A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on \(\mathbb S^1\) (Q2880710)

The memoir under review proves the almost global existence for the quasilinear Klein-Gordon equation on \(S^1\). Previously, the almost global existence for the semilinear Klein-Gordon equation on \(S^1\) had been obtained in [\textit{D. Bambusi}, Commun. Math. Phys. 234, No. 2, 253--285 (2003; Zbl 1032.37051)]. Hence this paper is to generalize the semilinear case to the quasilinear case. More precisely, the main result is stated as follows: Consider the initial value problem NEWLINE\[NEWLINE(\partial_t^2-\partial_x^2+m^2)v=\frac{\partial}{\partial x}[H'_3(x,v,v_x)]-H'_2(x,v,v_x),NEWLINE\]NEWLINE NEWLINE\[NEWLINEv|_{t=0}=\epsilon v_0, v_t|_{t=0}=\epsilon v_1.NEWLINE\]NEWLINE NEWLINEThere is a subset \(N\subset (0,+\infty)\) of zero measure such that for any \(m\in (0,+\infty)\setminus N\) and for any \(M\in \mathbb N\) there is \(s_0\in\mathbb N\) such that for any \(s\geq s_0\) there exist \(\epsilon_0\in (0,1)\), \(c>0\) satisfying the following: For any \(\epsilon\in(0,\epsilon_0)\), and any pair \((v_0,v_1)\) in the unit ball of \(H^{s+1/2}(S^1;\mathbb R)\times H^{s-1/2}(S^1;\mathbb R)\) the above equation has a unique solution \(v\) defined on \((-T_\epsilon,T_\epsilon)\times S^1\) with \(T_\epsilon\geq c\epsilon^{-M}\) and belonging to the space \(C_b^0((-T_\epsilon,T_\epsilon),H^{s+1/2}(S^1;\mathbb R))\times C_b^1((-T_\epsilon,T_\epsilon),H^{s-1/2}(S^1;\mathbb R))\), where \(C_b^j((-T_\epsilon,T_\epsilon),E)\) denotes the space of \(C^j\) functions on the interval \((-T_\epsilon,T_\epsilon)\) with values in the space \(E\) whose derivatives up to order \(j\) are bounded in \(E\) uniformly on \((-T_\epsilon,T_\epsilon)\).