On the Stability of swelling porous elastic soils with a single internal fractional damping

We study polynomial stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils with an internal fractional damping. We establish an optimal decay result by frequency domain methodComment: 22 page

Strong asymptotic stability for a coupled system of degenerate wave equations with only one fractional feedback

This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.We prove the well-posedness and study the strong asymptotic stability of a coupled system of degenerate wave equations with a fractional feedback acting on one end only.Publisher's Versiio

Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback

In this paper, we consider the wave equation with a weak internal constant delay term: $$u''(x, t)-\Delta_{x}u(x, t)+ \mu_1(t) \ u'(x, t) + \mu_2(t) \ u'(x, t-\tau) = 0$$ in a bounded domain. Under appropriate conditions on $\mu_1$ and $\mu_2$, we prove global existence of solutions by the Faedo-Galerkin method and establish a decay rate estimate for the energy using the multiplier method

Blowup of solutions of a nonlinear wave equation

We establish a blowup result to an initial boundary value problem for the nonlinear wave equation utt−M(‖B1/2u‖ 2) Bu+kut=|u| p−2, x∈Ω, t>0

GENERAL DECAY OF SOLUTION FOR COUPLED SYSTEM OF VISCOELASTIC WAVE EQUATIONS OF KIRCHHOFF TYPE WITH DENSITY IN Rn

A system of viscoelastic wave equations of Kirchhoff type is considered. For a wider class of relaxation functions, we use spaces weighted by the density function to establish a very general decay rate of the solution

NULL CONTROLLABILITY OF DEGENERATE NONAUTONOMOUS PARABOLIC EQUATIONS

In this paper we are interested in the study of the null controllability for the one dimensional degenerate non autonomous parabolic equation$$u_{t}-M(t)(a(x)u_{x})_{x}=h\chi_{\omega},\qquad  (x,t)\in Q=(0,1)\times(0,T),$$ where $\omega=(x_{1},x_{2})$ is asmall nonempty open subset in $(0,1)$, $h\in L^{2}(\omega\times(0,T))$, the diffusion coefficients $a(\cdot)$ isdegenerate at $x=0$ and $M(\cdot)$ is non degenerate on $[0,T]$. Also the boundary conditions are considered tobe Dirichlet or Neumann type related to the degeneracy rate of $a(\cdot)$. Under some conditions on the functions$a(\cdot)$ and $M(\cdot)$, we prove some global Carleman estimates which will yield the  observability inequalityof the associated adjoint system and equivalently the null controllability of our parabolic equation

Well-posedness and asymptotic stability for the Lamé system with internal distributed delay

In this work, we consider the Lamé system in 3-dimension bounded domain with distributed delay term. We prove, under some appropriate assumptions, that this system is well-posed and stable. Furthermore, the asymptotic stability is given by using an appropriate Lyapunov functional

Energy decay for wave equations of phi-Laplacian type with weakly nonlinear dissipation

In this paper, first we prove the existence of global solutions in Sobolev spaces for the initial boundary value problem of the wave equation of $phi$-Laplacian with a general dissipation of the form $$(|u'|^{l-2}u')'-Delta_{phi}u+sigma(t) g(u')=0 quadext{in } Omegaimes mathbb{R}_+ ,$$ where $Delta_{phi}=sum_{i=1}^n partial_{x_i}igl(phi (|partial_{x_i}|^2)partial_{x_i}igr)$. Then we prove general stability estimates using multiplier method and general weighted integral inequalities proved by the second author in [18]. Without imposing any growth condition at the origin on $g$ and $phi$, we show that the energy of the system is bounded above by a quantity, depending on $phi$, $sigma$ and $g$, which tends to zero (as time approaches infinity). These estimates allows us to consider large class of functions $g$ and $phi$ with general growth at the origin. We give some examples to illustrate how to derive from our general estimates the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize many existing results in the literature, and generate some interesting open problems

Existence of global solutions to a quasilinear wave equation with general nonlinear damping

In this paper we prove the existence of a global solution and study its decay for the solutions to a quasilinear wave equation with a general nonlinear dissipative term by constructing a stable set in $H^{2}cap H_{0}^{1}$. Submitted July 02, 2002. Published October 26, 2002. Math Subject Classifications: 35B40, 35L70, 35B37. Key Words: Quasilinear wave equation; global existence; asymptotic behavior; nonlinear dissipative term; multiplier method