New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

• Main Authors: Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Salim A. Messaoudi
• Format: Article
• Language: English
• Published: SpringerOpen 2020-11-01
• Series: Boundary Value Problems
• Subjects: Viscoelasticity; Stability; Nonlocal boundary conditions; Relaxation function; Convex functions
• Online Access: http://link.springer.com/article/10.1186/s13661-020-01467-5

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 $k_{i}$ , namely, k i ′ ( t ) ≤ − ξ i ( t ) Ψ i ( k i ( t ) ) , i = 1 , 2 . $$\begin{aligned} k_{i}^{\prime }(t)\le -\xi _{i}(t) \Psi _{i} \bigl(k_{i}(t)\bigr),\quad i=1,2. \end{aligned}$$ 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 $k_{i}(s) = s^{p}$ and p covers the full admissible range [ 1 , 2 ) $[1, 2)$ .