  The paper is a comprehensive study of the existence, uniqueness, blow up and
regularity properties of solutions of the Burgers equation with fractional
dissipation. We prove existence of the finite time blow up for the power of
Laplacian $\alpha < 1/2,$ and global existence as well as analyticity of
solution for $\alpha \geq 1/2.$ We also prove the existence of solutions with
very rough initial data $u_0 \in L^p,$ $1 < p < \infty.$ Many of the results
can be extended to a more general class of equations, including the surface
quasi-geostrophic equation.
