Global existence of solutions of a nonlinear dispersive model (Q1910038)

The authors study the existence of global solutions in \(H^{\mu/2}(\mathbb{R})\) of the initial value problem \[ u_t-D^\mu u_x+ u^p u_x=0,\quad x\in\mathbb{R},\quad t\geq 0,\quad p>1,\quad \mu\geq 1,\quad u(x,0)=u_0(x),\tag{1} \] where \(D^\mu=-(\partial^2_x)^{\mu/2}\), which describe the propagation of weakly nonlinear waves in a dispersion medium. Denote \(|\cdot|_{s,2}\) the norm of the Sobolev space \(H^s(\mathbb{R})\), \(|\cdot|_2\) the norm of the Lebesgue space \(L^2(\mathbb{R})\) and for \(f:\mathbb{R} \times[0,T]\to\mathbb{R}\), \(T>0\), let \[ |f|_{L^p_x L^q_T}=\Biggl(\int^{+\infty}_{-\infty} \Biggl(\int^T_0|f(x,t)|^q dt\Biggr)^{p/q}dx\Biggr)^{1/p},\quad x\in\mathbb{R}. \] Suppose \(2<\mu\leq 3\) and \(u_0\in H^{\mu/2}(\mathbb{R})\). The authors establish two existence results. The local existence result states that if \((\mu+1)/4<s\leq \mu/2\), \(\beta=s+1-\mu/2\), then there exist \(T(|u_0|_{\mu/2,2})>0\), a unique solution \(u\) of (1) satisfying \[ u\in C([0,T],H^{\mu/2}(\mathbb{R})),\;|D^\beta D^{\mu/2}u|_{L_xL^2_T}<+\infty,\;|u_x|_{L^4_TL_x}\leq+\infty, |u|_{L^2_xL_T}\leq+\infty\tag{2} \] and a neighbourhood \(U_0\) of \(u_0\) such that the function from \(U_0\) into the class of functions \(u\) defined by (2) is Lipschitz. The proof of this result combines some linear estimates given by the authors with a contraction mapping principle. The global result states that the solution given in the local existence result can be extended for any \(T>0\) and under some conditions; moreover, if \(p\geq2\mu\) and \(T^*\) is the maximum value such that for all \(T\in[0,T^*)\) the solution of (1), with initial data \(u_0\), lies in \(C([0,T],H^{\mu/2}(\mathbb{R}))\), then either \(T^*=+\infty\) and the solution is global, or else \(\lim_{t\to T^*} |u(\cdot,t)|_\infty=+\infty\).