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Motivated by a classical work of Erd} os we give rather precise necessary and sucient growth conditions on the nonlinearity in a semilinear wave equation in order to have global existence for all initial data. Then we improve some former exact controllability theorems of Imanuvilov and Zuazua. R esum e .M otiv e par un travail classique d'Erd} os on donne des conditions n ecessaires et susantes de croissance de la non lin earit e dans une equation des ondes semilin eaire pour l'existence des solu- tions globales pour toutes les donn ees initiales. Ensuite on am eliore certains th eor emes ant erieurs de contr^ olabilit e exacte de Imanuvilov et de Zuazua.

Well posedness and control of semilinear wave equations with iterated logarithms