**Alexander Domoshnitsky**

## About Asymptotic and Oscillation
Properties of the Dirichlet

Problem for Delay Partial Differential Equations

**abstract:**

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.