Multiple impulse solutions to McKean's caricature of the nerve equation. II. Stability

We study McKean's caricature of a nerve conduction equation where H is the Heaviside function. It is proved that an n-ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n - 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to the spacings. The n amplitude modes always exist. We prove also that for an n-ple impulse solution resembling the superposition of n stable solitary impulses, there are at most n - 1 unstable modes and all of them are of spacing type.     

Stabilization of a locally damped incompressible wave equation

We show that the solutions of an incompressible vector wave equation with a locally distributed nonlinear damping decay in an algebraic rate to zero, that is, denoting by E(t) the total energy associated to the system, there exist positive constants C (which depends on E(0)) and γ satisfying, for t?0: E(t)?C(1+t)^−γ.     

Stabilization of a transmission wave/plate equation

We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchhoff plate equation. We prove an exponential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate.     

Multiple impulse solutions to McKean's caricature of the nerve equation. I—existence

McKean's caricature of the nerve equation: is considered. The H in (1) is the Heaviside function. We prove the existence of multiple impulse solutions consisting of any finite number of pulses. These solutions are referred to as n-ple impulse solutions, where n is an arbitrary positive integer.     

Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation     

Stabilization of energy for a wave equation with memory

Translated from Matematicheskie Zametki, Vol. 50, No. 4, pp. 38–46, October, 1991.     

Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback

We consider the stabilization of the wave equation with variable coefficients and a delay in the dissipative boundary feedback. By virtue of the Riemannian geometry methods, the energy-perturbed approach and the multiplier skills, we establish the uniform stability of the energy of the closed-loop system.     

Stabilization of the wave equation with acoustic and delay boundary conditions

In this paper, we consider the wave equation on a bounded domain with mixed Dirichlet-impedance type boundary conditions coupled with oscillators on the Neumann boundary. The system has either a delay in the pressure term of the wave component or the velocity of the oscillator component. Using the velocity as a boundary feedback it is shown that if the delay factor is less than that of the damping factor then the energy of the solutions decays to zero exponentially. The results are based on the energy method, a compactness-uniqueness argument and an appropriate weighted trace estimate. In the critical case where the damping and delay factors are equal, it is shown using variational methods that the energy decays to zero asymptotically.     

Stabilization of the nonlinear damped wave equation via linear weak observability

We consider the problem of energy decay rates for nonlinearly damped abstract infinite dimensional systems. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely a weak observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine the optimal-weight convexity method of Alabau-Boussouira (Appl Math Optim 51:61–105, 2005) and a methodology of Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001) for weak stabilization by observability. Our results extend to nonlinearly damped systems, those of Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001). At the end, we give an appendix on the weak stabilization of linear evolution systems.     

Stabilization of the Kirchhoff type wave equation with locally distributed damping

We derive energy decay estimates of the Kirchhoff type wave equation with a localized damping term in a bounded domain. The damping coefficient function may act alive only on a neighborhood of some part of the boundary.     

Stabilization and control for the subcritical semilinear wave equation

In this paper, we analyze the exponential decay property of solutions of the semilinear wave equation in with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions on the nonlinearity we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p<5. The method of proof combines classical energy estimates for the linear wave equation allowing to estimate the total energy of solutions in terms of the energy localized in the exterior of a ball, Strichartz's estimates and results by P. Gérard on microlocal defect measures and linearizable sequences. We also give an application to the stabilization and controllability of the semilinear wave equation in a bounded domain under the same growth condition on the nonlinearity but provided the nonlinearity has been cut-off away from the boundary.     

A relation between two simple localized solutions of the wave equation

A relation between two previously known exact solutions of the wave equation that describe propagation of localized waves is found.     

Traveling wave solutions to a reaction-diffusion equation

In this paper, we restrict our attention to traveling wave solutions of a reaction-diffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.     

Traveling wave solutions to a reaction-diffusion equation

In this paper, we restrict our attention to traveling wave solutions of a reaction-diffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.      

Stabilization and regularization for solutions of an ill posed problem for the wave equation

In this paper we study the problem of finding u: [O, T] → D(A), D(A) ? H, H a Hibert space such that: Is a linear positive, self-adjoint operator with a compact inverse. The problem is well known to be illposed because uniqueness and existence generally fail. We restore the stability with an a priori bound on ∥ du(0)/dt ∥ for some particular values of T.     

Radial solutions to the wave equation

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂ _t ²-▵ _x )u(t,x)=0 and meanwhile we give a new proof of the solution formula. Received: July 7, 1998?Published online: March 19, 2002     

Stabilization of the solution to a heat equation in the exterior of a sphere with boundary control

The problem of boundary control stabilization of the solution to the heat equation defined in the exterior of a sphere is studied in the paper. The boundary control function stabilizing the solution to zero with the rate 1/t^k is constructed for any k > 0.