We show that the spherically symmetric Einstein-scalar-field equations for wave-like decaying initial data at null infinity have unique local solutions and unique global solutions for small initial data. We also generalize Christodoulou's global generalized solutions to the wave-like decaying initial data. We emphasize that this decaying condition is sharp. 

　　Publication: 

　　Advances in Mathematics, Volume 409, Part A, 19 November 2022, 108642 

　　DOI: 10.1016/j.aim.2022.108642 

　　Author: 

　　Chuxiao Liu 

　　School of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, PR China 

　　Xiao Zhang 

　　Guangxi Center for Mathematical Research, Guangxi University, Nanning, Guangxi 530004, PR China 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China 

　　Email: xzhang@gxu.edu.cn; xzhang@amss.ac.cn