Loss of regularity for super critical wave equations. (Perte de régularité pour les équations d’ondes sur-critiques.) (French. English summary) Zbl 1071.35020

Lebeau, Gilles

Bull. Soc. Math. Fr. 133, No. 1, 145-157 (2005).

Summary: We prove that the local Cauchy problem for the supercritical wave equation in \(\mathbb{R}^d\), \(\square u+ u^p= 0\), with \(d\geq 3\), \(p> 3\) and \(p> (d+ 2)/(d- 2)\), is ill-posed in \(H^\sigma\) for every \(\sigma\in ]1,\sigma_c[\), where \(\sigma_c= d/2- 2/(p- 1)\) is the critical exponent.

MSC:

35B33	Critical exponents in context of PDEs
35L15	Initial value problems for second-order hyperbolic equations
35L70	Second-order nonlinear hyperbolic equations
35R25	Ill-posed problems for PDEs

Keywords:

local Cauchy problem

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