On uniqueness of weak solutions to semilinear wave equations

Abstract

Weak solutions to Cauchy problems of the form are considered for n≧3 , p > 1 . The data are assumed to be only of (locally) finite energy, and solutions u are to satisfy Uniqueness is obtained under any of the following three condition i) ii)solutions are radial and satisfy a certain well-known space-time estimate, and iii) for n = 3 and 1 < p < 5 , solutions which satisfy either an appropriate “cone estimate” or the abovementioned space-time estimate (due to Morawetz).