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Weighted Strichartz estimates for the wave equation in even space dimensions. (English) Zbl 1063.35042
The authors establish the weighted Strichartz estimate \[ \left\| \left| t^2 - | x| ^2\right| ^a w \right\| _{L^q(\mathbb R^{1+n}_+)} \leq C \left\| \left| t^2 - | x| ^2\right| ^b F \right\| _{L^{q'}(\mathbb R^{1+n}_+)} \] whenever \(w\) is spherically symmetric and solves the inhomogeneous Cauchy problem \(\square w = F\) with zero initial data \(w(0) = w_t(0) = 0\), \(2 < q < 2(n+1)/(n-1)\), the dimension \(n \geq 2\) is even and \(a,b\) obey the estimates \[ a-b + \frac{n+1}{q} = \frac{n-1}{2},\qquad \frac{n}{q} - \frac{n-1}{2} < b < \frac{1}{q}. \] This theorem was previously demonstrated for odd \(n\) by V. Georgiev, H. Lindblad and C. D. Sogge [Am. J. Math. 119, 1291–1319 (1997; Zbl 0893.35075)]. As a consequence the authors construct global solutions to the nonlinear wave equation \(\square u = F(u)\), where \(F(u)\) scales like \(| u| ^p\) for some \(p_0(n) < p < (n+3)/(n-1)\), \(p_0(n)\) being the Strauss exponent for global existence of small data solutions, and the initial position is a small multiple of the scale-invariant \(| x| ^{-2/(p-1)}\) (and initial velocity a small multiple of \(| x| ^{-2/(p-1)-1})\). This was achieved in odd dimensions by the authors [Indiana Univ. Math. J. 52, 1615–1630 (2003; Zbl 1053.35029)]. The methods are based on an explicit representation of the fundamental solution, which is more difficult in even dimensions than in odd.

MSC:
35B45 A priori estimates in context of PDEs
35L05 Wave equation
35L70 Second-order nonlinear hyperbolic equations
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