Abstract. In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schrödinger equation with a defocusing energysubcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767-808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 2015) to four dimension without radial assumption and the proof of scattering is based on the interaction Morawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schröinger problem in dimension d = 4 for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.
The connection between the leg members and diagonal members of the urban transmission line tower is mostly in the form of single-limb connection. This paper puts forward a new connection form of pipe double-limb double-plate connection joint, which is based on the model of key joints in an urban narrow base tower structure. The traditional pipe single-limb single-plate and new pipe double-limb double-plate joint are analyzed and studied from three aspects of theory, numerical simulation and experimental study. Through finite element analysis, it is obtained that the section stress of angle steel under eccentric load is 2.05 times of that under axial load, which is basically consistent with the 2.5 times of the theoretical calculation. This shows that the stress of the angle steel in the pipe double-limb double-plate joint is greatly reduced as the axial stress component, which can ensure the safety of the angle steel. Based on the theoretical analysis of the tensile force of two kinds of joints, through the test research and corresponding numerical simulation of pipe single-limb single-plate and pipe double-limb double-plate joints, under the same load, compared with pipe single-limb single-plate joints, the pipe double-limb double-plate joints designed in this paper can greatly reduce the stress of connection plates and members, and compared with the existing joint forms, the bending stress of joint plates can be reduced by about four times, which greatly improves the bearing capacity of the joint. The research on the pipe double-limb double-plate connection joint will provide the basis for the design of new connection joints of narrow base towers in urban areas.
In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov spaceḂ 3 1,1 ×Ḃ 2 1,1 (R 5 ). This is the five dimensional analogue of [6], which is the first result on the global well-posedness and scattering of the energy subcritical nonlinear wave equation without the uniform boundedness assumption on the critical Sobolev norms employed as a substitute of the missing conservation law with respect to the scaling invariance of the equation. The proof is based on exploiting the structure of the radial solution, developing the Strichartz-type estimates and incorporation of the strategy in [6], where we also avoid a logarithm-type loss by employing the inhomogeneous Strichartz estimates.2000 Mathematics Subject Classification. Primary: 35B40, 35Q40.
We revisit the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schrödinger equationwhen 0 < b < 1 and d = 2 in the 2D case. We use a modified version of Arora-Dodson-Murphy's approach [1] to give a new proof that extends the scattering results of [2] and avoids concentration compactness.
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