Search

In this paper, we are interested in some theoretical and numerical studies of a special case of a degenerate nonlinear Schrödinger equation namely the so-called Gross-Pitaevskii Equation(GPE). More precisely, we will treat in a first time the well-posedness of GPE model with a degeneracy occurring in the interior of the space variable domain, i.e ∃x 0 ∈ (0, L), s. t k(x 0) = 0, where k stands for the diffusion coefficient and L is a positive constant. Thereafter, we will focus ourselves on some numerical simulations showing the influence of a different parameters, especially the interior degeneracy, on the behavior of the wave solution corresponding to our model in a special case of the function k namely k(x) = |x − x 0| α, α ∈ (0, 1).

This item is concerned with a controllability result of a population dynamics cascade model with one control force and two different scattering coefficients. Such coefficients are allowed to be null on the left-hand side of the gene-type domain. To this end, we follow the classical track consisting to prove the observability inequality for the full associated adjoint system using the so-called Carleman estimates and the semigroups theory.

In this paper, we are interested in the null controllability of a linear population dynamics model with degenerate dispersion coefficient. We develop first a Carleman-type inequality for its adjoint system, and then an observability inequality which allows us to establish the existence of a control acting on a subset of the space domain which steers the population of a certain age to extinction in a finite time.