The large amplitude solution of the Boltzmann equation with soft potential

In this paper, we deal with (angular cut-off) Boltzmann equation with soft potential ($-3<\gamma <0$). In particular, we construct a unique global solution in $L_{x,v}^{\mathrm{\infty }}$ which converges to global equilibrium asymptotically provided that initial data has a large amplitude but with sufficiently small relative entropy. Because frequency multiplier is not uniformly positive anymore, unlike hard potential case, time-involved velocity weight will be used to derive sub-exponential decay of the solution. Motivated by recent development of $L^2-L^{\mathrm{\infty }}$ approach also, we introduce some modified estimates of quadratic nonlinear terms. Linearized collision kernel will be treated in a subtle manner to control singularity of soft potential kernel.