On a hierarchy of models for electrical conduction in biological tissues

In this paper we derive a hierarchy of models for electrical conduction in a biological tissue, which is represented by a periodic array of period ε of conducting phases surrounded by dielectric shells of thickness εη included in a conductive matrix. Such a hierarchy will be obtained from the Maxwell equations by means of a concentration process η → 0, followed by a homogenization limit with respect to ε. These models are then compared with regard to their physical meaning and mathematical issues. Copyright © 2005 John Wiley & Sons, Ltd.     

Applications of homogenization techniques to the electrical conduction in biological tissues

We study a family of problems set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. We prove a unified derivation of these different schemes from the Maxwell equations in the quasi-stationary approximation and we investigate the behaviour of the corresponding macroscopic models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A general formula for decay rates of nonlinear dissipative systems

This work is concerned with stabilization of hyperbolic systems by a nonlinear feedback which can be localized on part of the boundary or locally distributed. We present here a general formula which gives the energy decay rates in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We give also two other significant examples of nonpolynomial growth at the origin. We also show that we either obtain or improve significantly the decay rates of Lasiecka and Tataru (Differential Integral Equations 8 (1993) 507–533) and Martinez (Rev. Mat. Comput. 12 (1999) 251–283). To cite this article: F. Alabau-Boussouira, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Remarks on the energy decay for nonlinear wave equations with nonlinear localized dissipative terms

We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation with a nonlinear localized dissipation. To overcome a difficulty related to derivative-loss mechanism we employ a ‘loan’ method.     

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H² solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.     

具时间依赖系数和耗散边界的非线性波方程的能量指数衰减性

利用能量法证明了具耗散边界条件和时间依赖系数的非线性波方程的能量指数衰减性.     

Asymptotic stability and decay rate of solutions for a nonlinear fourth order wave equation

This paper is concerned with investigating the global asymptotic behavior of the zero solution of the initial-boundary value problem for a nonlinear fourth order wave equation. Moreover an estimate of the rate of decay of the solutions is obtained.     

General energy decay of solutions for a weakly dissipative Kirchhoff equation with nonlinear boundary damping

In this article, we study the weak dissipative Kirchhoff equation \({u_{tt}} - M\left( {\left\| {\nabla u} \right\|_2^2} \right)\Delta u + b\left( x \right){u_t} + f\left( u \right) = 0\), under nonlinear damping on the boundary \(\frac{{\partial u}}{{\partial v}} + \alpha \left( t \right)g\left( {{u_t}} \right) = 0\). We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping. Our result extends and improves some results in the literature such as the work by Zhang and Miao (2010) in which only exponential energy decay is considered and the work by Zhang and Huang (2014) where the energy decay has been not considered.     

Asymptotic stability and decay rate of solutions for a nonlinear wave equation with variable coefficients

This paper is concerned with investigating the global asymptotic behavior of the solution to a nonlinear wave equation with variable coefficients. Moreover an estimate of the rate of decay of the solution is obtained.     

Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms

This paper is concerned with the asymptotic behavior of solutions of a stochastic nonlinear wave equation with dispersive and dissipative terms defined on an unbounded domain. It is proved that the random dynamical system generated by the equation has a random attractor in a Sobolev space. To overcome the difficulty caused by the non-compactness of Sobolev embeddings on unbounded domains, a cut-off method and a decomposition trick are combined to prove the asymptotic compactness of the solutions.     

decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂_t)u=0 in with (u,∂_tu)|_t=0=(u₀,u₁). Because the L² estimates and the L^∞ estimates of the solution u(t) are well known, in this paper we pay attention to the L^p estimates with 1p<2 (in particular, p=1) of the solution u(t) for t0. In order to derive L^p estimates we first give the representation formulas of the solution u(t)=∂_tS(t)u₀+S(t)(u₀+u₁) and then we directly estimate the exact solution S(t)g and its derivative ∂_tS(t)g of the dissipative wave equation with the initial data (u₀,u₁)=(0,g). In particular, when p=1 and n1, we get the L¹ estimate: u(t)_L¹Ce^−t/4(u_0W^n,1+u_1W^n−1,1)+C(u_0L¹+u_1L¹) for t0.     

Stability and optimality of decay rate for a weakly dissipative Bresse system

This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic and exponential decay in mean square for delay geometric Brownian motion

Applications of Mathematics - We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions...