Slow Blow-up Solutions for the H(r) Critical Focusing Semi-linear Wave Equation

Given ν > 1 2 and δ > 0 arbitrary, we prove the existence of energy solutions of (0.1) ∂ttu−∆u− u = 0 in R3+1 that blow up exactly at r = t = 0 as t → 0−. These solutions are radial and of the form u = λ(t) 1 2W (λ(t)r)+η(r, t) inside the cone r ≤ t, where λ(t) = t−1−ν , W (r) = (1 + r2/3)− 1 2 is the stationary solution of (0.1), and η is a radiation term with Z [r≤t] ` |∇η(x, t)| + |ηt(x, t)| + |η(x, t)| ́ dx→ 0, t→ 0 Outside of the light-cone there is the energy bound Z [r>t] ` |∇u(x, t)| + |ut(x, t)| + |u(x, t)| ́ dx < δ for 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).