In this article we consider the (complex) Ginzburg-Landau equation, we discretize in time using the implicit Euler scheme, and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the global attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time-step approaches zero.

Considering the acoustic source scattering problems, when the source is non-radiating/invisible, we investigate the geometrical characterization for the underlying sources at polyhedral and conical corner. It is revealed that the non-radiating source with Hölder continuous regularity must vanish at the corner. Using this kind of geometrical characterization of non-radiating sources, we establish local and global unique determination for a source with the polyhedral or corona shape support by a single far field measurement. Uniqueness by a single far field measurement constitutes of a long standing problem in inverse scattering problems.

In this paper, we consider a mixed boundary value problem for the stationary Kirchhoff-type equation containing $p(·)$-Laplacian. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of at least one, two or infinitely many nontrivial weak solutions according to hypotheses on given functions.

As an application of the theoretical results, in this paper, we study the symmetric reduction and Hamilton-Jacobi theory for the underwater vehicle with two internal rotors as a regular point reducible RCH system, in the cases of coincident and non-coincident centers of the buoyancy and the gravity. At first, we give the regular point reduction and the two types of Hamilton-Jacobi equations for a regular controlled Hamiltonian (RCH) system with symmetry and a momentum map on the generalization of a semidirect product Lie group. Next, we derive precisely the geometric constraint conditions of the reduced symplectic forms for the dynamical vector fields of the regular point reducible controlled underwater vehicle-rotor system, that is, the two types of Hamilton-Jacobi equations for the reduced controlled underwater vehicle-rotor system, by calculations in detail. The work reveals the deeply internal relationships of the geometrical structures of the phase spaces, the dynamical vector fields and the controls of the system.