Hyperbolic conormal spaces and semilinear wave equation (Q1887192)

This paper deals with the semilinear Cauchy problem for the multidimensional wave equation. The initial data \(g_j\) are assumed to be conormal with respect to the origin, and the source term is polynomial with respect to the solution \(u\) and \(\nabla u\). More precisely, \(g_j\in I^{-s-{n\over 4}+ j}(\mathbb{R}^n_x,\{0\})\), \(j= 0,1\), \(s-j\) being the corresponding Besov index of the functional space \(^\infty H^{s-j}(\mathbb{R}^n)\) and \(s> {n+1\over 2}+ 1\). The conormal bundle of \(0\) in \(T^*(\mathbb{R}^{n+1})\setminus \{0\}\) is denoted by \(\Lambda_0\) and \(\Lambda_1\) stands for the closure in \(T^*(\mathbb{R}^{n+1})\setminus \{0\}\) of the conormal bundle to the light cone \(\Gamma\setminus\{0\}\). Then it is proved that microlocally in \(\Lambda_0\setminus\Lambda_1\), \(u\) belongs to the conormal Besov type space \(^\infty H^{s_0}(\mathbb{R}^{n+1},\{0\})\), with \(s_0= 2s- {n\over 2}+{1\over 2}\). This way, several known results on nonlinear type singularities to the solution \(u\) are improved.