377 research outputs found
Long time existence for semilinear wave equations on asymptotically flat space-times
We study the long time existence of solutions to nonlinear wave equations
with power-type nonlinearity (of order ) and small data, on a large class of
-dimensional nonstationary asymptotically flat backgrounds, which
include the Schwarzschild and Kerr black hole space-times. Under the assumption
that uniform energy bounds and a weak form of local energy estimates hold
forward in time, we give lower bounds of the lifespan when and is
not bigger than the critical one. The lower bounds for three dimensional
subcritical and four dimensional critical cases are sharp in general. For the
most delicate three dimensional critical case, we obtain the first existence
result up to , for many space-times including the
nontrapping exterior domain, nontrapping asymptotically Euclidean space and
Schwarzschild space-time.Comment: Final version, to appear in Communications in Partial Differential
Equations. 24 page
Recent works on the Strauss conjecture
In this review paper, we summarize the current state-of-art on the Strauss
conjecture with nontrapping obstacles. Among others, three essential estimates
are emphasized and presented: Morawetz-KSS estimates (also known as local
energy estimates), weighted Strichartz estimates and generalized Strichartz
estimates.Comment: 21 pages, no figures. No changes in content, but disable the usage of
the package showkey
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