In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions, which states that, for the nonlinear wave equations of the form $\Box u=|\nabla u|^p$, the critical exponent to admit global small solutions is given by $p_c=1+\frac{2}{n-1}$. Moreover, we are able to prove the existence results with low regularity assumption on the initial data and extend the solutions to the sharp lifespan. The main idea is to exploit the trace estimates and KSS type estimates.Comment: 28 page

In this paper, we establish an optimal dual version of trace estimate involving angular regularity. Based on this estimate, we get the generalized Morawetz estimates and weighted Strichartz estimates for the solutions to a large class of evolution equations, including the wave and Schrödinger equation. As applications, we prove the Strauss' conjecture with a kind of mild rough data for 2 ≤ n ≤ 4, and a result of global well-posedness with small data for some nonlinear Schrödinger equation with L 2 -subcritical nonlinearity.

We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of L ∞ x estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate L 4 t L ∞ x for n = 2.

We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. 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 a localized energy estimate on the black hole background, which handles the behavior in the remaining compact set.Comment: 21 pages, no changes in contents, fix a technical problem in pdf file it generate

This paper investigates the combined effects of two distinctive power-type nonlinear terms (with parameters p, q > 1) in the lifespan of small solutions to semilinear wave equations. We determine the full region of ( p, q) to admit global existence of small solutions, at least for spatial dimensions n = 2, 3. Moreover, for many ( p, q) when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan. Mathematics Subject ClassificationIn memory of Rentaro Agemi.The authors are very grateful to the referees for their helpful comments.