On the well-posedness of higher order viscous Burgers' equations

We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the L²$L^2$-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 L²$L^2$. 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$k+1$, satisfy a relation p=2k+1$p=2k+1$.