Vol. 4, No. 2, 2011

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Well- and ill-posedness issues for energy supercritical waves

Slim Ibrahim, Mohamed Majdoub and Nader Masmoudi

Vol. 4 (2011), No. 2, 341–367

We investigate the initial value problem for some energy supercritical semilinear wave equations. We establish local existence in suitable spaces with continuous flow. The proof uses the finite speed of propagation and a quantitative study of the associated ODE. It does not require any scaling invariance of the equation. We also obtain some ill-posedness and weak ill-posedness results.

nonlinear wave equation, well-posedness, ill-posedness, finite speed of propagation, oscillating second order ODE
Mathematical Subject Classification 2000
Primary: 34C25, 35L05, 49K40, 65F22
Received: 6 December 2009
Revised: 31 May 2010
Accepted: 29 June 2010
Published: 18 November 2011
Slim Ibrahim
Department of Mathematics and Statistics
University of Victoria
PO Box 3060 STN CSC
Victoria V8P 5C3
Mohamed Majdoub
Department of Mathematics
University of Tunis El Manar
Campus Universitaire
2092 Tunis
Nader Masmoudi
Courant Institute for Mathematical Sciences
New York University
New York, NY 10012-1185
United States