Focusing of spherical nonlinear pulses in \({\mathbb R}^{1+3}\) (Q2759030)

This paper constructs families of approximate radial solutions of \(u_{tt}-\Delta u + a|u_t|^{p-2}u_t=0\) in three space dimensions, with \(1<p<2\), \(a\in {\mathbb C}\), which depend on a small parameter \(\varepsilon\). The objective is to show that they are close to the exact solutions with Cauchy data of the form \((\varepsilon U_0(r,(r-r_0)/\varepsilon),U_1(r,(r-r_0)/\varepsilon)\), where the \(U_j\) (\(j=0\), 1) are compactly supported in their second argument. The approximate solutions are obtained by patching solutions of ODEs with solutions of the wave equation; their Cauchy data are determined by the leading part of the exact data in characteristic coordinates. The error in such an approximation is \(O(\varepsilon)\) in \(L^\infty\) for \(t\leq r_0-\delta\), \(\delta>0\), but only \(O(\varepsilon^{2-p})\) afterwards.