Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation

Abstract

Abstract. Given ν > 1 2 and δ > 0 arbitrary, we prove the existence of energy solutions ofin R 3+1 that blow up exactly at r = t = 0 as t → 0−. These solutions are radial and of the form u = λ(t)2 is the stationary solution of (0.1), and η is a radiation term with ZOutside of the light-cone there is the energy bound Zfor all small t > 0. The regularity of u increases with ν. As in our accompanying paper on wave-maps [10], the argument is based on a renormalization method for the 'soliton profile' W (r).

Global dynamics away from the ground state for the energy-critical nonlinear wave equation

“…which follows from L λ ΛW λ = 0 and L λ ρ λ = −k 2 λ 2 ρ λ , and ρ ∈ S. Such λ(u) is uniquely determined at least in the region 12) by the implicit function theorem, since…”

“…Since ∇u 2 2 ∼ ∇W 2 2 in the hyperbolic region I H , we thus replace (5.6) with 12) and so from (5.5) we obtain…”

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

“…which is a stationary solution of (1.1). The construction of [KST09] relies on an elaborate fixed point argument which yields the following description of the solution: In this work, we investigate the converse problem: if we consider an arbitrary type II radial blow-up solution, does such a decomposition hold?…”

Blow up and near soliton dynamics for the L 2 critical gKdV equation

“…In [7], an explicit formula on the growth of the solution at infinity is given directly in terms of the initial data. Continuums of blow up rates were also observed in pioneering works by Krieger, Schlag and Tataru [15], [16] for energy critical wave problems, see also Donninger and Krieger [3]. All these results point out that the critical topology is not enough by itself to classify the flow near the ground state.…”

“…(Blow up) For all t ∈ [0, T ), u(t) ∈ T α * and the solution blows up in finite time T < +∞ in the regime described by Theorem 1 (14), (15), (16).…”