Decay of solutions of some nonlinear equations (Q1429017)

The object of the paper is the equation \[ u_t-\alpha u_{xx}+\beta u_{xxx}+\gamma u_{xxxxt}+ (g(u))_x= 0, \] where \(t> 0\), \(x\in\mathbb{R}\), \(\alpha\), \(\beta\), \(\gamma\) are real parameters and \(g(u)\) is a real function. For different particular values of the parameters, the above equation becomes a particular well known one. For instance, when \(\alpha> 0\), \(\beta= 0\) it becomes a Burgers equation, when \(\alpha= 0\), \(\beta> 0\), a Korteweg-de Vries equation, when \(\alpha> 0\), \(\beta> 0\), a Korteweg-de Vries-Burgers equation. The author considers the written equation as a general form of the mentioned particular ones. The principal results are contained in two sections of the paper. In section 2, the author studies the equation \[ u_t-\alpha u_{xx}+ u_{xxx}+ (g(u))_x= 0 \] (\(\beta= 1\), \(\gamma= 0\), \(\alpha> 0\)), for \(g\in \mathbb{C}^2\) and gives a criterion for the existence of traveling wave solutions of the form \(u(x,t)= \phi(x- ct)\). In section 3, he studies the asymptotic behaviour of the solution to the Rosenau-Burgers equation (\(\beta= 0\), \(\gamma=1\)), \(u_t-\alpha u_{xx}+ u_{xxxxt}+ (u^{p+1}/p+1)_x= 0\) for \(x\in\mathbb{R}\), \(t> 0\), in the initial condition \(u(x,0)= u_0(x)\to 0\) as \(x\to\pm\infty\); \(\alpha> 0\) and \(p\geq 1\), an integer. This part of the paper contains the main results of the study: the global existence of the Rosenau-Burgers equation solution and the properties of the global solution. [Cf. the editorial notice ibid. 65, No. 4, 569 (2008; Zbl 1149.35399)].