On the well-posedness for higher order parabolic equations with rough coefficients

In the first part we study the existence and uniqueness of solutions to the higher order parabolic Cauchy problems on the upper half space, given by ${\partial }_{t}u=(-1{)}^{m+1}{\text{div}}_{m}A(t,x){\mathrm{\nabla }}^{m}u$ and $L^p$ initial data space. The (complex) coefficients are only assumed to be elliptic and bounded measurable. Our approach follows the recent developments in the field for the case $m=1$.

In the second part we consider the $BMO$ space of initial data. We will see that the Carleson measure condition $$\underset{x\in {\mathbb{R}}^n}{sup}\underset{r>0}{sup}\frac{1}{|B(x,r)|}{\int }_{B(x,r)}{\int }_0^r|t^m{\mathrm{\nabla }}^{m}u(t^{2m},x){|}^2\frac{dxdt}{t}<\mathrm{\infty }$$

provides, up to polynomials, a well-posedness class for $BMO$. In particular, since the operator $L$ is arbitrary, this also leads to a new, broad Carleson measure characterization of $BMO$ in terms of solutions to the parabolic system.