Class Of Second Order Evolution Equations Research Articles

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

SemiGlobal Exact Controllability of Nonlinear Plates

The exact controllability problem for several semilinear thin plate models is considered. A distributed control affects a collar of the plate's boundary. The result is semiglobal in the sense that there is no restriction on the size of the initial and the target states, but the controllability time is uniform only with respect to a given bounded set containing these states. Both (i) closed-loop-based and (ii) “pure” open-loop constructions are discussed. Strategy (i) describes exact controls for an abstract class of second-order evolution equations. It applies to the Berger plate with small in-plane stresses (and, depending on some open questions, possibly to the von Kármán model). This method partly relies on uniform stabilization and offers no apparent leeway to improve the controllability time. Strategy (ii) is demonstrated on the example of a Kirchhoff model with a dissipative polynomial source term. Such a source serves as a prototype for ultimately considering the same control construction for the Berger and von Kármán equations. The bound on the control time in this case is still open, but this method offers a conjectural possibility to sharpen the minimal control time estimate.

Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization

Global existence of solutions for a class of second-order evolution equations with damping is shown by proving convergence of a full discretization. The discretization combines a fully implicit time stepping with a Galerkin scheme. The operator acting on the zero-order term is assumed to be a potential operator where the potential may be nonconvex. A linear, symmetric operator is assumed to be acting on the first-order term. Applications arise in nonlinear viscoelasticity and elastodynamics.

Conditional Lie–Bäcklund symmetries and sign-invariants to second-order evolution equations

The present paper discusses a class of second-order evolution equations which generalize reaction-diffusion–convection equations. It is shown that some equations admit of second-order conditional Lie–Bäcklund symmetries and first-order Hamilton–Jacobi sign-invariants, which preserve both signs, ⩾0 and ⩽0, on the solution manifold. Some examples are given to present the corresponding symmetry reductions, which reduce the considered equations to two-dimensional dynamical systems.