Abstract
In the present paper, we consider an important problem from the application perspective in science and engineering, namely, one-dimensional porous–elastic systems with nonlinear damping, infinite memory and distributed delay terms. A new minimal conditions, placed on the nonlinear term and the relationship between the weights of the different damping mechanisms, are used to show the well-posedness of the solution using the semigroup theory. The solution energy has an explicit and optimal decay for the cases of equal and nonequal speeds of wave propagation.
Highlights
As introduced in [1], the one-dimensional porous–elastic model constitutes a system of two partial differential equations with unknown (u, φ) given by Academic Editor: Zhuojia FuReceived: 30 August 2021 Accepted: 13 October 2021 Published: 16 October 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.ρ0utt = μuxx + βφx, in (0, l) × (0, L), ρ0kφtt = αφxx − βux − τ φt − ξ φ, in (0, l) × (0, L), (1)where l, L > 0 the constant ρ is the mass density, κ is the equilibrated inertia and the constants μ, α, β, τ, ξ are assumed to satisfy the appropriate conditions
Where l, L > 0 the constant ρ is the mass density, κ is the equilibrated inertia and the constants μ, α, β, τ, ξ are assumed to satisfy the appropriate conditions. This type of problem has been studied by many authors and a lot of results have been shown
The basic evolution equations for one-dimensional theories of porous materials with memory effect are given by ρutt = Tx, Jφtt = Hx + G, (2)
Summary
As introduced in [1], the one-dimensional porous–elastic model constitutes a system of two partial differential equations with unknown (u, φ) given by Academic Editor: Zhuojia Fu. System (4) subjected Neumann–Dirichlet boundary conditions, where g is the relaxation function; the authors obtained a general decay result for the case of equal speeds of wave propagation (See [12,13]). In [15] the authors considered the following system with memory and distributed delay terms ρutt. The exponential stability results of systems with memory and distributed delay terms, for the case of equal speeds of wave propagation under a suitable assumptions, are proved. Motivated by all the above papers, we investigate the well-posedness and stability results with distributed delay for the cases of equal and nonequal speeds of wave propagation, under additional conditions of the following system ρutt − μuxx − bφx = 0. The main results in this manuscript are as follows: Theorem 1 for the existence and uniqueness of solution and Theorem 2 for the general stability estimates
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