scholarly journals On the Cauchy problem for a certain nonlinear hyperbolic partial differential equation of the second order

1970 ◽  
Vol 46 (2) ◽  
pp. 162-167
Author(s):  
Masayoshi Tsutsumi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Second Order ◽  
Hyperbolic Partial Differential Equation ◽  
Partial Differential ◽  
The Cauchy Problem

Estimates for Solutions of Wave Equations with Vanishing Curvature

1985 ◽  
Vol 37 (6) ◽  
pp. 1176-1200 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Linear Combination ◽  
Initial Value Problem ◽  
Wave Equations ◽  
Hyperbolic Partial Differential Equation ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Image Position

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the formFor example, the solution of the initial value problemis given by u(x, t) = Ttf(x) wherePeral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only ifFrom the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

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