An energy inequality for higher order linear parabolic operators and its applications

Abstract

A generalization of the classical energy inequality is obtained for evolution operators $(\partial /\partial t)I - H(t){\Lambda ^{2k}} - J(t)$, associated with higher order linear parabolic operators with variable coefficients. Here $H(t)$ and $J(t)$ are matrices of singular integral operators. The key to the result is an algebraic inequality involving matrices similar to the symbol of $H(t)$ having their eigenvalues contained in a fixed compact subset of the open left-half complex plane. Then a sharp estimate on the norms of certain imbedding maps is obtained. These estimates along with the energy inequality is applied to the Cauchy problem for higher order linear parabolic operators restricted to slabs in ${R^{n + 1}}$.