A priori estimates of strong solutions of semilinear parabolic equations

We study an initial boundary value problem for the semilinear parabolic equation
$$ \frac{\partial u}{\partial t} +\sum_{|\alpha|\le2b}a_\alpha(x,t)D^\alpha u =f(x,t,u,Du,\dots,D^{2b-1}u), $$
where the left-hand side is a linear uniformly parabolic operator of order $2b$. We prove sufficient growth conditions on the function $f$ with respect to the variables $u,Du,\dots,D^{2b-1}u$, such that the apriori estimate of the norm of the solution in the Sobolev space $W_p^{2b,1}$ is expressible in terms of the low-order norm in the Lebesgue space of integrable functions $L_{l,m}$.