On the solvability of the Cauchy problem for the Hopf equation corresponding to a nonlinear hyperbolic equation

1982 ◽  
pp. 161-184 ◽  
Author(s):  
M. I. Višik ◽  
A. I. Komeč
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Nonlinear Hyperbolic Equation ◽  
Hopf Equation ◽  
The Cauchy Problem

scholarly journals Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yu-Zhu Wang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Hyperbolic Equation ◽  
Contraction Mapping ◽  
Dimensional Space ◽  
Contraction Mapping Principle ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped nonlinear hyperbolic equation inn-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

scholarly journals The Cauchy problem for a second-order nonlinear hyperbolic equation with initial data on a line of parabolicity

1979 ◽  
Vol 20 (3) ◽  
pp. 345-365
Author(s):  
John M.S. Rassias
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Initial Conditions ◽  
Small Neighborhood ◽  
Second Order ◽  
Nonlinear Hyperbolic Equation ◽  
Linear Hyperbolic Equation ◽  
The Cauchy Problem ◽  
Upper Half Plane ◽  
Image Position

In this paper we study the Cauchy problem for the second order nonlinear hyperbolic partial differential equationwith initial conditionswhereand |u|, |ux|, |uy| < ∞, y ≥ 0, r = r(x) ∈ C4(·), ν = ν(x) ∈ C4(·).These conditions on k, H, f, r, and ν are assumed to be satisfied in some sufficiently small neighborhood of the segment I, y = 0, in the upper half-plane y > 0This paper generalizes the results obtained by N.A. Lar'kin (Differencial'nye Uravnenija 8 (1972), 76–84), who has treated the special case H = H(x, y, u); that is, the quasi-linear hyperbolic equation (*).

scholarly journals Cauchy problem for a substantially loaded hyperbolic equation

2021 ◽  
pp. 10-15
Author(s):  
А.Х. Аттаев
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Equation ◽  
Vibration Equation ◽  
One Dimensional ◽  
The Cauchy Problem

В работе проводится исследование задачи Коши для существенно нагруженного уравнения колебания одномерной струны. Приводятся примеры характеристических многообразий, для которых задача Коши поставлена корректно, а также нехарактеристических многообразий, для которых задача Коши поставлено некорректно. In this work, we study the Cauchy problem for a substantially loaded vibration equation of a one-dimensional string. Examples are given of characteristic manifolds for which the Cauchy problem is posed correctly, as well as non-characteristic manifolds for which the Cauchy problem is posed incorrectly.

Sign in / Sign up

Export Citation Format

Share Document