一类具弱阻尼的奇性扰动Boussinesq型方程的位势井方法

Potential Well Method for the Damped Singularly Perturbed Boussinesq-type Equation

采用位势井方法研究一类具弱阻尼的奇性扰动Boussinesq型方程的初边值问题utt-uxx-αux4-βux6+but=σ(u)xx,x∈Ω,t>0,u(0,t)=u(1,t)=uxx(0,t)=uxx(1,t)=ux4(0,t)=ux4(1,t)=0,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,其中uxi=ixui,σ(s)是一个已知的非线性函数,α和β是两个正的实常数,b≥0是任意实数,Ω=(0,1).得到了相应初边值问题整体广义解的存在唯一性.

The initial boundary value problem for the damped singularly perturbed Boussinesq-type equation utt-uxx-αux4-βux6+but=σ(u)xx,x∈Ω,t>0,u(0,t)=u(1,t)=uxx(0,t)=uxx(1,t)=ux4(0,t)=ux4(1,t)=0,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,is studied by the potential well method,where uxi=iuxi,σ(s) is a given nonlinear function,α and β are two positive constants,b≥0 is a real number,and Ω=(0,1).The existence and uniqueness of the global generalized solution to the problem are obtained by the potential well method.