scholarly journals Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping

2020 ◽  
Vol 9 (2) ◽  
pp. 359-373 ◽  
Author(s):  
Jun-Ren Luo ◽  
◽  
Ti-Jun Xiao
Keyword(s):  
Hilbert Spaces ◽  
Evolution Equations ◽  
Second Order ◽  
Decay Rates ◽  
Time Dependent

scholarly journals Optimal energy decay rates for abstract second order evolution equations with nonautonomous damping

2021 ◽  
Author(s):  
J. R. Luo ◽  
T. J. Xiao
Keyword(s):  
Evolution Equation ◽  
Polynomial Growth ◽  
Evolution Equations ◽  
Energy Decay ◽  
Second Order ◽  
Decay Rates ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Energy Decay Rates ◽  
Autonomous Evolution

We consider an abstract second order non-autonomous evolution equation in a Hilbert space $H:$ $u''+Au+\gamma(t) u'+f(u)=0,$ where $A$ is a self-adjoint and nonnegative operator on $H$, $f$ is a conservative $H$-valued function with polynomial growth (not necessarily to be monotone), and $\gamma(t)u'$ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient $\gamma(t)$ and the exponent associated with the nonlinear term $f$? There seems to be little development on the study of such problems, with regard to {\it non-autonomous} equations, even for strongly positive operator $A$. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of $\gamma(t)$ and $f$.  As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when $f$ is a monotone operator.

Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density

2017 ◽  
Vol 6 (2) ◽  
pp. 121-145 ◽  
Author(s):  
Marcelo M. Cavalcanti ◽  
Valéria N. Domingos Cavalcanti ◽  
Irena Lasiecka ◽  
Claudete M. Webler
Keyword(s):  
Evolution Equations ◽  
Variable Density ◽  
Second Order ◽  
Decay Rates ◽  
Equation Modeling ◽  
Relaxation Kernel ◽  
Time Behavior ◽  
Content Type ◽  
Frictional Damping ◽  
Definite Operator

AbstractWe consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form$\frac{d}{dt}\rho(u_{t})+Au_{tt}+\gamma A^{\theta}u_{t}+Au-\int_{0}^{t}g(s)Au(t% -s)=0,$where A is a self-adjoint, positive definite operator acting on a Hilbert space H, ${\rho(s)}$ is a continuous, monotone increasing function, and the relaxation kernel ${g(s)}$ is a continuous, decreasing function in ${L_{1}(\mathbb{R}_{+})}$ with ${g(0)>0}$. Of particular interest is the case when ${A=-\Delta}$ with appropriate boundary conditions and ${\rho(s)=|s|^{\rho}s}$. This model arises in the context of solid mechanics accounting for variable density of the material. While finite energy solutions of the underlying PDE solutions exhibit exponential decay rates when strong damping is active (${\gamma>0,\theta=1}$), this uniform decay is no longer valid (by spectral analysis arguments) for dynamics subjected to frictional damping only, say, ${\theta=0}$ and ${g=0}$. In the absence of mechanical damping (${\gamma=0}$), the linearized version of the model reduces to a Volterra equation generated by bounded generators and, hence, it is exponentially stable for exponentially decaying kernels. The aim of the paper is to study intrinsic decays for the energy of the nonlinear model accounting for large classes of relaxation kernels described by the inequality ${g^{\prime}+H(g)\leq 0}$ with H convex and subject to the assumptions specified in (1.13) (a general framework introduced first in [1] in the context of linear second-order evolution equations with memory). In the context of frictional damping, such a framework was introduced earlier in [15], where it was shown that the decay rates of second-order evolution equations with frictional damping can be described by solutions of an ODE driven by a suitable convex function H which captures the behavior at the origin of the dissipation. The present paper extends this analysis to nonlinear equations with viscoelasticity. It is shown that the decay rates of the energy are intrinsically described by the solution of the dissipative ODE${S_{t}+c_{1}H(c_{2}S)=0}$with given intrinsic constants ${c_{1},c_{2}>0}$. The results obtained are sharp and they improve (by introducing a novel methodology) previous results in the literature (see [20, 19, 21, 6]) with respect to (i) the criticality of the nonlinear exponent ρ and (ii) the generality of the relaxation kernel.

scholarly journals ON DIFFERENTIAL-NON-AUTONOMOUS REPRESENTATION INTEGRATIVE ACTIVITY OF NEUROPOPULATION BILINEAR SECOND-ORDER MODEL WITH A DELAY

2021 ◽  
Vol 23 (2) ◽  
pp. 115-126
Author(s):  
A.V. Daneev ◽  
◽  
A.V. Lakeev ◽  
V.A. Rusanov ◽  
P.A. Plesnev ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Evolution Equations ◽  
Basic Research ◽  
Second Order ◽  
Infinite Dimensional ◽  
Machine Interface ◽  
Additive Combination ◽  
Basic Research Project ◽  
Principle Of Maximum

For neuromorphic processes specified by the behavior of a local neuropopulation (for example, processes induced by a brain-machine interface platform of the Neuralink type), we study the solvability of the problem of the existence of a differential realization of these processes in the class of bilinear nonstationary ordinary differential equations of the second order (with delay) in separable Hilbert space. This formulation belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The metalanguage of the theory being developed is the constructions of tensor products of Hilbert spaces, lattice structures with orthocompletion, the functional apparatus of the nonlinear Rayleigh-Ritz operator, and the principle of maximum entropy. It is shown that the property of sublinearity of this operator, allows you to obtain conditions for the existence of such differential realizations; along the way, metric conditions for the continuity of the projectivization of this operator are substantiated with the calculation of the fundamental group of its compact image. This work was financially supported by the Russian Foundation for Basic Research (project no. 19-01-00301).

Sign in / Sign up

Export Citation Format

Share Document