Evolution of semilinear waves with swallowtail singularities (Q1341284)

The author considers the formation of caustics for bounded solutions of a semilinear wave equation \(Pu = f(z,u)\), \(z \in \Omega \subset \mathbb{R}^n\), \(f\) smooth, with \(P\) a second order strictly hyperbolic differential operator, and shows that in the case of the swallowtail caustic and conormal initial data the only additional singularities of the solution \(u\) due to the nonlinearity of the equation lie on the forward characteristic cone over the swallowtail tip. Firstly the formation of caustics for the linear wave equation is considered. Then many propositions and theorems are demonstrated after introduction of a family of spaces \(J_K (\Omega)\), \(K \in\mathbb{N}\), satisfying particular algebraical conditions.