  We consider higher order viscous Burgers' equations with generalized
nonlinearity and study the associated initial value problems for given data in
the $L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces
to derive multilinear estimates and use them in the contraction mapping
principle argument to prove local well-posedness for data with Sobolev
regularity below $L^2$. We also prove ill-posedness for this type of models and
show that the local well-posedness results are sharp in some particular cases
viz., when the orders of dissipation $p$, and nonlinearity $k+1$, satisfy a
relation $p=2k+1$.
