Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case
The author deals with the fourth-order Schrödinger equation of the form \[ i\partial_t u+\Delta^2 u +\varepsilon\Delta u +\lambda| u| ^{p-1}u=0,\quad (x, t)\in {\mathbb R}^{n+1}, \tag \(*\) \] where \(\lambda\in {\mathbb R}\), \(p\in (1, m-1]\), \(m=2n / (n-4)\) is the critical exponent for the embedding of \(H^2\) into Lebesgue's space. The main result is the next Theorem. Let \(n\geq 5\), \(\varepsilon\in {\mathbb R}\), \(\lambda\in {\mathbb R}_+\), and \(p\in (1, m-1]\). For any radially symmetrical data \(u_0\in H^2\) there exists a unique global solution \(u\in C({\mathbb R}; H^2)\) of \((*)\) such that \(u(0)=u_0\).
energy-critical fourth order Schrödinger equation
scientific article; zbMATH DE number 5290376