Existence of spherically symmetric global solution to the semi-linear wave equation \(u_{tt}-\Delta u=au^ 2_ t+b(\nabla u)^ 2\) in five space dimensions (Q1056830)

The author shows the existence of global classical solutions to the Cauchy problem for the wave equation mentioned in the title in the case of spherically symmetric, smooth and small data. The main point is that the nonlinearity vanishes of second order at the origin and is independent of u. In this case the result was known before only if the space dimension is at least 6. Meanwhile, however, the result was shown to hold even without spherical symmetry, in any dimension \(n\geq 4\), and for a more general class of nonlinearities which vanish of second order at the origin [see \textit{S. Klainerman}, Commun. Pure Appl. Math. 38, 321-332 (1985)].