Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation. (1997) - Varlamov, Vladimir V.
Abstract : For the damped Boussinesq equationu t t − 2 b u t x x = − α u x x x x + u x x + β ( u 2 ) x x, x ∈ ( 0, π ), t > 0 ; α, b = c o n s t > 0, β = c o n s t ∈ R 1, the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtained in the explicit form and the question of the blow up of the solution in a certain case is examined. The possibility of passing to the limitb → + 0 in the constructed solution is investigated.