The hyperbolic semilinear Cauchy problem for a scalar operator when the initial data are discontinuous on a submanifold (Q1900614)

Let \(P= P(t, x, D_t, D_x)\) be a strictly hyperbolic (in time direction) differential operator of second order with \(C^\infty\) coefficients. The author studies the existence and regularity of local solution for the Cauchy problem associated with the semilinear equation \(Pu= f(t, x, u, D_t u, D_x u)\) for the case when the initial data have singularities of conormal type on a submanifold of \(\mathbb{R}^n\). The solution is constructed as the limit of the following iterative scheme: Solve the Cauchy problem with the same initial data (as the original problem) associated with the linear equation \(Pu_{v+ 1}= f(t, x, u_v, D_t u_v, D_x u_v)\), \(v\geq 0\), \(Pu_0= 0\). A study of the linear problem is also done. To prove convergence of \(u_v\), certain estimates for \(f(t, x, u, D_t u, D_x u)\) in terms of \(u\) are needed. This requires the use of Gagliardo-Nirenberg inequalities and extensions of \(u\) and \(\nabla u= (D_t u, D_x u)\) to the whole \(\mathbb{R}^{n+ 1}\).