Solutions with prescribed local blow-up surface for the nonlinear wave equation
| Autor: | Yvan Martel, Thierry Cazenave, Lifeng Zhao |
|---|---|
| Přispěvatelé: | Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Versailles (LMV), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, Chinese Academy of Sciences [Changchun Branch] (CAS), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Surface (mathematics)
Work (thermodynamics) Trace (linear algebra) General Mathematics 010102 general mathematics Statistical and Nonlinear Physics 01 natural sciences Blowing up 010101 applied mathematics Mathematics - Analysis of PDEs Hypersurface Compact space FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Ball (mathematics) Differentiable function 0101 mathematics Mathematics Mathematical physics Analysis of PDEs (math.AP) 35L05 35B44 35B40 |
| Zdroj: | Advanced Nonlinear Studies Advanced Nonlinear Studies, Walter de Gruyter GmbH, 2019, ⟨10.1515/ans-2019-2059⟩ |
| ISSN: | 1536-1365 |
| DOI: | 10.1515/ans-2019-2059⟩ |
| Popis: | 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 . 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. |
| Databáze: | OpenAIRE |
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