Abstract
This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function k_{i}, namely, ki′(t)≤−ξi(t)Ψi(ki(t)),i=1,2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} k_{i}^{\\prime }(t)\\le -\\xi _{i}(t) \\Psi _{i} \\bigl(k_{i}(t)\\bigr),\\quad i=1,2. \\end{aligned}$$ \\end{document} We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when k_{i}(s) = s^{p} and p covers the full admissible range [1, 2).
Highlights
In this paper, we consider the following system: ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨vutttt(xx(xx∈∈,tt))1 x (xux (x, t))x +, t > 0, (xvx(x, t))x, t > 0, t 0 k1 (t s) (x, s))x ds
Initial boundary value problems for second-order evolution partial differential equations and systems having nonlocal boundary conditions can be encountered in many scientific domains and many engineering models and are widely applied in heat transmission theory, underground water flow, medical science, biological processes, thermoelasticity, chemical reaction diffusion, plasma physics, chemical engineering, heat conduction processes, population dynamics, and control theory
Motivated by the above work, we prove a general stability result of system (1) replacing the condition (6) used in [21] by a more general assumption of the form: ki(t) ≤ –ξi(t) i ki(t), i = 1, 2
Summary
We consider the following system:. Al-Gharabli et al Boundary Value Problems (2020) 2020:170 where = (0, L), ki : [0, +∞) −→ (0, +∞), (i = 1, 2), are non-increasing differentiable functions satisfying more general conditions to be mentioned later and. Initial boundary value problems for second-order evolution partial differential equations and systems having nonlocal boundary conditions can be encountered in many scientific domains and many engineering models and are widely applied in heat transmission theory, underground water flow, medical science, biological processes, thermoelasticity, chemical reaction diffusion, plasma physics, chemical engineering, heat conduction processes, population dynamics, and control theory See in this regard the work by Cannon [1], Shi [2], Capasso and Kunisch [3], Cahlon and Shi [4], Ionkin and Moiseev [5], Shi and Shilor [6], Choi and Chan [7], and Ewing and Lin [8].
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