Diffusion phenomenon for abstract linear wave equations with time decaying coefficients of propagation and dissipation

2021 ◽  
pp. 1-53
Author(s):  
Taeko Yamazaki
Keyword(s):  
Wave Equations ◽  
Propagation Speed ◽  
Linear Wave ◽  
Diffusion Phenomenon ◽  
Linear Wave Equations

We show diffusion phenomenon for linear abstract dissipative wave equations with time dependant coefficients of propagation speed and dissipation. Coefficients are decaying in time but not assumed to be monotone.

Uniqueness for stochastic non-linear wave equations

2007 ◽  
Vol 67 (12) ◽  
pp. 3287-3310 ◽  
Author(s):  
Martin Ondreját
Keyword(s):  
Wave Equations ◽  
Linear Wave ◽  
Non Linear ◽  
Linear Wave Equations

An inverse scattering theorem for (1 + 1)-dimensional semi-linear wave equations with null conditions

2021 ◽  
Vol 18 (01) ◽  
pp. 143-167
Author(s):  
Mengni Li
Keyword(s):  
Inverse Scattering ◽  
Wave Equations ◽  
Scattering Problem ◽  
Weighted Sobolev Spaces ◽  
One Dimension ◽  
Energy Estimates ◽  
Small Data ◽  
Linear Wave ◽  
Global Existence Of Solutions ◽  
Linear Wave Equations

We are interested in the inverse scattering problem for semi-linear wave equations in one dimension. Assuming null conditions, we prove that small data lead to global existence of solutions to [Formula: see text]-dimensional semi-linear wave equations. This result allows us to construct the scattering fields and their corresponding weighted Sobolev spaces at the infinities. Finally, we prove that the scattering operator not only describes the scattering behavior of the solution but also uniquely determines the solution. The key ingredient of our proof is the same strategy proposed by Le Floch and LeFloch [On the global evolution of self-gravitating matter. Nonlinear interactions in Gowdy symmetry, Arch. Ration. Mech. Anal. 233 (2019) 45–86] as well as Luli et al. [On one-dimension semi-linear wave equations with null conditions, Adv. Math. 329 (2018) 174–188] to make full use of the null structure and the weighted energy estimates.

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