Type II Blow up Manifolds for the Energy
                    Supercritical Semilinear Wave Equation

Collot, Charles

doi:10.1090/memo/1205

Cited by 34 publications

(47 citation statements)

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“…(1.1) is conjectured in dimensions 5 ≤ d < 13 based on numerical experiments performed by Kycia [43]. In higher space dimensions d ≥ 13, a result by Collot [14] implies the existence of smooth type II blowup solutions of the form (1.7) with blowup rates λ(t) ∼ (T − t) α for > α(d) > 2. In this case, the corresponding ground state Q is not known explicitly and the constructed solutions are co-dimension − 1 stable.…”

Section: 1mentioning

confidence: 99%

Threshold for blowup for the supercritical cubic wave equation

2020

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In this paper, we discuss singularity formation for the focusing cubic wave equation in the energy supercritical regime. For this equation an explicit nontrivial self-similar blowup solution was recently found by the first and third author in [27]. In the seven dimensional case it was proven to be stable along a co-dimension one manifold of initial data. Here, we provide numerical evidence that this solution is in fact a critical solution at the threshold between finite-time blowup and dispersion. Furthermore, we discuss the spectral problem arising in the stability analysis in general dimensions d ≥ 5.

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Section: 1mentioning

confidence: 99%

Threshold for blowup for the supercritical cubic wave equation

2020

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“…We would like to mention that this kind of method has been successfully applied for various nonlinear evolution equations. In particular in the dispersive setting for the nonlinear Schrödinger equation both in the mass critical [43,44,42,41] and mass supercritical [47] cases; the mass critical gKdV equation [40,39,38]; the energy critical [17], [29] and supercritical [10] wave equation; the two dimensional critical geometric equations: the wave maps [53], the Schrödinger maps [46] and the harmonic heat flow [54,56]; the semilinear heat equation (1.16) in the energy critical [58] and supercritical [9] cases; and the two dimensional Keller-Segel model [55,25]. In all these works, the method relies on two arguments:…”

Section: )mentioning

confidence: 99%

On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

2019

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We will study the problem under an additional assumption of 1-corotational symmetry, with the following corotational ansatz Φ(x, t) = cos(u(|x|, t))x |x| sin(u(|x|, t)) 2 √ κ , where κ is an upper bound on the sectional curvature of the target manifold M (see Jost [31] and Lin-Wang [33]). Without these assumptions, the solution u(r, t) may develop singularities in some finite time (see for examples, Coron and Ghidaglia [14], Chen and Ding [11] for d ≥ 3, Chang, Ding and Yei [12] for d = 2). In this case, we say that u(r, t) blows up in a finite time T < +∞ in the sense that lim t→T ∇u(t) L ∞ = +∞.Here we call T the blowup time of u(x, t). The blowup has been divided by Struwe [60] into two types: u blows up with type I if: lim sup t→T (T − t) 1 2 ∇u(t) L ∞ < +∞, u blows up with type II if: lim sup t→T (T − t) 1 2 ∇u(t) L ∞ = +∞. 1-COROTATIONAL HARMONIC MAP HEAT FLOW IN SUPERCRITICAL DIMENSIONS 4 d−4−2 √ d−1 for d ≥ 11, correspond to the cases d = 2 and d = 7 in the study of equation (1.4) respectively.

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“…Theorem 3.4 (Type II blow up for the energy super critical wave equation, [22]). Let d ≥ 11, p JL be given by (3.10) and a nonlinearity p = 2q + 1, q ∈ N * , p > p JL .…”

Section: )mentioning

confidence: 99%