Asymptotic Analysis

We obtain a dispersive long-time decay in weighted energy norms for solutions to the 3D wave equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schrödinger equation.

We study the resonant set of a two-level Schrödinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr–Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.

We study an explicit interior observability estimate for the Schrödinger equation with potential in a bounded domain of R^N (N≥1), and with zero Dirichlet boundary condition. The method combines Carleman inequality and multiplier techniques. Moreover, by microlocal analysis tools, we prove a precised observability estimate in high frequencies.

In this paper, we study the asymptotic behavior of solutions to a chemotaxis model with volume-filling effect subject to the homogeneous Neumann boundary conditions. We establish a non-smooth version of Łojasiewicz–Simon inequality, and we prove that as time goes to infinity the solution to our system converges to an equilibrium in W^1,p(Ω)×W^1,p(Ω), p>max (n, 2), Ω⊂Rⁿ. We also obtain an estimate of the decay rate to equilibrium.

We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity. We reveal that the depth of the potential well plays an important role in the estimate. Particularly, in one-dimensional case, we give a much more elaborate characterization for the eigenvalue. Some numerical examples are also shown.