Yue Ma, Xi'an Jiao Tong University
601 Yue-Kong Pao Library
We introduce a generalization of the hyperboloidal foliation method in order to remove the main restriction on the support of the initial data. The key point is to make a smooth gluing together asymptotically Euclidean hyper-surfaces and asymptotically hyperboloidal hyper-surfaces. Well-chosen frames of vector fields (null-semi-hyperboloidal frame, Euclidean-hyperboloidal frame) allow us to exhibit clearly the structure of the wave-Klein-Gordon system under consideration and then analyze the decay of solutions in time-like and in space-like directions. New Sobolev inequalities valid in positive cones and in each domain of Euclidean-hyperboloidal foliation are established. Based on these constructions, we establish a first set of results including global existence of Einstein-positive scalar field model system and a type of wave-Klein-Gordon system in 1+1 dimension.