Finite-time blow-up for the real-valued defocusing energy-supercritical quintic NLW

Abstract

We consider the defocusing quintic wave equation $\partial_t^2u-Δu+u^5=0$ for real-valued functions on $\mathbb{R}^{1+10}$, which is energy-supercritical. We construct a discretely self-similar solution in a backward light cone, of the form $$u(t,x)=(T-t)^{-1/2}W^*\big(ω^*\log\frac1{T-t},\frac{|x|}{T-t}\big),$$ where the nonzero profile $W^*$ is $2π$-periodic in its first variable and real-analytic up to and across the light cone. Cutting off its initial data gives smooth, compactly supported, radial data whose solution coincides with the discretely self-similar solution in the backward light cone, is smooth on $[0,T]\times\mathbb{R}^{10}$ except at the vertex $(T,0)$ of the cone, and blows up there at the self-similar rate. The solutions we construct here appear to be the first real-valued blow-up solutions of defocusing energy-supercritical NLW. Existence of the real profile is proved by a computer-assisted Newton--Kantorovich argument in a Banach algebra of Fourier--Chebyshev coefficients, with rigorous error control. We find the approximate profile by numerically continuing rotating self-similar profiles of the complex-valued equation. This step is not essential for the blow-up proof, so it is left as non-rigorous numerics. The main analytic ingredients are an exact description of the mode operators for the linear part, which are hypergeometric and upper triangular in the Chebyshev basis, and a bound for their inverses that is uniform at high frequencies. All computer-assisted proofs, including the requisite codes and the approximate objects, are accessible on GitHub at \cite{code}.