On the Dirichlet problem for fourth-order linear hyperbolic equations

2002 ◽  
Vol 49 (2) ◽  
pp. 197-219 ◽  
Author(s):  
T. Kiguradze ◽  
V. Lakshmikantham
Keyword(s):  
Dirichlet Problem ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Order Linear ◽  
Linear 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.

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

1999 ◽  
Vol 6 (6) ◽  
pp. 537-552
Author(s):  
T. Kiguradze
Keyword(s):  
Dirichlet Problem ◽  
Fourth Order ◽  
Unique Solvability ◽  
Hyperbolic Equations ◽  
Small Perturbations ◽  
Order Linear ◽  
The Stability ◽  
Effective Conditions

Abstract In the rectangle 𝐷 = (0, 𝑎) × (0, 𝑏) with the boundary Γ the Dirichlet problem 𝑢(𝑥, 𝑦) = 0 for (𝑥, 𝑦) ∈ Γ is considered, where 𝑝 and 𝑞 : 𝐷 → ℝ are locally summable functions and may have nonintegrable singularities on Γ. The effective conditions guaranteeing the unique solvability of this problem and the stability of its solution with respect to small perturbations of the coefficients of the equation under consideration are established.

Sign in / Sign up

Export Citation Format

Share Document