Let $\mathcal{M}(x,D)$ and $\mathcal{L}(x,D)$ be linear partial differential operators of order $2m$ with complex-valued coefficients defined on a bounded region $\Omega $ in $R^n$ and suppose $\mathcal{M}$ is elliptic in $\Omega $. Necessary and sufficient conditions are given in order that solutions of $\mathcal{M}(x,D){{\partial u} / {\partial t}} - \mathcal{L}(x,D)u = 0$ in the cylinder $\Omega \times [0,\infty )$ which satisfy general boundary conditions on the wall of the cylinder satisfy inequalities of the form $\| {u(t)} \|_{2m} \leqq Ce^{ - at} \| {u(0)} \|_{2m} $ and $| {u(t)} |_{2m + \rho } \leqq Ce^{ - at} | {u(0)} |_{2m + \rho } $, $t > 0$, with positive constants a and C independent of u. $\| \cdot \|_{2m} $ and $| \cdot |_{2m + \rho } $ denote the customary norms in the spaces $H^{2m,2} (\Omega )$ and $C^{2m + \rho } (\bar \Omega )$, $0 < \rho < 1$, respectively.