The Lifespan for Nonlinear Wave Equation Outside of Star-Shaped Obstacle in Three Space Dimensions

Concerning the Strauss Conjecture and Almost Global Existence for Nonlinear Dirichlet-Wave Equations in 4-Dimensions

“…Recently, the first and last authors [3] were able to show that Lindblad's sharp existence theorem, involving general quadratic nonlinearities of all three types, Q(u, u ′ , u ′′ ), for R + × R 3 extends to Dirichlet wave equations in R + × R 3 \K if K is star-shaped. This was done by coupling Hardy-type estimates with a variant the weighted local energy estimates of Keel, Smith and one of the authors [7], [9].…”

“…Therefore, for n ≥ 4, (2.11) and the proof of Theorem 2.3 shows that if w is as in (2.5), then (2.12) (log(2 + T )) [3], Du and Zhou used the sharp Huygens principle and arguments of [7] to show that in this case (2.12) is a consequence of (2.7).…”

“…This follows from the fact that the nonobstacle variant of this inequality holds together with the arguments from [7] that we used here to show that the nonobstacle inequality (2.10) yields the obstacle version (2.15). If we combine the aforementioned variant of (2.15) with Lemmas 4.2 and 4.3, then the arguments from the last section provide a somewhat simpler proof of the existence results in [3] that avoids the use of the scaling vector field L = t∂ t + x, ∇ x .…”

“…In [3] the first and last authors showed that when n = 3 and K is a star-shaped obstacle the analog of (1.5) has solutions with lifespan c/ε 2 if the 3-dimensional analog of (1.8) holds. The proof was based on a variant of (2.15) which says that…”

“…To prove results for quasilinear equations we shall couple these constant coefficient estimates with variable coefficient weighted L 2 estimates of the second and third authors [13]. This allows a somewhat simpler approach than the one in [3] since one does not have to use the scaling vector field L = t∂ t + x, ∇ x . An apparent compromise, though, is that, while we can show that the natural variant of the Strauss conjecture is valid outside of nontrapping obstacles, our almost global existence results for quasilinear equations are for the starshaped case.…”