The Strauss conjecture on Kerr black hole backgrounds

Lindblad, Hans; Metcalfe, Jason; Sogge, Christopher D.; Tohaneanu, Mihai; Wang, Chengbo

doi:10.1007/s00208-014-1006-x

Cited by 53 publications

(69 citation statements)

References 57 publications

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“…Moreover, Wang [36] showed global existence when n ≥ 4 and p = 2 and derived sharp lifespan estimates when 1 < p ≤ p 0 (n) and n = 3, 4. The blow-up result for Schwarzschild spacetime was obtained by Catania & Georgiev [3] when n = 3 and 1 < p < p 0 (3 [25] showed global existence in the supercritical case p > p 0 (3).…”

Section: Introductionmentioning

confidence: 95%

Blow-up of solutions to critical semilinear wave equations with variable coefficients

Wakasa

Yordanov

2019

Journal of Differential Equations

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We verify the critical case p = p 0 (n) of Strauss' conjecture [31] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in R n , where n ≥ 2. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when p = p 0 (n). The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [44] and Zhou & Han [46]: exponential "eigenfunctions" of the Laplacian [38] are used to construct the test function φq for linear wave equation with variable coefficients and John's method of iterations [14] is augmented with the "slicing method" of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case.

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Section: Introductionmentioning

confidence: 95%

Blow-up of solutions to critical semilinear wave equations with variable coefficients

Wakasa

Yordanov

2019

Journal of Differential Equations

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“…The key difference between the current result and [25] is the approach to proving the weighted Strichartz estimates. In [25] a duality argument of [49] is used to "divide through" by a derivative in the localized energy estimate. That process required that the data be compactly supported.…”

Section: Introductionmentioning

confidence: 96%

“…and typical examples include F p (u) = ±|u| p and F p (u) = ±|u| p−1 u. In the process, we shall also weaken some hypotheses that were made on the data in [25]. We shall consider asymptotically flat space-times that permit a localized energy estimate.…”

Section: Introductionmentioning

confidence: 99%

The Strauss Conjecture on Asymptotically Flat Space-Times

Metcalfe¹,

Wang²

2017

SIAM J. Math. Anal.

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By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angular momentum) black hole backgrounds, long range asymptotically Euclidean spaces, and small time-dependent asymptotically flat perturbations of Minkowski space-time. We also permit lower order perturbations of the wave operator. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and the assumed localized energy estimate, which is used in the remaining compact set.2010 Mathematics Subject Classification. 35L70, 35L15.

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“…Morawetz type estimates and local energy estimates have rich history and a large body of related literature. We refer the interested readers to [28,25] for more exhaustive treatment of such estimates.…”

Section: 2mentioning

confidence: 99%

Weighted fractional chain rule and nonlinear wave equations with minimal regularity

et al. 2019

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We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data:It has been known that the problem is well-posed for s ≥ 2 and ill-posed for s < 3/2. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for s > 3/2 and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.1.1. Quadratic semi-linear wave equation . In [19], by using bilinear estimates, together with standard energy-type estimates, Klainerman and Machedon proved the local well-posedness for the problem (1.1) with a + b = 0 (thus, satisfying the null condition) in H 2 , which was later improved to H s for s > s c , see [20], [21], [22] and references therein. Moreover, even without the null condition, their bilinear

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The Strauss conjecture on Kerr black hole backgrounds

Cited by 53 publications

References 57 publications

Blow-up of solutions to critical semilinear wave equations with variable coefficients

Blow-up of solutions to critical semilinear wave equations with variable coefficients

The Strauss Conjecture on Asymptotically Flat Space-Times

Weighted fractional chain rule and nonlinear wave equations with minimal regularity

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