In this paper, the L p convergence rates of planar diffusion waves for multi-dimensional Euler equations with damping are considered. The analysis relies on a newly introduced frequency decomposition and Green function based energy method. It is a combination of the L p estimate on the low frequency component by using an approximate Green function and L 2 estimate on the high frequency component through the energy method. By noticing that the low frequency component in the approximate Green function has the algebraic decay which governs the large time behavior, while the high frequency component has the exponential decay but with singularity, their combination leads to a global algebraic decay estimate. To use the decay property only of the low frequency component in the approximate Green function avoids the singularity in the high frequency component so that it simplifies and improves the

The purpose of this paper is to introduce a new characterization of the characteristic functions for the study on the measure valued solution to the homogeneous Boltzmann equation so that it precisely captures the moment constraint in physics. This significantly improves the previous result by Cannone and Karch (2010) [2] in the sense that the new characterization gives a complete description of infinite energy solutions for the Maxwellian cross section. In addition, the global in time smoothing effect of the infinite energy solution is justified as for the finite energy solution except for a single Dirac mass initial datum.

We consider the Schrödinger equation with a nondispersive logarithmic nonlinearity and a repulsive harmonic potential. For a suitable range of the coefficients, there exist two positive stationary solutions, each one generating a continuous family of solitary waves. These solutions are Gaussian, and turn out to be orbitally unstable. We also discuss the notion of ground state in this setting: for any natural definition, the set of ground states is empty.

The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in varies settings when the initial energy is finite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev-Cercignani. The purpose of this paper is to rigorously justify this for the spatially homogeneous problem with Maxwellian molecule type cross section without angular cutoff.

In [9] Anderson’s conjecture was proven by comparing values of Bloch functions with the variation of the function. We extend that result on Bloch functions from two to arbitrary dimension and prove that ∫_{[0,x]}|∇b(ζ)|e^{b(ζ)}d|ζ|<∞.. In the second part of the paper, we show that the area or volume integral ∫_{B^d}|∇u(w)|p(w,θ)dA(w) for positive harmonic functions u is bounded by the value cu(0) for at least one θ. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.