Communications on Pure & Applied Analysis
July 2020 , Volume 19 , Issue 7
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We have in a previous study introduced a novel elliptic operator
In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under
In this paper, we consider a viscoelastic rotating Euler-Bernoulli beam that has one end fixed to a rotated motor in a horizontal plane and to a tip mass at the other end. For a large class relaxation function
We provide a new proof of the Hardy–Moser–Trudinger inequality and the existence of its extremals which are established by Wang and Ye ("G. Wang, and D. Ye, A Hardy–Moser–Trudinger inequality, Adv. Math, 230 (2012) 294–230.") without using the blow-up analysis method. Our proof is based on the transformation of functions via the transplantation of Green functions. This method enables us to compute explicitly the concentrating level of the Hardy–Moser–Trudinger functional over the normalizing concentrating sequences which is crucial to prove the existence of extremals for the Hardy–Moser–Trudinger inequality. Some comments on the applications of this approach to some other Moser–Trudinger type inequalities are given.
In this paper, we give a new and more general sufficient condition for exponential stability of thermoelastic Bresse systems with heat flux given by Cattaneo's law acting in shear and longitudinal motion equations. This condition, which we also prove to be necessary in some special cases, is given by a relation between the constants of the system and generalizes the well-known equal wave speed condition.
In this paper, we consider the following equation with the higher-order fractional Laplacian
In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We illustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that
We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.
This paper is concerned with a class of free boundary models with "nonlocal diffusions'' and different free boundaries, which are natural extensions of free boundary problems of reaction diffusion systems with different free boundaries in [M.X.Wang and Y.Zhang, J. Differ. Equ., 264 (2018), 3527-3558] and references therein. These different free boundaries, which may intersect each other as time evolves, are used to describe the spreading front of the species. We prove that such kind of nonlocal diffusion problems has a unique global solution. Moreover, we investigate the long time behavior of global solution and criteria of spreading and vanishing for the classical Lotka-Volterra competition, prey-predator and mutualist models.
In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in
In this paper, we are concerned with the following two-component system of Schrödinger equations with Hartree nonlinearity:
The aim of this paper is to study symmetry and monotonicity for positive solutions to fractional equations. We first consider the following problems in bounded domains in the sense of distributions
In this paper, we consider the following fractional critical Schrödinger-Poisson system without perturbation terms
This paper is concerned with blow-up solution for the Cauchy problem of two-component Camassa-Holm equation with generalized weak dissipation. By Kato's theorem and monotonicity, we investigate the local well-posedness of Cauchy problem and establish the blow-up criteria and the blow-up rate. Moreover, the property of blow-up points set is characterized.
Chen and Zhang [
In this paper, we first establish the existence of trajectory attractors for the 3D smectic-A liquid crystal flow system and 3D smectic-A liquid crystal flow-
In this paper we study algebraic structures of the classes of the
In this paper, we investigate the chemotaxis-fluid system with singular sensitivity and logistic source in bounded convex domain with smooth boundary. We present the global existence of very weak solutions under appropriate regularity assumptions on the initial data. Then, we show that system possesses a global bounded classical solution. Finally, we present a unique globally bounded classical solution for a fluid-free system. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature, and partially results are new.
We consider the semilinear wave equation with time-dependent damping
We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the
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