Scattering of solutions with group invariance for the fourth-order nonlinear Schrödinger equation (Q6566566)

The authors consider the focusing fourth-order nonlinear Schrödinger equation\N\[\N i\partial_t u -\Delta^2 u +\mu\Delta u = -|u|^{p-1}u,\quad x\in \mathbb R^d,\tag{1}\N\]\Nwhere \(\mu\ge 0\) and the nonlinearity is \(L^2\)-supercritical and \(\dot H^2\)-subcritical,\N\[\N1+\frac{8}{d}<p< \begin{cases} \infty& \text{if }1\le d\le 4,\\\N1+\frac{8}{d-4}& \text{if }d\ge 5. \end{cases}\N\]\NSeveral previous results consider this equation in the case of radial initial data. In order to relax this assumption, the authors consider a subgroup \(G\) of \(\mathbb R/2\pi\mathbb Z\times O(d)\) (equipped with the binary operation \((+,\cdot)\)), where \(O(d)\) denotes the set of \(d\times d\) (real) orthogonal matrices. The Sobolev space with \(G\)-invariance is defined by\N\[\NH_G^k=\{f\in H^k(\mathbb R^d)\quad | f=\mathcal G f,\ \forall \mathcal G\in G\},\N\]\Nwhere \(\mathcal G f(x) = e^{-i\theta} f( \mathcal G^{-1}x)\) whenever \(\mathcal G=(\theta,\mathcal G)\in G\).\N\NFor \(\omega>0\), the action is defined by \(S_\omega(f) = E(f)+\frac{\omega}{2}M(f)\), where the mass \(M\) and the energy \(E\), which are preserved by the flow of (1), are defined by\N\begin{align*}\N& M(f) = \int_{\mathbb R^d}|f(x)|^2dx,\\\N& E(f) = \frac{1}{2}\int_{\mathbb R^d} |\Delta fx)|^2dx +\frac{\mu}{2} \int_{\mathbb R^d} |\nabla f(x)|^2dx -\frac{1}{p+1}\int_{\mathbb R^d} |f(x)|^{p+1}dx.\N\end{align*}\NA function \(Q_\omega\) is a ground state to\N\[\N\tag{2} -\omega\phi-\Delta^2\phi+\mu\Delta \phi+|\phi|^{p-1}\phi=0\N\]\Nif\N\[\NS_\omega(Q_\omega) = \inf \{S_\omega(\phi)\ |\ \phi\in H^2(\mathbb R\setminus\{0\},\ \phi\text{ solves }(2)\}.\N\]\NThe virial functional \(K\) is defined by\N\begin{align*}\NK(f) & = \partial_{\lambda =1}S_\omega\left( \lambda^{d/2} f(\lambda x)\right) \\\N& =2 \int_{\mathbb R^d} |\Delta f(x)|^2dx +\mu \int_{\mathbb R^d}|\nabla f(x)|^2dx -\frac{d(p-1)}{2(p+1)} \int_{\mathbb R^d} |f(x)|^{p+1}dx,\N\end{align*}\Nand \(S_\omega(Q_\omega) \) satisfies\N\[\NS_\omega(Q_\omega) = \inf \{S_\omega(\phi)\ |\ \phi\in H^2(\mathbb R\setminus\{0\},\ K(\phi)=0\}.\N\]\NThe conjecture stated by the authors reads as follows: If \(u_0\in H^2(\mathbb R^d)\) satisfies\N\[\NS_\omega(u_0)<S_\omega(Q_\omega)\quad\text{and}\quad K(u_0)>0\N\]\Nfor some \(\omega>0\), then there exists a unique global solution \(u\in C(\mathbb R,H^2(\mathbb R))\) to (1) with initial data \(u(0)=u_0\). Moreover, \(u\) scatters forward and backward in time, in the sense that there exist \(\phi^\pm\in H^2(\mathbb R^d)\) such that\N\[\N\lim_{t\to \pm \infty} \|u(t) - e^{-it(\Delta^2-\mu\Delta)}\phi^\pm\|_{H^2}=0.\N\]\NThe main result of the paper consists in proving scattering of group-invariant solutions below the ground state threshold, under the hypothesis that the threshold for group-invariant solutions is less that a certain value, whose expression requires the introduction of several other notations.\N\NUsing Strichartz estimates, scattering is characterized by the finiteness of \(\|u\|_{X(\mathbb R)}\), where, for \(I\subset \mathbb R\), the scattering size is defined by\N\[\N\|u\|_{X(I)}= \|u\|_{L^{q_1}_tL^r(I\times \mathbb R^d)},\quad r=p+1,\ q_1 = \frac{4(p-1)(p+1)}{8-(d-4)(p-1)}.\N\]\NFor \(\omega>0\) and \(L\ge 0\), define\N\[\NC_\omega(L) = \sup\{\|u\|_{X(I)},\quad S_\omega(u_0)\le L,\ K(u_0)>0\},\N\]\Nwhere \(u\) solves (1) on \(I\times \mathbb R^d\), with initial data \(u_0\in H^2(\mathbb R^d)\), and\N\[\NL_\omega^* = \sup\{L\in [0,\infty),\quad C_\omega(L)<\infty\}.\N\]\NOne always has \(L_\omega^*\le S_\omega(Q_\omega)\). The above conjecture is equivalent to the identity \(L_\omega^* =S_\omega(Q_\omega)\). The criterion for \(G\)-invariant solutions is defined by\N\[\NC_{G,\omega}(L) = \sup\{\|u\|_{X(I)},\quad S_\omega(u_0)\le L,\ K(u_0)>0\},\N\]\Nwhere the initial data are now restricted to \(u_0\in H_G^2\). Similarly,\N\[\NL_{G,\omega}^* = \sup\{L\in [0,\infty),\quad C_{G,\omega}(L)<\infty\}.\N\]\NSince \(C_{G,\omega}(L)\le C_\omega(L)\), one always has \(L_{G,\omega}^* \ge L_\omega^*\). The main result can then be stated more precisely: If \(L_{G,\omega}^*<m_{G,\omega}\), then \(L_{G,\omega}^*\ge S_\omega(Q_\omega)\).\N\NThe quantity \(m_{G,\omega}\) is defined by\N\[\Nm_{G,\omega} =\inf_{|x|=1}(\sharp Gx)L_{G_x,\omega}^*,\N\]\Nwhere \(Gx = \{\mathcal G x\ | \ \mathcal G\in G\}\) and \(G_x = \{\mathcal G\in G\ |\ \mathcal Gx=x\}\).\N\NThe scheme of the proof follows a rather standard path (technical details remain demanding), relying on small data theory, long-time perturbation argument, existence of a critical solution leading to a contradiction. A central role is played by a profile decomposition for sequences of \(G\)-invariant functions (uniformly bounded in \(H^2(\mathbb R^d)\)).