The Initial-Value Problem For A Mildly Perturbed Wave Equation

2018 ◽  
pp. 27-33
Author(s):  
Alain Haraux
Keyword(s):  
Wave Equation ◽  
Initial Value Problem ◽  
Initial Value

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.

Global well-posedness and linearization of a semilinear 2D wave equation with exponential type nonlinearity

2010 ◽  
Vol 17 (3) ◽  
pp. 543-562 ◽  
Author(s):  
Olfa Mahouachi ◽  
Tarek Saanouni
Keyword(s):  
Wave Equation ◽  
Exponential Type ◽  
Initial Value Problem ◽  
Energy Estimate ◽  
Linear Wave ◽  
Well Posedness ◽  
Two Dimensional ◽  
Initial Value ◽  
Linear Wave Equation ◽  
Funct Anal

Abstract We consider the initial value problem for a two-dimensional semi-linear wave equation with exponential type nonlinearity. We obtain global well-posedness in the energy space. We also establish the linearization of bounded energy solutions in the spirit of Gérard [J. Funct. Anal. 141: 60–98, 1996]. The proof uses Moser–Trudinger type inequalities and the energy estimate.

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