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

Well- and ill-posedness issues for energy supercritical waves