On the equation \(\square u=|\nabla u|^2\) in \(5+1\) dimensions
In this article, the local well-posedness of the Cauchy problem for the semilinear hyperbolic equation \(u_{tt}-\Delta u=G(u)|\nabla u|^2\), \((x,t)\in {\mathbb R}^n\times {\mathbb R}\), is studied. Using the spherical dyadic partition of the Fourier space, the author defines a space \(F^s\) having several useful properties which allow to obtain appropriate estimates for the nonlinear term in the equation. Using this space it is proved, under the assumptions \(n\geq 5\) and \(s>n/2\) and by standard iterations, that given initial data \((u_0,u_1)\in H^s({\mathbb R})\times H^{s-1}({\mathbb R})\) there exists a unique \(H^s\) local solution within the space \(F^s\). Furthermore, the solution depends analytically on the initial data.
scientific article; zbMATH DE number 1489539