Abstract:
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 $L_{l,m}$.
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