Global existence for rough solutions of a fourth-order nonlinear wave equation
Consider the Cauchy problem of the defocusing cubic forth-order wave equation \[ \partial^2_t u+\Delta^2 u+u+|u|^2u=0, \quad u|_{t=0}=u_0, \partial_t u|_{t=0}=u_1, \qquad (t,x)\in \mathbb R\times \mathbb R^n\;. \] It is easy to check that, for regular solutions, we have the conserved energy quantity \[ E(t)=\int_{\mathbb R^n} \left(|\partial_t u|^2+|\Delta u|^2+|u|^2+\tfrac{1}{2}|u|^4 \right)\,dx\simeq \|u\|_{H^2}^2+\|\partial_t u\|_{L^2}^2+\|u\|_{L^4}^4, \] which, as usual, can be exploited to prove the global well-posedness for the problem in \(H^2\times L^2\), when \(n<8\) (see \textit{B. Pausader} [J. Differ. Equations 241, No. 2, 237--278 (2007; Zbl 1145.35090)] for \(n\geq 5\) with general \(p\)). In the case \(n\leq 6\) (\(H^2\subset L^6\)), it follows from the classical energy argument without using the Strichartz type estimates. In this paper, the author studied the global well-posedness in \(H^s\times H^{s-2}\). The main result states that when \(3\leq n< 8\) and \(\min(\frac{n-2}{2},\frac{n}{4})<s<2\), the problem is globally well-posed in \(H^s\times H^{s-2}\). The proof follows from \textit{J. Bourgain}'s Fourier truncation idea [Int. Math. Res. Not. 1998, No. 5, 253--283 (1998; Zbl 0917.35126)], together with the Strichartz estimates for the corresponding linear equation in [\textit{B. Pausader}, loc. cit.].
defocusing cubic forth-order wave equation
scientific article; zbMATH DE number 5757905
scientific article