Weighted fractional chain rule and nonlinear wave equations with minimal regularity

• Kunio Hidano ^[1] ; Jin-Cheng Jiang ^[2] ; Sanghyuk Lee ^[3] ; Chengbo Wang ^[4]
  1. [1] Mie University

     Mie University

     Japón

     
  2. [2] National Tsing Hua University

     National Tsing Hua University

     Taiwán

     
  3. [3] Seoul National University

     Seoul National University

     Corea del Sur

     
  4. [4] Zhejiang University

     Zhejiang University

     China

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 2, 2020, págs. 341-356
• Idioma: inglés
• DOI: 10.4171/rmi/1130
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• Resumen

  • We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: It has been known that the problem is well-posed for s≥2 and ill-posed for s<3/2. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for s>3/2 and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.