On Blowup in Supercritical Wave Equations

Donninger, Roland; Schörkhuber, Birgit

doi:10.1007/s00220-016-2610-2

Cited by 48 publications

(75 citation statements)

References 65 publications

(96 reference statements)

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“…In the energy-critical case there are numerical evidences that generic blow-up solutions behave like y 0 (t) see Bizoń, Chmaj and Tabor [3]. The stability of y 0 in light cones, in the energy topology was proved by Donninger [8]; see also previous results in stronger topology by Donninger and Schörkhuber [11].…”

Section: Viii-1mentioning

confidence: 83%

Dynamics of the focusing critical wave equation

Duyckaerts¹

2016

Séminaire Laurent Schwartz — EDP Et Applications

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Section: Viii-1mentioning

confidence: 83%

Dynamics of the focusing critical wave equation

Duyckaerts¹

2016

Séminaire Laurent Schwartz — EDP Et Applications

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“…Furthermore, the continuity of the map follows from the triangle inequality and an approximation argument using the density of C ∞ (B 5 1+δ ) in H k (B 5 1+δ ). For a detailed proof see Lemma 5.8 in [15]. Now, one can apply Proposition 8.5 to get the following result.…”

Section: The Modulation Equationmentioning

confidence: 90%

“…which follows from semigroup theory, see [17], page 55, Theorem 1.10. For more details see Lemma 4.6 in [15]. Remark 6.9.…”

Section: Spectral Analysismentioning

confidence: 99%

Stable blowup for the cubic wave equation in higher dimensions

Chatzikaleas

Donninger

2019

Journal of Differential Equations

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We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in a backward light-cone and approaches the ODE blowup profile.Here, the ground-state solution (1.3) corresponds to f 0 = √ 2. Levine [23] used energy methods and a convexity argument to show that initial data with negative energy and finite L 2 −norm lead to blowup in finite time, see also [22] for generalizations to the Klein-Gordon equation. We also mention the works of Alinhac [2] and Caffarelli and Friedman [10], [9] where more blowup results can be found. The stability of the ground-state has been studied extensively by Schörkhuber and the second author in three space dimensions (in [13], [14] for radial initial data and in [15] without symmetry restrictions) and later in [16] for all space dimensions and for radial initial data. Some numerical results are available in a series of papers by Bizoń, Chmaj, Tabor and Zenginoğlu, see [4], [6], [8]. Furthermore, in the superconformal and Sobolev subcritical range, an upper bound on the blowup rate was proved by Killip, Stoval and Vişan in [22], then refined by Hamza and Zaag in [19]. In a series of papers [27], [33], [32], [26], [25], Merle and Zaag obtained sharp upper and lower bounds on the blowup rate of the H 1 −norm of the solution inside cones that terminate at the singularity, see also the work of Alexakis and Shao [1]. We also mention the recent work by Dodson-Lawrie [11] on large-data scattering for the cubic equation in five dimensions.

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“…• From now on we follow the argument introduced in our earlier works [11][12][13][14][15][16] on selfsimilar blowup for wave-type equations. We first show that the nonlinearity is locally Lipschitz on X .…”

Section: Outline Of the Proofmentioning

confidence: 95%

Stable self-similar blowup in the supercritical heat flow of harmonic maps

2017

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We consider the heat flow of corotational harmonic maps from R 3 to the threesphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators. Mathematics Subject Classification

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On Blowup in Supercritical Wave Equations

Abstract: We study the blowup behavior for the focusing energysupercritical semilinear wave equation in 3 space dimensions without symmetry assumptions on the data. We prove the stability in H 2 × H 1 of the ODE blowup profile.

Cited by 48 publications

References 65 publications

Dynamics of the focusing critical wave equation

Dynamics of the focusing critical wave equation

Stable blowup for the cubic wave equation in higher dimensions

Stable self-similar blowup in the supercritical heat flow of harmonic maps

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