Existence of self-similar solutions for arbitrary amplitudes

Determine whether the spatially inhomogeneous cutoff Maxwell-molecule Boltzmann equation admits a global forward self-similar mild solution with the prescribed singular homogeneous initial trace for every amplitude of a fixed nontrivial profile.

Background

The main theorem constructs a nonzero global forward self-similar mild solution only when the amplitude of the prescribed singular homogeneous profile is sufficiently small. The proof requires smallness both to construct a global solution of the gain-only equation and to close the mass argument showing that the Kaniel–Shinbrot lower and upper limits coincide.

The authors explicitly state that it is unresolved whether the same existence conclusion holds for arbitrary amplitudes. They also explain that the kinetic dilation preserves the homogeneous trace and therefore cannot be used to reduce the amplitude, while the restrictions in the proof do not imply nonexistence for large amplitudes.