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

Abstract

Abstract We prove that any sufficiently differentiable space-like hypersurface of ℝ 1 + N {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation ∂ t ⁢ t ⁡ u - Δ ⁢ u = | u | p - 1 ⁢ u {\partial_{tt}u-\Delta u=|u|^{p-1}u} on ℝ × ℝ N {{\mathbb{R}}\times{\mathbb{R}}^{N}} , for any 1 ≤ N ≤ 4 {1\leq N\leq 4} and 1 < p ≤ N + 2 N - 2 {1<p\leq\frac{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 {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 {t=0} . To obtain a finite-energy solution of the original problem from trace arguments, we need to work with H 2 × H 1 {H^{2}\times H^{1}} solutions for the transformed problem.