2024/10/23 by Ngoc, Tran Thi, Xuan, Pham Truong
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2410.17745
In this paper we establish a conformal scattering theory for the defocusing semilinear wave equations on the Schwarzschild spacetime. We combine the energy and pointwise decay results of solutions obtained in \citeYang and a Sobolev embedding on spacelike hypersurfaces to establish the energy estimates in two sides between the energy flux of solution through the Cauchy initial hypersurface Σ0=\ t=0 \ and the one through the null conformal boundaries \mathfrakH+∪ \scri+ (resp. \mathfrakH-∪ \scri-). Combining these estimates and the well-posedness of Cauchy and Goursat problems for nonlinear wave equations we provide a bounded linear and locally Lipschitz scattering operator which maps the past scattering data to the future scattering data. Our results extend also the previous works \citeBaSeZho1990,Jo2012,Jo2019 to an asymptotically flat spacetime such as Schwarzschild spacetime which is static and spherically symmetric.