Nonlinear model of quasi-stationary process in crystalline semiconductor

We consider the question of global existence and asymptotics of small, smooth, and localized solutions of a certain pseudoparabolic equation in one dimension, posed on half-line x > 0 , ⎪⎨ ⎪⎩ ( 1−∂ 2 x ) ut = ∂ 2 x (u+α2 (|u|2 u))+α1 |u|1 u, x ∈ R+, t > 0, u(0,x) = u0 (x) , x ∈ R+, u(0,t) = h(t), (0.1) where αi ∈ R,qi > 0, i = 1,2,u : Rx × R+ t ∈ C. This model is motivated by the a wave equation for media with a strong spatial dispersion, which appear in the nonlinear theory of the quasy-stationary processes in the electric media. We show that the problem (0.1) admits global solutions whose long-time behavior depend on boundary data. More precisely, we prove global existence and modified by boundary scattering of solutions. Mathematics subject classification (2010): 35Q35, 35B40.