and computation of periodic travelling waves for a 2D water wave model
José Raúl Quintero and Juan Carlos Muñoz Grajales
2018, 17(2): 557-578 doi: 10.3934/cpaa.2018030 +[Abstract](651) +[HTML](150) +[PDF](716.7KB)
Abstract:

In this work we establish the existence of at least one weak periodic solution in the spatial directions of a nonlinear system of two coupled differential equations associated with a 2D Boussinesq model which describes the evolution of long water waves with small amplitude under the effect of surface tension. For wave speed \begin{document} $0 < |c| < 1$ \end{document}, the problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space \begin{document} $H^{1}_k(\mathbb{R})$ \end{document} (\begin{document} $k$ \end{document}-periodic functions \begin{document} $f∈ L_k^2(\mathbb{R})$ \end{document} such that \begin{document} $f' ∈ L_k^2(\mathbb{R})$ \end{document}). For wave speed \begin{document} $|c|>1$ \end{document}, the result is a direct consequence of the Lyapunov Center Theorem since the nonlinear system can be rewritten as a \begin{document} $4× 4$ \end{document} system with a special Hamiltonian structure. In the case \begin{document} $|c|>1$ \end{document}, we also compute numerical approximations of these travelling waves by using a Fourier spectral discretization of the corresponding 1D travelling wave equations and a Newton-type iteration.

On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption
Igor Pažanin and Marcone C. Pereira
2018, 17(2): 579-592 doi: 10.3934/cpaa.2018031 +[Abstract](708) +[HTML](119) +[PDF](471.21KB)
Abstract:

Motivated by the applications from chemical engineering, in this paper we present a rigorous derivation of the effective model describing the convection-diffusion-reaction process in a thin domain. The problem is described by a nonlinear elliptic problem with nonlinearity appearing both in the governing equation as well in the boundary condition. Using rigorous analysis in appropriate functional setting, we show that the starting singular problem posed in a two-dimensional region can be approximated with one which is regular, one-dimensional and captures the effects of all physical processes which are relevant for the original problem.

Global existence for a coupled wave system related to the Strauss conjecture
Jason Metcalfe and David Spencer
2018, 17(2): 593-604 doi: 10.3934/cpaa.2018032 +[Abstract](669) +[HTML](109) +[PDF](455.15KB)
Abstract:

A coupled system of semilinear wave equations is considered, and a small data global existence result related to the Strauss conjecture is proved. Previous results have shown that one of the powers may be reduced below the critical power for the Strauss conjecture provided the other power sufficiently exceeds such. The stability of such results under asymptotically flat perturbations of the space-time where an integrated local energy decay estimate is available is established.

Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity
Dengfeng Lü
2018, 17(2): 605-626 doi: 10.3934/cpaa.2018033 +[Abstract](732) +[HTML](186) +[PDF](499.84KB)
Abstract:

In the present paper the following Kirchhoff-Schrödinger-Poisson system is studied:

where \begin{document}$a>0,b≥q0 $\end{document} are constants and \begin{document}$μ>0 $\end{document} is a parameter, \begin{document}$f∈ C(\mathbb{R},\mathbb{R}) $\end{document}. Without assuming the Ambrosetti-Rabinowitz type condition and monotonicity condition on \begin{document}$f $\end{document}, we establish the existence of positive radial solutions for the above system by using variational methods combining a monotonicity approach with a delicate cut-off technique. We also study the asymptotic behavior of solutions with respect to the parameter \begin{document}$μ $\end{document}. In addition, we obtain the existence of multiple solutions for the nonhomogeneous case corresponding to the above problem. Our results improve and generalize some known results in the literature.

A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates
Felipe Hernandez
2018, 17(2): 627-646 doi: 10.3934/cpaa.2018034 +[Abstract](558) +[HTML](111) +[PDF](440.33KB)
Abstract:

A new decomposition for frequency-localized solutions to the Schrodinger equation is given which describes the evolution of the wavefunction using a weighted sum of Lipschitz tubes. As an application of this decomposition, we provide a new proof of the bilinear Strichartz estimate as well as the multilinear restriction theorem for the paraboloid.

Approximation of a nonlinear fractal energy functional on varying Hilbert spaces
Simone Creo, Maria Rosaria Lancia, Alejandro Vélez-Santiago and Paola Vernole
2018, 17(2): 647-669 doi: 10.3934/cpaa.2018035 +[Abstract](679) +[HTML](108) +[PDF](580.81KB)
Abstract:

We study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a two dimensional domain with Koch-type fractal boundary. We consider suitable approximating pre-fractal problems in the corresponding pre-fractal varying domains. After proving existence and uniqueness results via standard semigroup approach, we prove that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals adapted by Tölle to the nonlinear framework in varying Hilbert spaces.

Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature
Brittany Froese Hamfeldt
2018, 17(2): 671-707 doi: 10.3934/cpaa.2018036 +[Abstract](668) +[HTML](106) +[PDF](701.0KB)
Abstract:

We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampère type. These surfaces need not be continuous up to the boundary of the domain and the Dirichlet boundary condition must be interpreted in a weak sense. As a consequence, sub-solutions do not always lie below super-solutions, standard comparison principles fail, and existing convergence theorems break down. By relying on a geometric interpretation of weak solutions, we prove a relaxed comparison principle that applies only in the interior of the domain. We provide a general framework for proving existence and stability results for consistent, monotone finite difference approximations and modify the Barles-Souganidis convergence framework to show convergence in the interior of the domain. We describe a convergent scheme for the prescribed Gaussian curvature equation and present several challenging examples to validate these results.

On the nonlocal curvatures of surfaces with or without boundary
Roberto Paroni, Podio-Guidugli Paolo and Brian Seguin
2018, 17(2): 709-727 doi: 10.3934/cpaa.2018037 +[Abstract](560) +[HTML](135) +[PDF](514.83KB)
Abstract:

For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.

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