In this paper, we study the asymptotic behavior of positive solutions of fractional Hardy-Henon equation with an isolated singularity at the origin. We give a classification of isolated singularities of positive solutions near x = 0. Further, we prove the asymptotic behavior of positive singular solutions. These results parallel those known for the Laplacian counterpart proved by Caffarelli, Gidas and Spruck (Caffarelli, Gidas and Spruck in Comm Pure Appl Math, 1981, 1989), but the technique is very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with a blow up (down) argument, the Kelvin transformation and uniqueness of solutions of related degenerate equations on semi spherical surface. We also investigate isolated singularities located at infinity of fractional Hardy-Henon equation.

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What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving explicitly this inverse potential problem.

Light bullets are spatially localized ultra-short optical pulses in more than one space dimensions. They contain only a few electromagnetic oscillations under their envelopes and propagate long distances without essentially changing shapes. Light bullets of femtosecond durations have been observed in recent numerical simulation of the full Maxwell systems. The sineGordon (SG) equation comes as an asymptotic reduction of the two level dissipationless MaxwellBloch system. We derive a new and complete nonlinear Schrdinger (NLS) equation in two space dimensions for the SG pulse envelopes so that it is globally well-posed and has all the relevant higher order terms to regularize the collapse of the standard critical NLS (CNLS). We perform a modulation analysis and found that SG pulse envelopes undergo focusingdefocusing cycles. Numerical results are in qualitative agreement with asymptotics and

The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or at some isolated interior points, it was unknown whether the entropy remains its boundedness. The results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy, locally or globally in time, if the vacuum occurs at the far field only and the density decays slowly enough at the far field. Precisely, we consider the Cauchy problem to the one-dimensional full compressible Navier-Stokes equations without heat conduction, and establish the local and global existence and uniqueness of entropy-bounded solutions, in the presence of vacuum at the far field only. It is also shown that, different from the case that with compactly supported initial density, the compressible Navier-Stokes equations, with slowly decaying initial density, can propagate the regularities in inhomogeneous Sobolev spaces.