No abstract

Abstract.The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear term f (y) grows slower than |y| log 3/2 (1 + |y|) at infinity. In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.Mathematics Subject Classification. 35B37.

Optical/IR images of transition disks (TDs) have revealed deep intensity decrements in the rings of HAeBes HD 142527 and HD 100453, that can be interpreted as shadowing from sharply tilted inner disks, such that the outer disks are directly exposed to stellar light. Here we report similar dips in SPHERE+IRDIS differential polarized imaging (DPI) of TTauri DoAr 44. With a fairly axially symmetric ring in the sub mm radio continuum, DoAr 44 is likely also a warped system. We constrain the warp geometry by comparing radiative transfer predictions with the DPI data in H band (Q φ (H)) and with a re-processing of archival 336 GHz ALMA observations. The observed DPI shadows have coincident radio counterparts, but the intensity drops are much deeper in Q φ (H) (∼88%), compared to the shallow drops at 336 GHz (∼24%). Radiative transfer predictions with an inner disk tilt of ∼ 30 ± 5 deg approximately account for the observations. ALMA long-baseline observations should allow the observation of the warped gas kinematics inside the cavity of DoAr 44.

S U M M A R YThe Chilean subduction zone presents a unique opportunity to study trench outer rise deformation of the subducting oceanic lithosphere at different thermal ages. The shape of the outer rise for plate ages ranging from 0 to 50 Ma is predicted by using an elastic plate model with variable elastic thickness T e (x) as a function of the distance measured from the trench axis. In addition, the uncertainties of our results are estimated by performing a Monte Carlo-type analysis and we considered explicitly the sediment loading effect on the lithospheric flexure in regions where the trench is heavily sedimented.The results show a systematic reduction in T e of up to 50 per cent (or reduction in the flexural rigidity D of up to ∼90 per cent) from the peak of the outer rise to the trench axis. The reduction in T e and D observed in most of the bathymetric profiles is coincident with (i) high plate curvatures (>5 × 10 −7 m −1 ), (ii) strong bending moments (>10 16 N m), (iii) pervasive fracturing and faulting of the oceanic basement (as imaged by high-resolution bathymetry data) and (iv) reduction in crustal and mantle seismic velocities. The reduction in flexural rigidity towards the trench suggests a weakening of the oceanic lithosphere and is interpreted to be caused partially by fracturing and a likely increase in fluid-pore pressure. In general, our estimates do not show consistent increases in elastic thickness as a function of plate age. This result suggests that either T e is independent of plate age or T e depends strongly on other factors. These factors could include lithospheric weakening due to hydro-fracturing and the loading history of the plate prior to subduction.

Baudouin and Puel (2002 Inverse Problems 18 1537-54), investigated some inverse problems for the evolution Schrödinger equation by means of Carleman inequalities proved under a strict pseudoconvexity condition. We show here that new Carleman inequalities for the Schrödinger equation may be derived under a relaxed pseudoconvexity condition, which allows us to use degenerate weights with a spatial dependence of the type ψ(x) = x • e, where e is some fixed direction in R N . As a result, less restrictive boundary or internal observations are allowed to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schrödinger equation from a single boundary or internal measurement.