Some Problems of the Generalized Kuramoto-Sivashinsky Type Equations with Dispersive Effects

Abstract

The Kuramoto-Sivashinsky equation has been studied by many authors [1–4]. In this note, firstly we consider the following initial value problem for the generalized KS type equations with the dispersive effects $${u_t} + f{(u)_x} + \alpha {u_{xx}} + \varphi {(u)_{xx}} + \beta {u_{xxx}} + \gamma {u_{xxxx}} = g(u),$$ (1) $$u(x,0) = {u_0}(x),x \in {R^1},$$ (2) where α,β and γ are constants. f (x), ø(x) and g(x) are known functions. In n(n≥ 2) dimensional case, the following initial value problem for the system of generalized K-S type equations has been studied: $${\vec u_t} + \sum\limits_{i = 1}^n {\frac{\partial }{{\partial {x_i}}}} grad\varphi ({u_1}, \ldots ,{u_N}) + \alpha \Delta \vec u + \beta {\Delta ^2}\vec u = \vec g(\vec u),$$ (3) $$\vec u(x,0) = {\vec u_0}(x),x \in {R^n},n \geqslant 2,$$ (4) where \(\vec u(x,t) = ({u_1}(x,t), \ldots ,{u_N}(x,t))\) is a N dimensional unknown functional vector, ø(s 1,..., s N ), is a scalar function of its variables. Under some conditions the existence, nonexistence and asymptotic behaviour as t → ∞ for the global smooth solutions of the problem (1) (2) and problem (3) (4) are obtained.