In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet boundary value problem for the heat equation: the maximum principle is not valid, unbounded solutions appear while all solutions of the classical Dirichlet problem tend to zero at infinity, for ``narrow enough zones'' all solutions oscillate instead of being positive. We establish that the Dirichlet problem for the wave equation with delay can possess unbounded solutions. We estimate zones of positivity of solutions for hyperbolic equations.