Long time behaviour of solutions of abstract inequalities: Applications to thermo-hydraulic and magnetohydrodynamic equations

We study some scalar inequalities of parabolic type and we give the leading term of an asymptotic expansion as t → ∞ for solutions of thermo-hydraulic equations without external excitation. A phenomenon of resonance is pointed out. We also treat M. H. D. equations and Navier-Stokes equations on a Riemannian manifold.     

Numerical solution of non-linear Volterra integral equations

This paper deals with non-linear Volterra integral equations of the type $\text{y(x)}\text{=}\text{f(x)}\text{+ ?}{}_0{}^{\mathrm{<mtext>x</mtext>}}\text{H[t, x, y (t), y (x)] dt}$. Convergence criteria are given (in the same sense of the maximum and C^a norms) for the numerical solution of this type of Volterra integral equation. Several numerical methods are compared.     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

A solution to Schrödinger's problem of non-linear integral equations

A solution of Schrödinger's system of non-linear integral equations determines the rate function of a large deviation principle for kernels with prescribed marginal distributions. This kind of large deviation principle has some meaning in quantum mechanics.Diffusion equations associated with Schrödinger equations have typically transition functions with singular creation and killing. Hence they provide measurable non-negative generally unbounded kernels which may vanish on sets with positive measure and which can possess infinite mass.For Schrödinger systems with such kernels, a solution is proved to exist uniquely in terms of a product measure. It is obtained from a variational principle for the local adjoint of a product measure endomorphism. The generally unbounded factors of the solution are characterized by integrability properties.     

Linearly implicit time discretization of non-linear parabolic equations

We give a stability and error analysis of linearly implicitone-step methods for time discretization of non-linear parabolicequations. We derive precise error bounds for Rosenbrock andW-methods, and we explain the error reduction by Richardsonextrapolation of the linearly implicit Euler method which occursin spite of the breakdown of asymptotic expansions. The parabolicequations are studied in a Hilbert space framework that includessemilinear and quasilinear parabolic equations, and also stiffreaction-diffusion equations with reactions at different timescales.     

Long time dynamics of layered solutions to the shallow water equations

We study the existence of a positive connection, i.e. a stationary solution connecting the boundary data, for the initial-boundary value problem for the viscous shallow water system in a bounded interval (??, ?) of the real line. Subsequently, we investigate the asymptotic behavior of the time-dependent solutions, showing that they first develop into a layered function and then they drift towards the steady state in an exponentially long time interval. The main tool of our analysis is given by the derivation of an ODE for the interface location.     

Asymptotic behaviour of non-linear parabolic equations with monotone principal part

We consider non-linear parabolic equations with subdifferential principal part and give conditions under which they posses global attractors in spite of considering non-Lipschitz perturbations. The case of globally Lipschitz perturbations of a maximal monotone operator has been addressed in Boll. Un. Mat. Ital. B (8) 2 (2000) 693–706. In the case of perturbations which are not globally Lipschitz, the main difficulty is the lack of uniqueness of solutions which at first does not even allow us to define attractors. We overcome this difficulty for problems enjoying certain regularity and absorption properties that allow uniqueness of solutions after some time has been elapsed. The results developed here are applied to the case when the subdifferential operator is the p-Laplacian to obtain existence of attractors and the existence of periodic solutions.     

Long time behaviour of a stochastic nanoparticle

In this article, we are interested in the behaviour of a single ferromagnetic mono-domain particle submitted to an external field with a stochastic perturbation. This model is the first step toward the mathematical understanding of thermal effects on a ferromagnet. In a first part, we present the stochastic model and prove that the associated stochastic differential equation is well defined. The second part is dedicated to the study of the long time behaviour of the magnetic moment and in the third part we prove that the stochastic perturbation induces a non-reversibility phenomenon. Last, we illustrate these results through numerical simulations of our stochastic model.     

Global smooth solution for a coupled non-linear wave equations

In this paper, the global existence and uniqueness of smooth solution to the initial-value problem for coupled non-linear wave equations are studied using the method of a priori estimates.     

An Approximate solution for a class of non-linear hill's equations

The object of this paper is to prove the existence of an approximate solution in the mean for some non-hear differential equations. Further, we investigate the behavior of the class or solutions which may be associated with the differential equalion.     

Long time behavior of solutions of Davey-Stewartson equations

1. IntroductionIn the present paper we study the following Davey-Stewartson systemsupplemented with boundary conditionsand initial conditionwhere a = al la2, b = hi fo2, p = gi iap2, 7 = 71 iap and X = FI ixZ are complexconstallts, fi C RZ is a smooth bounded domain. The system was derived by Davey etalll] to model the evolution of a three-dimensional disturbance in the nonlinear regime ofplane Poiseuille flow (fully developed steady flow under a constallt pressure gradient betwe…     

Long time behavior of non-Fickian delay reaction-diffusion equations

In this paper, we investigate the long time behavior of non-Fickian delay reaction-diffusion equations. These kinds of Volterra integro-differential equations are derived by combining a time memory term in the flux and a delay parameter in the reaction term. Energy estimates, dissipativity, asymptotic stability, and contractivity of the problems are obtained. Moreover, we prove that the numerical method discussed in the present paper has the ability to preserve stability and contractivity of the underlying systems. Some confirmations of these are illustrated by using the numerical method on two biological models.     

Long time behaviour of stochastic interest rate models

In this paper, we study the long time behaviour of two classes of stochastic interest rate models. Suppose that x(t) is a one-factor interest rate model with positive jumps. For a suitable constant we prove that converges almost surely as t→∞. A similar result is also proved for a two-factor affine model.     

Long time behaviour of the solutions of nonlinear wave equation

In this paper, we consider the nonlinear wave equation $$u_{tt}-\Delta u+mu+f(x,u)=0,\ x\in\mb{T}^{d}:=(\mb{R}/2\pi\mb{Z})^{d},$$ where $m>0$ and $f$ is an analytic function of order at least two in $u$. The long time behaviour of its solutions is proved by Birkhoff normal form.     

Directional fields algebraic non-linear solution equations for mobile robot planning

In this study, a novel approach to robot navigation/planning by using half-cell electrochemical potentials is presented. The half-cell electrode’s potential is modelled by the Nernst equation to yield automatic search/detection of pipeline flaws by using the direct current voltage gradient (DCVG) technique. We introduce a theory of spherical volumetric electric density in the soil to sustain our postulates for navigational potential fields. The Nernst potential is correlated with the distance to a pipe’s flaw by proposing a fitted theoretical-empirical nonlinear regression model. From this, volumetric derivatives are solved as gradient-based fields to control wheeled robot’s motion. A nonlinear system for trajectory planning is proposed, and analytically solved by an algebraic solution. This solution directly adjust robot’s speed kinematic values to lead it toward the flaw. The inverse/forward kinematic constraints are non-holonomic, and are recursively integrated into the general potential equation. Analytical modelling is reported, and a set of numerical simulations are presented to prove the feasibility of the proposed formulations.     

The numerical solution of the non-linear integro-differential equations based on the meshless method

This article investigates the numerical solution of the nonlinear integro-differential equations. The numerical scheme developed in the current paper is based on the moving least square method. The moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. It consists of a local weighted least square fitting, valid on a small neighborhood of a point and only based on the information provided by its n closet points. Hence the method is a meshless method and does not need any background mesh or cell structures. The error analysis of the proposed method is provided. The validity and efficiency of the new method are demonstrated through several tests.     

On the long time behaviour of solutions to nonlinear diffusion equations on Rn

Received September 7,1995