Long-time behavior of solutions of a BBM equation with generalized damping

Abstract

We study the long-time behavior of the solution of a damped BBM equation ut + ux − uxxt + uux + Lγ (u) = 0. The proposed dampings Lγ generalize standards ones, as parabolic (Lγ (u) = −∆u) or weak damping (Lγ (u) = γu) and allows us to consider a greater range. After establish the local well-posedness in the energy space, we investigate some numerical properties.

The symbol approximation method: a numerical approach to the approximation of the symbol of self-adjoint operators

“…We find in the literature different choices for L, depending on the physical situations, see [8,13,14,15,16,20,21,22,23,36,37] and [29] for physical experiments.…”

Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

“…At this point, we use ( 40) and obtain Unfortunately, in practice, this fixed point method converges only for very small values of ∆t. To enhance the stability region, and then to allow to take larger values of ∆t, we use the ∆ κ acceleration procedure introduced in [11] and applied in [1,2,19] for Allen-Cahn's, weakly damped Schrödinger and BBM equations respectively. In two words, the ∆ κ procedure consists in replacing the Picard iterates by…”

Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

“…The proof will be done by using a fixed point argument. Therefore, the application of the following lemma, proved in [4], will be needed: and A is the compact operator defined by (10). Thus, we obtain that the solution of (74) is given by…”

“…Despite of the developments obtained for Boussinesq systems, there are many issues still open that deserves further attention, specially when dissipative mechanisms are incorporated to the models. In real physical situations, dissipative effects are often as important as nonlinear and dispersive effects (see, for instance, [5,6,10]) and this fact has given currency to the study of water wave model in nonlinear dispersive media. Indeed, it was clear from the experimental outcomes that damping effects must be accounted in addition to those of nonlinearity and dispersion for good quantitative agreement with model predictions.…”