The initial-value problem for the generalized Burger's equation (Q1374486)

The author investigates the Cauchy problem for the generalized Burgers' equation \[ \partial_tu+\partial_x(u^{k+1})=\partial_x^2u,\quad x,t,\in\mathbb{R},\;k=1,2,\dots. \] Assuming initial data to be in the homogeneous Sobolev space \(\dot {L}^{p,s}(\mathbb R)\) the author shows in two main theorems the well-posedness of the problem for the critical value \(s=1/p-1/k\), \(1\leq p<\infty\). Discussing cases depending on values of parameters, the author obtains local and global existence and uniqueness results in various Bochner spaces of continuous/integrable functions.