Construction of multi-bubble solutions for the energy-critical wave equation in dimension 5

Abstract

We prove the existence of a global solution of the energy-critical focusing wave equation in dimension 5 blowing up in infinite time at any K given points z k of R 5 , where K ≥ 2. The concentration rate of each bubble is asymptotic to c k t −2 as t → ∞, where the c k are positive constants depending on the distances between the blow-up points z k . This result complements previous constructions of blow-up solutions and multi-solitons of the energycritical wave equation in various dimensions N ≥ 3. 7 3 on R 5 .Up to scaling and translation invariance, W is the unique positive solution of (1.4). In particular, u(t, x) = (W (x), 0) is a stationary solution of (1.1) and other explicit solutions of (1.1) are deduced by the sign, scaling, translation and Lorentz invariances of the equation: u(t, x) = ± (W ℓ,λ (x − ℓt − x 0 ), −(ℓ · ∇)W ℓ,λ (x − ℓt − x 0 )) ,

Construction of excited multi-solitons for the focusing 4D cubic wave equation

“…Final argument on the unstable parameters. Let The proof is based on a standard compactness argument (see for example [6,24,38]). The main point is the following proposition of weak continuity of the flow near a compact set.…”

“…In the same direction, recall that for the 6D energy-critical wave equation, Jendrej [23] proved the existence of radial two-bubble solutions. Later, for the 5D energy-critical wave equation, Jendrej-Martel [24] proved the existence of N -bubble solutions for any N ≥ 2. For the 3 and 5D cases, Krieger-Nakanishi-Schlag [26] constructed a center-stable manifold of the ground state.…”

Nondegeneracy of the positive solutions for critical nonlinear Hartree equation in $\R^6$

“…is crucial in the construction of blow-up solutions to the equations (1.9) and (1.10) (see, for instance, [6][7][8][9][10]13]). With the help of Theorem 1.1, we are able to construct blow-up solutions to the Ḣ1 -critical nonlinear Schrödinger equation with Hartree terms,…”

Global, Non-scattering solutions to the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$

“…Multisoliton solutions to the 5 `1 dimensional critical wave equation were constructed in [19] [23]. Infinite time blow-up solutions involving multiple dynamically rescaled solitons for the 5 `1 dimensional critical wave equation were constructed in [8]. Two bubble solutions to the 6 `1 dimensional critical wave equation were constructed in [7].…”