TY - JOUR

T1 - About Asymptotic and Oscillation Properties of the Dirichlet Problem for Delay Partial Differential Equations

AU - Domoshnitsky, Alexander

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - Delay partial differential equations

KW - oscillation

KW - unboundedness of solutions

KW - zone of positivity

UR - http://www.scopus.com/inward/record.url?scp=26844462242&partnerID=8YFLogxK

U2 - 10.1515/GMJ.2003.495

DO - 10.1515/GMJ.2003.495

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AN - SCOPUS:26844462242

SN - 1072-947X

VL - 10

SP - 495

EP - 502

JO - Georgian Mathematical Journal

JF - Georgian Mathematical Journal

IS - 3

ER -