We consider the plasma confined in a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, and look at a certain class of equilibria, assuming axisymmetry in the problem. We prove a sharp criterion of spectral stability under these settings. Moreover, we provide several explicit families of stable/unstable equilibria using this criterion.

The Benjamin Ono equation with a slowly varying potential isand H denotes the Hilbert transform. The soliton profile is Q a,c (x) = cQ(c(x − a)), where Q(x) = 4 1+x 2 and a ∈ R, c > 0 are parameters. For initial condition u 0 (x) to (pBO) close in H 1/2 x to Q 0,1 (x), we show that the solution u(x, t) to (pBO) remains close in H 1/2 x to Q a(t),c(t) (x) and specify the (a, c) parameter dynamics on an O(h −1 ) time scale.

This paper is a continuation of the previous work [3] by the first two authors. We focus on 1 dimensional quadratic Klein-Gordon equations with a potential, under some assumptions that are less general than [3], but allow us to present some simplifications in the proof of global existence with decay for small solutions. In particular, we can propagate a stronger control on a basic L 2 -weighted type norm while providing some shorter and less technical proofs for some of the arguments.Contents 2.1. Linear scattering theory 2.2. Exceptional and generic potentials 2.3. Flat and distorted Fourier transform 2.4. The case of even and odd functions 3. Linear decay estimates 4. The quadratic spectral distribution 5. The main nonlinear decomposition and bootstrap 5.1. The equation on the profile 5.2. Normal form transformation and renormalized profile 5.3. The equation for the renormalized profile 5.4. Bootstrap and basic a priori bounds 6. The main regular interaction 7. The main singular interaction 8. The remainder terms 8.1. Weighted norm and bootstrap for g 8.2. Weighted estimates for remainder terms 8.3. The Sobolev norm 8.4. Fourier L ∞ estimates and asymptotics

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