Global Well-posedness for the Logarithmically Energy-Supercritical Nonlinear Wave Equation with Partial Symmetry

Abstract

We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and extend work of Tao in the radially symmetric setting. The techniques involved include weighted versions of Morawetz and Strichartz estimates, with weights adapted to the partial symmetry assumptions.In an appendix, we establish a corresponding quantitative result for the energy-critical problem.

Scattering for focusing supercritical wave equations in odd dimensions

“…We note that the methods in [11] apply to all supercritical exponents, yielding analogous results to [16]. For results addressing the nonradial setting, see [15,7] and the references cited therein. Also, we refer the reader to [37] for a corresponding result addressing the focusing nonlinear Schrödinger equation.…”

Global existence of solutions of a loglog energy-supercritical Klein-Gordon equation

“…In particular C and c do not depend on a and b3 In view of what is written above, if (a, b, d) ∈ R 3 , then a b d+ means that there exists a constant C > 0 that may depend on ǫ and such that a ≤ Cb d+ǫ 4. in particular if 0 < γ < γn, with γn defined in the statement of Theorem 45 More precisely, the computation holds for smooth solutions (i.e solutions in H p with exponents p large enough).…”

“…Since for all ǫ > 0 there exist c ǫ > 0 such that |u| n−2 +ǫ u|) then the nonlinearity of ( 1) is said to be barely energysupercritical. Barely-supercritical equations have been studied extensively in the literature: see e.g [3,8,14,15,16,17,18]. We write below a local-wellposedness result:…”

“…If a ∈ R then a := 1 + a 2 It may be that the constants C or c depend on some parameters α 1 , ..., α m : unless otherwise specified, we do not mention them, for sake of simplicity. We define b+ to be a number b + ǫ for some 0 < ǫ ≪ 1 3 . Unless otherwise specified, we let in the sequel f (resp.…”

Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds

“…Other results concerning possible blow-up in the energy-supercritical regime include studies of global well-posedness for logarithmically supercritical nonlinearities [27] (see also related works [32,28,29,30,6,11] for the nonlinear wave equation), and constructions of large data global solutions [1]. We remark that there are also well-developed constructions of blow-up solutions for focusing nonlinearities (where |u| 6 u is replaced by −|u| p u for p > 4 d−2 , with the problem posed on R d ); see, e.g. [9,13,24], and the references cited therein.…”