Single-Pulse Solutions for Oscillatory Coupling Functions in Neural Networks

We study the existence and linear stability of stationary pulse solutions of an integro-differential equation modeling the coarse-grained averaged activity of a single layer of interconnected neurons. The neuronal connections considered feature lateral oscillations with an exponential rate of decay and variable period. We identify regions in the parameter space where solutions exhibit areas of excitation with single- and dimpled-pulses. When the gain function reduces to the Heaviside function, we establish existence of single-pulse solutions analytically. For a more general gain function, we include numerical support of the existence of pulse-like solutions. We then study the linear stability behavior of these solutions.     

Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.     

Existence of local and global solutions of fractional-order differential equations

We study the existence of local and global mild solutions of the fractional-order differential equations in an arbitrary Banach space by using the semigroup theory and the Schauder fixed-point theorem. We also give some examples to illustrate the applications of abstract results.     

Exponential decay in a thermoelastic mixture of solids

In this paper, we investigate the asymptotic behaviour of solutions to the initial boundary value problem for a one-dimensional mixture of thermoelastic solids. Our main result is to establish a necessary and sufficient condition over the coefficients of the system to get the exponential stability of the corresponding semigroup. We also prove the impossibility of time localization of solutions.     

Persistence of Exponential Trichotomy for Linear Operators: A Lyapunov–Perron Approach

In this article we revisit the perturbation of exponential trichotomy of linear difference equation in Banach space by using a Perron–Lyapunov fixed point formulation for the perturbed evolution operator. This approach allows us to directly re-construct the perturbed semiflow without using shift spectrum arguments. These arguments are presented to the case of linear autonomous discrete time dynamical system. This result is then coupled to Howland semigroup procedure to obtain the persistence of exponential trichotomy for non-autonomous difference equations.     

Viscoelastic Solids of Exponential Type. II. Free Energies,Stability and Attractors

The asymptotic behavior for solutions of the semilinear motion equation of a linear viscoelastic solid of exponential type (VSET) is studied and the existence of a global attractor is proved. These results are obtained by means of a suitable class of quadratic free energies defined on the minimal state space and making use of semigroup techniques. This is the second part of a plan which was started in a previous paper [6] by the study of state-space representation, minimality and controllability for VSET.     

Exponential stability in thermoviscoelastic mixtures of solids

In this paper we investigate the asymptotic behavior of solutions to the initial boundary value problem for a one-dimensional mixture of thermoviscoelastic solids. Our main result is to establish the exponential stability of the corresponding semigroup and the lack of exponential stability of the corresponding semigroup.     

Some extensions of the initial value problem for second order evolution equations

The initial problem for second order linear evolution equation systems is discussed by using the contraction semigroup theory. A kind of initial value problem for second order is also discussed with variable coefficients for evolution equations by using the analytical semigroup theory, and is unified with the solutions of the initial value problem for this class of equations and those of first order temporally inhomogeneous evolution equations. This is an important class of equations in mathematical mechanics.     

A NOTE ON STOCHASTIC OPTIMAL CONTROL OF REFLECTED DIFFUSIONS WITH JUMPS

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The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium

We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in \({L^1_vL^\infty_x(m)}\), where \({m\sim (1+ |v|^k)}\) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an \({L^2-L^\infty}\) theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the \({L^1_vL^\infty_x}\) framework is dealt with for any \({k > k_0}\), recovering the optimal physical threshold of finite energy \({k_0=2}\) in the particular case of a multi-species hard spheres mixture with the same masses.     

A note on strochastic optimal control of reflected diffusions with jumps

Stochastic optimal control problems for a class of reflected diffusion with Poisson jumps in a half-space are considered. The nonlinear Nisio’s semigroup for such optimal control problems was constructed. A Hamilton-Jacobi-Bellman equation with the Neumann boundary condition associated with this semigroup was obtained. Then, viscosity solutions of this equation were defined and discussed, and various uniqueness of this equation was also considered. Finally, the value function was such optimal control problems is shown to be a viscosity solution of this equation.     

Spectrum Analysis of Some Kinetic Equations

We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with \({\gamma\geqq-2}\). As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate \({t^{-5/4}}\)) as \({t\to\infty}\) to that of the compressible Navier–Stokes equations for initial data around an equilibrium state.     

Inequalities of Korn’s Type and Existence Results in the Theory of Cosserat Elastic Shells

This paper is concerned with the linear theory of anisotropic and inhomogeneous Cosserat elastic shells. We establish the inequalities of Korn’s type which hold on Cosserat surfaces. Using these inequalities, we prove the existence of the solution to the variational equations in the elastostatics of Cosserat shells. For the dynamic problems, we employ the semigroup of linear operators theory to obtain the existence, uniqueness and continuous dependence of solution.      

Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C ^∞ regularity for any positive time.     

Input-to-state stability of impulsive reaction–diffusion neural networks with infinite distributed delays

In this paper, we focus on the global existence–uniqueness and input-to-state stability of the mild solution of impulsive reaction–diffusion neural networks with infinite distributed delays. First, the model of the impulsive reaction–diffusion neural networks with infinite distributed delays is reformulated in terms of an abstract impulsive functional differential equation in Hilbert space and the local existence–uniqueness of the mild solution on impulsive time interval is proven by the Picard sequence and semigroup theory. Then, the diffusion–dependent conditions for the global existence–uniqueness and input-to-state stability are established by the vector Lyapunov function and M-matrix where the infinite distributed delays are handled by a novel vector inequality. It shows that the ISS properties can be retained for the destabilizing impulses if there are no too short intervals between the impulses. Finally, three numerical examples verify the effectiveness of the theoretical results and that the reaction–diffusion benefits the input-to-state stability of the neural-network system.

     

Characteristic solutions of scalar newton equations in one space dimension

Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions.     

Impulsive Semilinear Functional Differential Equations

In this paper the Leray–Schauder nonlinear alternative combined with semigroup theory is used to investigate the existence of mild solutions for first-order impulsive semilinear functional differential equations in Banach spaces.     

Smooth soliton and kink solutions for a new integrable soliton equation

Nonlinear Dynamics - We establish a relationship between a new integrable soliton equation and Gardner’s equation by a transformation. Then, we use this transformation and solutions of...     

Dirichlet Problem of a Delayed Reaction–Diffusion Equation on a Semi-infinite Interval

We consider a nonlocal delayed reaction–diffusion equation in a semi-infinite interval that describes mature population of a single species with two age stages (immature and mature) and a fixed maturation period living in a spatially semi-infinite environment. Homogeneous Dirichlet condition is imposed at the finite end, accounting for a scenario that boundary is hostile to the species. Due to the lack of compactness and symmetry of the spatial domain, the global dynamics of the equation turns out to be a very challenging problem. We first establish a priori estimate for nontrivial solutions after exploring the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. Using the estimate, we are able to show the repellency of the trivial equilibrium and the existence of a positive heterogeneous steady state under the Dirichlet boundary condition. We then employ the dynamical system arguments to establish the global attractivity of the heterogeneous steady state. As a byproduct, we also obtain the existence and global attractivity of the heterogeneous steady state for the bistable evolution equation in the whole space.     

Nonlinear Differential Equations in {\mathcal{B}(\mathcal{H})} Motivated by Learning Models

We study a system of ordinary differential equations in B(H){\mathcal{B}(\mathcal{H})} , the space of all bounded linear operators on a separable Hilbert space H{\mathcal{H}} . The system considered is a natural generalization of the Oja–Cox–Adams learning models. We establish the local existence of solutions and solve explicitly the system for a class of initial conditions. For such cases, we also characterize the asymptotic behavior of solutions.