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In this paper, we study the ill-posdness of the Cauchy problem for semilinearwave equation with very low regularity, where the nonlinear term depends on $u$and $\partial_t u$. We prove a ill-posedness result for the "defocusing" case,and give an alternative proof for the supercritical "focusing" case, whichimproves our previous result in Chin. Ann. Math. Ser. B 26(3), 361-378, 2005.Comment: 11 pages, no figures; Mathematische Zeitschrift, accepte

Ill-posedness for semilinear wave equations with very low regularity