On the role of the Besov spaces for the solutions of the generalized burgers equation in homogeneous Sobolev spaces

Abstract

The generalized Burgers equation ∂ t u− ∂ xx u+ ∂ x u k+1 =0, with initial data u 0 in homogeneous Sobolev spaces are investigated. The starting point of this work is the construction of solutions in L ∞([0,∞), B ̇ ∞ −1/k,∞) for small size initial data in B ̇ ∞ −1/k,∞ . If in addition, the initial data belongs to L ̇ p,s then the obtained solution is actually in L ∞([0,∞), L ̇ p,s) .