Weighted Strichartz estimates and global existence for semilinear wave equations

Abstract

In this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power-type nonlinearities. The proof is based on a weighted Strichartz estimate involving powers of the Lorentz distance.

Critical Exponent for Semilinear Wave Equations with Space-Dependent Potential

“…The proof of this fact, famous as Strauss' conjecture [31], took almost 20 years and the e¤ort of many mathematicians, beginning with John [13], Glassey [3], Sideris [30], Strauss [32], Zhou [42], and ending with Lindblad and Sogge [19], Georgiev, Lindblad and Sogge [2] and Tataru [33]. The number p w ðNÞ is refered as Strauss critical exponent.…”

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

“…Estimates (1.3) were proved by Georgiev-Lindblad-Sogge [7] under the following conditions a n 1 2 n q b 1 q suppF ´t xµ; x t 1 (1.4)…”

“…In fact, in three space dimensions, Pecher [25] proved that if p 1 · Ô Then, p 0´3 µ 1 · Ô 2 and we expect p 0´n µ to be the critical power concerning the existence of selfsimilar solutions to the equation (3.1). We notice that p 0´n µ is the critical exponent concerning the existence of time-global solutions to the Cauchy problem of the equation (3.1) with small, smooth initial data (see John [11], Georgiev-Lindblad-Sogge [7] and references therein). So, it is natural to expect p 0´n µ to be the one because self-similar solutions are also time-global solutions.…”

On Systems of Semilinear Wave Equations with Unequal Propagation Speeds in Three Space Dimensions

“…For the critical case p ¼ 1 þ ffiffi ffi 2 p , the small data blowup of (1.3) was proved by Schae¤er [24]. See also [3,22,23,29] for other results in three space dimensions, and [8,9,10,25,27] for other dimensional case.…”