Communications on Pure and Applied Analysis
September 2017 , Volume 16 , Issue 5
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In this paper we consider the cubic Schrödinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness result in
In this paper, variational methods are used to establish some existence and multiplicity results and provide uniform estimates of extremal values for a class of elliptic equations of the form:
with Dirichlet boundary conditions, where
In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.
In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent
In this article we study the following quasilinear Schrödinger equation
The paper deals with the Cauchy problem of Navier-Stokes-Nernst-Planck-Poisson system (NSNPP). First of all, based on so-called Gevrey regularity estimates, which is motivated by the works of Foias and Temam [J. Funct. Anal., 87 (1989), 359-369], we prove that the solutions are analytic in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly obtain higher-order derivatives of solutions in Besov and Lebesgue spaces. Finally, we prove that there exists a positive constant
We consider the existence and concentration of solutions for the following Kirchhoff Type Equations
Under suitable conditions on the continuous functions
We study a damped wave equation with a nonlinear damping in the locally uniform spaces and prove well-posedness and existence of a locally compact attractor. An upper bound on the Kolmogorov's
In this paper, we consider positive solutions of a Cauchy problem for the following quasilinear degenerate parabolic equation with weighted nonlocal sources:
In this paper we consider the system involving fully nonlinear nonlocal operators:
A narrow region principle and a decay at infinity are established for carrying on the method of moving planes. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Furthermore non-existence of positive solutions to the system on a half space is derived.
We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let
In this paper we study the low Mach number limit of the full compressible Hall-magnetohydrodynamic (Hall-MHD) system in
We study the problem
In this paper, we prove the existence of multiple solutions to some intermediate local-nonlocal elliptic equation in the whole two dimensional space. The nonlinearities exhibit an exponential growth at infinity.
In this paper, we study the following nonlinear Schrödinger system in
We first consider the problem of characterizing the minimizers
Then we focus on the existence problem and assuming
that connects certain translates of
In this paper we compute the first and second general domain variation of the first eigenvalue of a fourth order Steklov problem. We study optimality conditions for the ball among domains of given measure and among domains of given perimeter. We show that in both cases the ball is a local minimizer among all domains of equal measure and perimeter.
We consider the 2D simplified Bardina turbulence model, with horizontal filtering, in an unbounded strip-like domain. We prove global existence and uniqueness of weak solutions in a suitable class of anisotropic weighted Sobolev spaces.
In this paper, we consider a microorganism flocculation model with time delay. In this model, there may exist a forward bifurcation/backward bifurcation. By constructing suitable positively invariant sets and using Lyapunov-LaSalle theorem, we study the global stability of the equilibria of the model under certain conditions. Furthermore, we also investigate the permanence of the model, and an explicit expression of the eventual lower bound of microorganism concentration is given.
In this paper we study two stochastic chemostat models, with and without wall growth, driven by a white noise. Specifically, we analyze the existence and uniqueness of solutions for these models, as well as the existence of the random attractor associated to the random dynamical system generated by the solution. The analysis will be carried out by means of the well-known Ornstein-Uhlenbeck process, that allows us to transform our stochastic chemostat models into random ones.
The structure-preserving finite difference schemes for the one dimensional Cahn-Hilliard equation with dynamic boundary conditions are studied. A dynamic boundary condition is a sort of transmission condition that includes the time derivative, namely, it is itself a time evolution equation. The Cahn-Hilliard equation with dynamic boundary conditions is well-treated from various viewpoints. The standard type consists of a dynamic boundary condition for the order parameter, and the Neumann boundary condition for the chemical potential. Recently, Goldstein-Miranville-Schimperna proposed a new type of dynamic boundary condition for the Cahn-Hilliard equation. In this article, numerical schemes for the problem with these two kinds of dynamic boundary conditions are introduced. In addition, a mathematical result on the existence of a solution for the scheme with an error estimate is also obtained for the former boundary condition.
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