## Gain of regularity for a Korteweg-de Vries–Kawahara type equation.(English)Zbl 1064.35172

The author investigates the existence of local and global solutions together with the gain of regularity of the initial value problem associated with the Korteweg-de Vries–Kawahara (KdVK) equation perturbed by a dispersive term which appears in several fluid dynamical problems. In Section 1 the KdVK equation under study is presented, i.e. $u_t+ \eta u_{xxxxx}+ uu_x= 0\tag{1}$ with $$-\infty< x<+\infty$$, $$t> 0$$ and $$\eta\in\mathbb{R}$$.
The main Theorem 1.1, which is proved in Section 8, states that if the initial data $$u(x, 0)$$ decays faster than polynomially on $$\mathbb{R}^+= \{x\in\mathbb{R};x> 0\}$$ and posesses certain initial Sobolev regularity, then the solution $$u(x, t)\in C^\infty$$ for all $$t> 0$$.
In Section 2 some preliminaries are described, while in Section 3 the main inequality is demonstrated. It Section 4 the author proves an important a priori estimate. In Section 5 a basic local-in-time existence and uniqueness theorem is shown. Section 6 contains the proof of a fundamental global existence theorem, while in Section 7 a series of estimates for solutions of (1) in weighted Sobolev norms is developed. These provide a starting point for the a priori gain of regularity.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 47J35 Nonlinear evolution equations 35B65 Smoothness and regularity of solutions to PDEs
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