scholarly journals Hörmander’s method for the characteristic Cauchy problem and conformal scattering for a nonlinear wave equation

2020 ◽  
Vol 110 (6) ◽  
pp. 1391-1423
Author(s):  
Jérémie Joudioux
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Characteristic Cauchy Problem

scholarly journals ON THE WAVE EQUATION WITH QUADRATIC NONLINEARITIES IN THREE SPACE DIMENSIONS

2011 ◽  
Vol 08 (01) ◽  
pp. 1-8 ◽  
Author(s):  
AXEL GRÜNROCK
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Parameter Range ◽  
Well Posedness ◽  
Quadratic Nonlinearities ◽  
The Cauchy Problem ◽  
Three Space

The Cauchy problem for the nonlinear wave equation [Formula: see text] in three space dimensions is considered. The data (u0, u1) are assumed to belong to [Formula: see text], where [Formula: see text] is defined by the norm [Formula: see text] Local well-posedness is shown in the parameter range 2 ≥ r > 1, [Formula: see text]. For r = 2 this coincides with the result of Ponce and Sideris, which is optimal on the Hs-scale by Lindblad's counterexamples, but nonetheless leaves a gap of ½ derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for □u = ∂u2 are obtained, too.

ON THE UNIFORM CONTINUITY OF THE SOLUTION MAP FOR A NONLINEAR WAVE EQUATION

2008 ◽  
Vol 05 (02) ◽  
pp. 381-397
Author(s):  
SVETLIN GEORGIEV GEORGIEV
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Nontrivial Solution ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Uniform Continuity ◽  
Solution Map ◽  
The Cauchy Problem ◽  
Uniformly Continuous

We construct a metric such that the Cauchy problem for the corresponding nonlinear wave equation admits a nontrivial solution but the solution map is not uniformly continuous.

scholarly journals Global classical solutions to the Cauchy problem for a nonlinear wave equation

1998 ◽  
Vol 21 (3) ◽  
pp. 533-548 ◽  
Author(s):  
Haroldo R. Clark
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Classical Solution ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Classical Solutions ◽  
Global Classical Solution ◽  
Global Classical Solutions ◽  
The Cauchy Problem

In this paper we consider the Cauchy problem{u″+M(|A12u|2)Au=0   in   ]0,T[u(0)=u0,       u′(0)=u1,whereu′is the derivative in the sense of distributions and|A12u|is theH-norm ofA12u. We prove the existence and uniqueness of global classical solution.

On a nonlinear wave equation with boundary conditions involved in a Cauchy problem

2011 ◽  
Vol 12 (1) ◽  
pp. 69-92
Author(s):  
Le Thi Phuong Ngoc ◽  
Le Khanh Luan ◽  
Tran Minh Thuyet ◽  
Nguyen Thanh Long
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Boundary Conditions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation
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