组合KdV-Burgers方程扭状孤波解的渐近稳定性

Asymptotic Stability of Kink Profile Wave Solutions of the Compound KdV-Burgers Equation

对组合KdV-Burgers方程单调递减扭状孤波解的渐近稳定性进行了研究。首先推导出该扭状孤波解的一阶、二阶导数的估计,然后再利用L^2能量估计方法和Young不等式,解决了方程中非线性项难以估计的问题,证明了该单调递减扭状孤波解在H^1中是渐近稳定的。进一步利用L^2估计方法和Gargliado-Nirenberg不等式,得到了扰动在L^2与L^∞范数意义下的衰减速率分别为(1+t)^-1/2和(1+t)^-1/4。

The asymptotic stability of monotone decreasing kink profile solitary wave solutions of the compound KdV-Burgers equation was studied.The estimate of the first-order and second-order derivatives of monotone decreasing kink profile solitary wave solutions was obtained and the difficulties caused by nonlinear terms in the compound KdV-Burgers equation in the estimation were overcome by using the L^2 energy estimate method and Young’s inequality.It is proved that the monotone decreasing kink profile solitary wave solution is asymptotically stable in H^1.Moreover,the decay rates of in the sense of L2 and L^∞ norm respectively are (1+t)^-1/2 and (1+t)^-1/4 by using the L^2 estimate method and Gargliado-Nirenberg inequality.