Global solutions for the KdV equation with unbounded data (Q1369875)

This paper is concerned with the global solvability of the initial value problem (IVP) associated to the KdV equation \[ \partial_t u_k+\partial_x^3 u+u\partial_x u=0\quad t>0,\;x\in \mathbb{R}\quad u(x,0)=u_0, \tag{1} \] with smooth unbouded initial data \(u_0\). The main result in this paper shows that for data \(u_0\) of the form \[ u_0(x)=a_kx^k+a_{k-1}x^{k-1}+\cdots+a_1x+a_0+f(x), \] with \(k\) odd, \(a_k>0\), and \(f\in{\mathcal S}(\mathbb{R})\) (Schwartz class) the IVP (1) has a global smooth solution, as shown in the following Theorem. For any data \(u_0\) in the class described above the IVP (1) has a unique global solution \(u(x,t)\) satisfying \(u\in C^\infty(\mathbb{R}\times[0,\infty))\), for any \(t>0\), \[ |u(x,t)|\leq c_T+\frac{|x|}{t},|\partial_xu(x,t)|\leq c_T+\frac{1}{t}\quad (x,t)\in\mathbb{R}\times (0,T), \] and \[ \int_0^T \left(\int_{-\infty}^\infty |(1+|x|^{j-2})\partial^j_xu(x,t)|^2dx\right)^{1/2} dt\leq c(T;j),\quad j=2,3,\cdots. \]