Riccati equations for abnormal time scale quadratic functionals

This paper focuses on developing new Riccati type conditions for an abnormal time scale symplectic system (S). These conditions provide characterizations of the nonnegativity (with and without a certain “image condition”) and positivity of the quadratic functionals associated with such a system. The novelty of these conditions rely on the natural conjoined basis (Xa,Ua) of (S) in which Xa(t) is not necessarily invertible, and thus the system (S) could be abnormal. These results are new even in the special case of continuous time, as are some of them in the discrete time setting.     

On progressive functions

Progressive functions at time t involve only the progressive functions at time before t and some nice compactly supported function at time t. We give sufficient conditions and explicit formulas to construct progressive functions with exponential decay and characterize the conditions on which the positive integer translates of a progressive function are orthonormal or a Riesz sequence. We provide explicit ways for construction of orthonormal progressive functions and for construction of the biorthogonal functions of nonorthogonal progressive functions. Such progressive functions can be used to construct wavelets with arbitrary smoothness on the half line if they are generated by a smooth refinable compactly supported function.     

Impulsive systems with hybrid delayed impulses: Input-to-state stability

This paper investigates input-to-state stability (ISS) and integral-input-to-state stability (iISS) of nonlinear impulsive systems with hybrid delayed impulses. Based on Lyapunov method, some sufficient conditions ensuring ISS and iISS of impulsive systems are obtained, where the time derivative of Lyapunov function is indefinite, and the hybrid effects of delayed impulses are also fully considered. It is shown that the impulsive system is ISS provided that the combined action of time delay existing in impulses, continuous dynamic, and the cumulative strength of hybrid impulses satisfies some conditions, even if the hybrid delayed impulses play a destabilizing effect on ISS. Examples and their simulations are presented to illustrate the applicability of the proposed results.     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I

We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived.     

Behaviour at time t = 0 of the solutions of semi-linear evolution equations

We investigate the regularity at time t = 0 of the solutions of linear and semi-linear evolutions equations (including the Stokes and Navier-Stokes equations), Necessary and sufficient conditions on the data for an arbitrary order of regularity are given (the classical “compatibility conditions”). In the case of the Stokes and Navier-Stokes equations the compatibility conditions which we find are global conditions on the data. The presentation given here seems to improve and generalize the known results even in the simplest case of linear evolution equations.     

Time reversed absorbing conditions

The aim of this note is to introduce the time reversed absorbing conditions (TRAC) in time reversal methods. These new boundary conditions enable one to “recreate the past” without knowing the source which has emitted the signals that are back-propagated. This new method does not rely on any a priori knowledge of the physical properties of the inclusion. We prove an energy estimate for the resulting non-standard boundary value problem. Two applications to inverse problems are given.     

A compactness result for perturbed semigroups and application to a transport model

In this paper we are concerned with the compactness properties of remainder terms of the Dyson-Phillips expansion of perturbed semigroups on general Banach spaces. More specifically, we derive conditions which ensure the compactness of the remainder term Rn(t) for some integer n. Our result applies directly to discuss the time asymptotic behaviour (for large times) of the solution of a one-dimensional transport equation with reentry boundary conditions on L¹-spaces without regularity conditions on the initial data.     

Sturm-Liouville operators with measure-valued coefficients

We give a comprehensive treatment of Sturm-Liouville operators whose coefficients are measures, including a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh-Kodaira theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm-Liouville operators, Sturm-Liouville operators with (local and non-local) δ and δ′ interactions or transmission conditions as well as eigenparameter dependent boundary conditions, Krein string operators, Lax operators arising in the treatment of the Camassa-Holm equation, Jacobi operators, and Sturm-Liouville operators on time scales as special cases.     

First order dynamic inclusions on time scales

In this paper, we study existence of solutions of first order dynamic inclusions on time scales with general boundary conditions. Both the ∇-derivative and Δ-derivative cases are considered. Examples are presented to illustrate that the Δ-derivative case needs more restrictive assumptions.     

Abstract time-dependent transport equations

We consider the abstract time-dependent linear transport equation as an initial-boundary value evolution problem in the Banach spaces L_p, 1 ⩽ p < ∞, or on a space of measures on a (possibly time-dependent) kinetic phase space. Existence, uniqueness, dissipativity, and positivity results are proved for very general, possibly time-dependent, transport operators and boundary conditions. When the phase space, boundary conditions, and transport operator are independent of time, corresponding results are obtained for the associated semigroup.     

Permanence of an N-species cooperation system with time delays and feedback controls on time scales

In this paper, an N-species cooperation system with time delays and feedback controls is proposed on time scales. In this process, some inequalities, which play a vital roles in the proof of main results, are firstly proved. Then, by applying the theory of differential inequality, sufficient conditions which guarantee the permanence of the system are obtained on time scales. Our results improve and complement some known results to some degree in the literature.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Permanence of a discrete nonlinear N-species cooperation system with time delays and feedback controls

A discrete nonlinear N-species cooperation system with time delays and feedback controls is considered in this paper. Sufficient conditions which ensure the permanence of the system are obtained. It is shown that these conditions are weaker than those of Chen [F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186(2007) 23-29], that is, our investigation shows that the additional condition in Chen’s paper is not necessary.     

Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

This paper deals for the first time with the Dirichlet problem for discrete (PD), discrete approximation problem on a uniform grid and differential (PC) inclusions of elliptic type. In the form of Euler-Lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.     

Characterizations of adjacency on the branching polyhedron

Given two distinct branchings of a directed graph G, we present several conditions which are equivalent to the corresponding incidence vectors of the branchings being adjacent on the branching polyhedron of G. The proof of these equivalences uses a “shrinking algorithm” which will determine in O(n²) time and space whether or not the incidence vectors are adjacent.     

Numerical generation of transparent boundary conditions on the side surface of a vertical transverse isotropic layer

We consider elastodynamics in transversely isotropic media with vertical symmetry axis. The governing equations are the two-dimensional second-order system for displacements. A numerical method for generating transparent boundary conditions on the cylindrical surface is proposed. The correspondent operator is non-local in both z-direction and time: it handles low-frequency spatial harmonics of the solution convolving their Fourier coefficients with sums-of-exponentials kernels with respect to time. Test calculations show high accuracy, efficiency, and stability of the proposed non-reflecting conditions even for those media parameters where PML fails.     

Dynamic boundary value problems of the second-order: Bernstein–Nagumo conditions and solvability

This article analyzes the solvability of second-order, nonlinear dynamic boundary value problems (BVPs) on time scales. New Bernstein–Nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the BVPs under consideration. Topological methods are then employed to gain solvability.     

Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in Rn, n=2,3, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in Rn. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1.     

Time scale symplectic systems without normality

We present a theory of the definiteness (nonnegativity and positivity) of a quadratic functional F over a bounded time scale. The results are given in terms of a time scale symplectic system (S), which is a time scale linear system that generalizes and unifies the linear Hamiltonian differential system and discrete symplectic system. The novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of F, and in establishing equivalent conditions for the nonnegativity of F without any normality assumption. To reach this goal, a new notion of generalized focal points for conjoined bases (X,U) of (S) is introduced, results on the piecewise-constant kernel of X(t) are obtained, and various Picone-type identities are derived under the piecewise-constant kernel condition. The results of this paper generalize and unify recent ones in each of the discrete and continuous time setting, and constitute a keystone for further development in this theory.