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This paper focuses on the study of existence and uniqueness of distributionaland classical solutions to the Cauchy Boltzmann problem for the soft potentialcase assuming $S^{n-1}$ integrability of the angular part of the collisionkernel (Grad cut-off assumption). For this purpose we revisit theKaniel--Shinbrot iteration technique to present an elementary proof ofexistence and uniqueness results that includes large data near a localMaxwellian regime with possibly infinite initial mass. We study the propagationof regularity using a recent estimate for the positive collision operator givenin [3], by E. Carneiro and the authors, that permits to study such propagationwithout additional conditions on the collision kernel. Finally, an$L^{p}$-stability result (with $1\leq p\leq\infty$) is presented assuming theaforementioned condition.

Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section