A generalization of the weighted Strichartz estimates for wave equations and an application to self‐similar solutions

Abstract

Weighted Strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without support restriction on the forcing term. The method of proof is based on the expansion by spherical harmonics and on the Sobolev space over the unit sphere, by which the required estimates are reduced to the radial case. As an application of the weighted Strichartz estimates, the existence and uniqueness of self-similar solutions to nonlinear wave equations is proved up to 5 space dimensions.

Weighted Strichartz estimates with angular regularity and their applications

“…Recently, there have been some interesting results in this direction, see e.g. Machihara-Nakamura-Nakanishi-Ozawa [24], Sterbenz [33], Kato-Nakamura-Ozawa [16], Cho-Ozawa [5]. In this paper, we make an attempt to get some systematic results in this direction.…”

On Hartree equations with derivatives

“…The above can be shown by the Leibniz rule and Sobolev embedding on the unit sphere (for instance see [25,26]). We now make use of the following Young inequality for radially symmetric functions which does not hold in general.…”

Global existence for the one-dimensional second order semilinear hyperbolic equations

“…More precisely, there exists no self-similar solution to one-dimensional wave equation while for two-and higher-dimensional wave equation the existence of such solutions was recently proved by Pecher [25,26], Hidano [12], Ribaud and Youssfi [29], Kato and Ozawa [13], Kato, Nakamura and Ozawa [15]. To show the nonexistence of self-similar solution we admit the non-smooth data, for example, homogeneous, like ψ 0 (x) = |x| −a and ψ 1 (x) = |x| −b .…”