This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.

The blowup is studied for the nonlinear Schrödinger equation $iu_{t}+\Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ blows up in finite or infinite time. A new proof is also presented for the previous result in [9], in which a similar result in a case of energy-subcritical was shown.

Of concern is the protection from overheating of an isotropically conducting body by an anisotropically conducting coating which is thin compared to the scale of the body. We assume either that the whole thermal tensor of the coating is small or that it is small in the directions normal to the body (a case we call “optimally aligned coating”). We study the asymptotic behavior of the solution to the heat equation with Dirichlet boundary conditions on the outer surface of the coating, as the thickness of the coating shrinks. We obtain the exact scaling relations between the thermal tensor and the thickness of the coating so that the effective (limiting) condition on the boundary of the body is of Dirichlet, Robin or Neumann type, with the last condition indicating good insulation.

In the field of aircraft manufacturing, the demand for high aerodynamic performance and long fatigue life is growing. This makes the high-precision assembly of large components and assembly coordination increasingly important. This is particularly prominent for wing-fuselage joining due to the large size, measurement difficulty, and multiconstraint coordination. In this article, we propose a novel optimization method for the wing-fuselage assembly coordination problem, as well as an assembly gap measurement approach. First, a gap measurement framework combining a laser tracker, 3-D scanner, and photogrammetry is adopted to solve the occlusion problem that makes measurement difficult in traditional methods. Compatible tooling is utilized to transform the scanned point cloud data in the local coordinate system to the global aircraft assembly coordinate system. Based on this, a comprehensive wing-fuselage pose coordination model is then established, which takes engineering constraints into account to realize assembly gap redistribution for wing-fuselage joining. The transformation parameters of pose adjustment are solved by the proposed multiconstraint optimization algorithm based on the Gauss–Newton method. Finally, experiments on synthetic data and actual data have verified the effectiveness of the coordination method. Results show that our method significantly optimizes the gap distribution within the given tolerance and outperforms traditional methods in both accuracy and efficiency. With advantages of high efficiency and accuracy, our method can be well applied to the wing-fuselage joining and satisfy the requirement of the large component assembly.

International Press of Boston - publishers of scholarly mathematical and scientific journals and books