This issuePrevious ArticleHausdorff dimension, strong hyperbolicity and complex dynamicsNext ArticleOn the behaviour of the solutions for a class of nonlinear elliptic problems in exterior domains
On the initial boundary value problem for the damped Boussinesq equation
The first initial-boundary value problem for the damped Boussinesq equation $
u_{t t}-2bu_{t x x}=-\alpha u_{x x x x}+u_{x x}+\beta (u^2)_{x x}, x\in (0,\pi
),\quad t>0,$ with $\alpha, b=const>0,\quad \beta =const\in R^1,$ is considered with
small initial data. For the most interesting case $\alpha >b^2$
corresponding to an infinite number of damped oscillations its solution is
constructed in the form of a Fourier series which coefficients in their own
turn are represented as series in small parameter present in the initial
conditions. The solution of the corresponding problem for the classical
Boussinesq equation on $[0,T],\quad T<+\infty,$ is obtained by means of passing
to the limit $b\rightarrow +0.$ Long-time asymptotics of the solution in
question is calculated which shows the presence of the damped oscillations
decaying exponentially in time. This is in contrast with the long time
behavior of the solution of the periodic problem studied in [30] which major
term increases linearly with time.