scholarly journals On the asymptotic of solution to the Dirichlet problem for hyperbolic equations in cylinders with edges

2014 ◽  
pp. 1-15
Author(s):  
Luong Vu Trong ◽  
Hue Nguyen Thi
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Equations

On the Correctness of the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Hyperbolic Equations

1999 ◽  
Vol 6 (5) ◽  
pp. 447-470
Author(s):  
T. Kiguradze
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Equation ◽  
Fourth Order ◽  
Trivial Solution ◽  
Hyperbolic Equations ◽  
Homogeneous Problem ◽  
Linear Hyperbolic Equation ◽  
Order Linear ◽  
Linear Hyperbolic Equations

Abstract It is proved that the Dirichlet problem is correct in the characteristic rectangle 𝐷𝑎𝑏 = [0, 𝑎] × [0, 𝑏] for the linear hyperbolic equation with the summable in 𝐷𝑎𝑏 coefficients 𝑝0, 𝑝1, 𝑝2, 𝑝3 and 𝑞 if and only if the corresponding homogeneous problem has only the trivial solution. The effective and optimal in some sense restrictions on 𝑝0, 𝑝1, 𝑝2 and 𝑝3 guaranteeing the correctness of the Dirichlet problem are established.

Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for Uxy = 0 in rectangles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 107-110 ◽  
Author(s):  
Ali I. Abdul-Latif
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Partial Answer ◽  
Neumann Boundary Value Problems ◽  
Well Posed

SynopsisIt is well known that the Dirichlet problem for hyperbolic equations is a classical “not well posed” problem. Here we consider the Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for the hyperbolic equation uxy = 0 in all positions of the square and a class of rectangles. We also get a partial answer to the problem which deals with a ray that moves from any point on the boundary of a rectangle and is reflected on the boundary such that the angle between every ray and its reflection is π/2.

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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