scholarly journals Fully integrable one-dimensional nonlinear wave equation: Solution of a general initial value problem

2022 ◽  
pp. 100268
Author(s):  
E.V. Trifonov
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Initial Value Problem ◽  
Initial Value ◽  
Equation Solution ◽  
One Dimensional

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in .H s × .H s-1, s> 1/2

2018 ◽  
Vol 2019 (21) ◽  
pp. 6797-6817
Author(s):  
Benjamin Dodson
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Three Dimensions ◽  
Well Posedness ◽  
Critical Space ◽  
Initial Value ◽  
Long Time ◽  
Well Posed

Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$ for some $s> \frac{1}{2}$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11].

The Pettis integration of a perturbed wave equation

1979 ◽  
Vol 86 (1) ◽  
pp. 145-159
Author(s):  
James P. Fink
Keyword(s):  
Wave Equation ◽  
Vector Field ◽  
Nonlinear Wave ◽  
Initial Value Problem ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Initial Value

AbstractIn this paper, we investigate the integrability of the vector field of the initial-value problem associated with certain nonlinear wave equations. This vector field involves translations and as such is not a strongly continuous or even strongly measurable L∞-valued function. It is shown that such a vector field, although not generally Pettis integrable, does turn out to be so in an important situation. We then indicate how this result can be used to obtain pseudo-solutions of the initial-value problem.

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