Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation

We prove that any sufficiently differentiable space-like hypersurface of R1+N coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation partial derivative(tt)u - Delta u = vertical bar u vertical bar(p-1)u on R x R-N, for any 1 <= N <= 4 and 1 < p <= N+2/N-2. We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at t = 0 for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at t = 0. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with H-2 x H-1 solutions for the transformed problem.