It is conjectured that for the Maxwell-Klein-Gordon equations having data with non-vanishing charge and arbitrary large size, the global solutions disperse as linear waves and enjoy the so-called peeling properties for pointwise estimates. We give a gauge independent proof of the conjecture.
The commutator of the Riesz transform (Hilbert transform in dimension 1) and a symbol $b$ is bounded on $L^2(\mathbb R^n)$ if and only if $b$ is in the BMO space BMO$(\mathbb R^n)$ (Coifman--Rochberg--Weiss). It is natural to ask whether it holds for commutator of Riesz transform on Heisenberg groups.
For a balanced bistable reaction-diffusion equation, an axisymmetric traveling front has been well known. We prove that an axially asymmetric traveling front with any positive speed does exist in a balanced bistable reaction-diffusion equation.