Stabilization of swelling porous elastic soils with fluid saturation, time varying-delay and time-varying weights

Abstract

In this article we study the well-posedness and exponential stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject with time-varying weights and time-varying delay. We prove existence of global solution for the problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the energy method without the equal wave speeds assumption.

A non‐homogeneous weakly damped Lamé system with time‐dependent delay

“…If there exists $$$ c>0 $$$, such that $$$$ \frac{{\left\Vert x\right\Vert}_t}{{\left\Vert x\right\Vert}_s}\le {e}^{c\left(t-s\right)},\kern0.30em x\in X-\left\{0\right\},\kern6.05pt s,t\in \left[0,T\right], $$$$ then the family is stable. We see in the Lemma 2.1 below an easier (elementary) sufficient condition for stability that can be applied to the framework of previous works 10,13,14,17,18 …”

“…Remark 2.1. In almost all papers about time-dependent delay systems in the framework of Fridman et al, 10 e.g., Kang, 13 Kirane et al, 14 and Nonato et al, 17 the norm in the delay space is given by…”

“…Remark In almost all papers about time‐dependent delay systems in the framework of Fridman et al, 10 e.g., Kang, 13 Kirane et al, 14 and Nonato et al, 17 the norm in the delay space is given by $$$$ {\left\Vert z\right\Vert}_{\xi}^2=\xi {\int}_0^1{\int}_{\Omega}{\left|z\left(x,\rho \right)\right|}^2\kern0.1em dxd\rho, $$$$ where $$$ \xi $$$ is some strategically fixed constant after assumptions on $$$ {\alpha}_1,{\alpha}_2 $$$, and $$$ d $$$. This is needed to prove dissipativeness of the system.…”

“…11,12 This method was applied to many other linear evolution equations, e.g. previous studies, [13][14][15][16][17][18] among others.…”

“…10,13,14,17,18 □ Let || • || t , t ∈ [0, T] be a family of uniformly equivalent norms in X; that is, there exist constants𝛾 1 , 𝛾 2 > 0 such that 𝛾 1 ||x|| X ≤ ||x|| t ≤ 𝛾 2 ||x||…”