Systems Of Differential Inequalities Research Articles

Blow-up for some nonautonomous differential equations and inequalities with deviating arguments

In this article, we provide sufficient conditions for the blow-up of solutions to certain nonautonomous differential equations and inequalities with advanced and delayed arguments. Furthermore, we consider the blow-up of solutions for some classes of nonautonomous coupled systems of differential inequalities.

The singular limit problem in a phase separation model with different diffusion rates $^*$

In this paper we study the singularly perturbed parabolic system ofcompeting species. This problem exhibit a ``phase separation'phenomena when the interaction between different species is verystrong. We are concerned with the case where different species mayhave different diffusion rates. We identify its singular limit withthe heat flow (i.e. gradient flow) of harmonic maps into a metricspace with non-positive curvature, by establishing the system ofdifferential inequalities satisfied by this heat flow and uniquenessof the solution to the corresponding initial-boundary value problem.

Planar undulator motion excited by a fixed traveling wave: Quasiperiodic averaging, normal forms, and the free electron laser pendulum

We present a mathematical analysis of planar motion of energetic electrons moving through a planar dipole undulator, excited by a fixed planar polarized plane wave Maxwell field in the X-Ray Free Electron Laser (FEL) regime. Our starting point is the 6D Lorentz system, which allows planar motions, and we examine this dynamical system as the wavelength of the traveling wave varies. By scalings and transformations the 6D system is reduced, without approximation, to a 2D system in a form for a rigorous asymptotic analysis using the Method of Averaging (MoA), a long time perturbation theory. The two dependent variables are a scaled energy deviation and a generalization of the so-called ponderomotive phase. As the wavelength varies the system passes through resonant and nonresonant (NR) zones and we develop NR and near-to-resonant (NtoR) normal form approximations. For a special initial condition and on resonance, the NtoR normal form reduces to the well-known FEL pendulum system. We then state and prove NR and NtoR first-order averaging theorems which give near optimal error bounds for the MoA approximations. The proofs are novel in that they do not use a near identity transformation and they use a system of differential inequalities. The NR case is an example of quasiperiodic averaging where the small divisor problem enters in the simplest possible way. To our knowledge the planar problem has not been analyzed with the generality we aspire to here nor has the standard FEL pendulum system been derived with associated error bounds as we do here.

Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system

This paper deals with the blow-up properties and asymptotic behavior of solutions to a semilinear integrodifferential system with nonlocal reaction terms in space and time. The blow-up conditions are given by a variant of the eigenfunction method combined with new properties on systems of differential inequalities. At the same time, the blow-up set is obtained. For some special cases, the asymptotic behavior of the blow-up solution is precisely characterized.

Nonlinear systems of differential inequalities and solvability of certain boundary value problems

Nonlinear systems of differential inequalities and solvability of certain boundary value problems

Nonexistence results of solutions to systems of semilinear differential inequalities on the Heisenberg group

We establish nonexistence results to systems of differential inequalities on the (2N + 1)‐Heisenberg group. The systems considered here are of the type (ESm). These nonexistence results hold for N less than critical exponents which depend on pi and γi, 1 ≤ i ≤ m. Our results improve the known estimates of the critical exponent.

Blow-up in Nonlocal Reaction-Diffusion Equations

We present new blow-up results for reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of the unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. For each type of problems, we give finite time blow-up results which significantly improve or extend previous results of several authors. In some cases, when the nonlocal source term is in competition with a local dissipative or convective term, optimal conditions on the parameters for finite time blow-up or global existence are obtained. Our proofs rely on comparison techniques and on a variant of the eigenfunction method combined with new properties on systems of differential inequalities. Moreover, a unified local existence theory for general nonlocal semilinear parabolic equations is developed.

Global asymptotic stability of a class of feedback neural networks with an application to optimization problems

By applying results from homotopy theory, new conditions are obtained for the existence and uniqueness of an equilibrium for a class of continuous-time feedback neural networks which contains the Hopfield model as a special case. Next, new criteria are established for the global asymptotic stability of the unique equilibrium of this class of neural networks by utilizing Lur'e-type Lyapunov functions and the stability theory for systems of differential inequalities. Several practical stability testing conditions are given. As a special case, criteria are derived for the global asymptotic stability of Hopfield neural networks. This is followed by a robustness analysis of the class of neural networks considered. The results obtained are then applied to an optimization problem.

NSpecies Competition for a Spatially Heterogeneous Prey with Neumann Boundary Conditions: Steady States and Stability

The article considers a reaction-diffusion equation coupled with a system of ordinary differential equations describing Volterra–Lotka type interactions. The reaction-diffusion equation is related to the prey with spatially heterogeneous growth rates, and subjected to homogeneous Neumann boundary conditions. The system of ordinary differential equations describes N competing predators or capital investments exploiting the prey.Careful analysis of an iterative scheme proves the existence of solutions to the initial value problem in different appropriate function spaces for the prey and predators. Existence of a positive steady state is proved by means of upper and lower solutions. A criterion of stability for the steady state is deduced by means of considering systems of differential inequalities.

Differential inequalities for systems of degenerate parabolic operators

The Prodi-Westphal theorem is generalized to systems of differential inequalities for nonlin ear nonuniformly parabolic operators of the form [d] The function F[d] is defined for (x,t) in a hypercylinder [d] where G is a bounded open domain in Rm. Suppose that F[d] is elliptic with respect to [d] and that [d] Since it is admissible that [d] at some points of [d] may be nonuniformly parabolic. Two additional theorems on differential inequalities for systems are proved and the results are applied to the paroblem of investigating stability of solutions of certain systems of degenerate parabolic equations,[d]

Systems of differential inequalities and stochastic differential equations. III

Consider the system of stochastic differential equations x′ (t,ω) =f (t,x (t,ω),ω), x (t0,ω) =x0(ω), where f (t,x (t,ω),ω) is a product measurable n-dimensional random vector function whenever x (t,ω) is a product measurable random function, and it satisfies the desired regularity conditions to ensure the existence of solution process. By developing systems of random differential inequalities, a very general comparison theorem in the framework of a vector Lyapunov function is developed, and furthermore sufficient conditions are given for the stability of solutions in probability, in the mean and with probability one.

Systems of differential inequalities and stochastic differential equations. IV

Consider the system of stochastic functional differential equations x′ (t,ω) =f (t,x (t,ω), xt(ω),ω), xt0(ω) =φ0(ω), where f (t,x (t,ω), xt(ω),ω) is a product measurable n-dimensional random vector functional whenever x (t,ω) is a product measurable random function, and it satisfies the desired regularity conditions to assure the existence of solution process. By developing systems of random differential inequalities, very general comparison theorems in the framework of a vector Lyapunov function are obtained, and, furthermore, sufficient conditions are given for the stability of solutions in probability, in the mean and with probability one.

Bemerkung zu Einem Vergleichssatz f�r Gew�hnliche Lineare Differentialgleichungen

In this note we prove a comparison theorem for linear systems of differential inequalities, which includes the special result of Rusmann [1].