Communications on Pure and Applied Analysis
June 2020 , Volume 19 , Issue 6
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In this paper, we mainly investigate the classification of singular sets of solutions to elliptic equations. Firstly, we define the
In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity:
By using the property known as Federer-Fleming conjecture (cf. [
holds for all
We will use such a theorem to provide a simple new proof of a well-known property of Sobolev functions.
This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time.
In this article we study the large-time behavior of perturbative classical solutions to the Fokker-Planck-Boltzmann equation for non-cutoff hard potentials. When the initial data is a small pertubation of an equilibrium state, global existence and temporal decay estimates of classical solutions are established.
In this paper, we study the backward compactness of random attractors, which describes the compactness of the union
We investigate the existence of radial solutions for a class of Hénon type systems with nonlinearities reaching the critical growth and interacting with the spectrum of the operator with the possibility of double resonance. The proof is made using variational methods, combining Brézis and Nirenberg arguments with Ni compactness result and Rabinowitz linking theorem.
In this paper we study the robustness of the stability in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel [
As counterparts, e show first in another example that it is possible to stabilize an unstable system by using a perturbation with a large period and a small mean value, and finally we give an example where we stabilize an unstable linear ODE with a small perturbation in infinite dimensions by using some ideas developed in Rodrigues & Solà-Morales [
In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya [
In this paper, we consider Ricci curvature of conformal deformation on compact 2-manifolds. And we prove that, by the conformal deformation, the resulting manifold is an Einstein manifold.
In this paper, we study the formation of singularities in a finite time for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first order spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by using a Lyapunov functional approach. It is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.
This paper studies some population dynamics of a diffusive Lotka-Volterra competition advection model under no-flux boundary condition. We establish the main results that determine the stability of semi-trivial steady states.
In 1976 Nazarenko proposed studying the delay differential equation
under the assumptions that
In this work, we apply the 'disconjugacy theory' and Elias's spectrum theory to study the disconjugacy
We prove a low regularity local well-posedness result for the Maxwell-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces
Spatial heterogeneity and movement of population play an important role in disease spread and control in reality. This paper concerns with a spatial Susceptible-Infected-Susceptible epidemic model with spontaneous infection and logistic source, aiming to investigate the asymptotic profiles of the endemic steady state (whenever it exists) for large and small diffusion rates. We firstly establish uniform upper bound of solutions. By studying the local and global stability of the unique constant endemic equilibrium when spatial environment is homogeneous, we apply the well-known Leray-Schuauder degree index formula to confirm the existence of endemic steady state. Our theoretical results suggest that spontaneous infection and varying total population strongly enhance the persistence of disease spread in the sense that disease component of the endemic steady state will not approach zero whenever the large and small diffusion rates of the susceptible or infected population is used. This gives new insights and aspects for infectious disease modeling and control.
We study the optimal order of natural analogues of Sobolev embedding properties within the framework of compact matrix quantum groups of Kac type. One of the main results of this paper is that the optimal order is given by the polynomial growth order of dual discrete quantum groups in a broad class, which covers all connected compact Lie groups, duals of polynomially growing discrete groups,
Chen and Zhang [
We are concerned with the shock reflection in gas dynamics for the pressure gradient system. Experimental and computational analysis has shown that two patterns of regular reflection may occur: supersonic and subsonic reflection. In this paper we establish the global existence of solutions for both configurations. The ideas and techniques developed here will be useful for the two-dimensional Riemann problems for hyperbolic conservation laws.
Consider the quasilinear Schrödinger equation
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