scholarly journals Periodic-Dirichlet boundary value problem for semi-linear dissipative hyperbolic equations

1990 ◽  
Vol 148 (2) ◽  
pp. 371-377 ◽  
Author(s):  
Norimichi Hirano ◽  
Wan Se Kim
Keyword(s):  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Dirichlet Boundary Value Problem

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

2003 ◽  
Vol 10 (3) ◽  
pp. 495-502
Author(s):  
Alexander Domoshnitsky
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Dirichlet Boundary Value Problem ◽  
The Maximum Principle ◽  
Unbounded Solutions ◽  
All Solutions

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.

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