Search

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples. [ABSTRACT FROM AUTHOR]

In this work, a type III thermo‐porous elastic system is considered. First, we use the semigroup theory to demonstrate that the system is well‐posed. Second, we show that the system is exponentially stable under a natural constraint on its parameters. Finally, we provide various numerical examples to illustrate numerically our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a piezoelectric beam model with nonlinear source terms and investigate the interaction between a viscoelastic damping and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we establish explicit formulae for the energy decay rates of this system and prove that the energy decays at a rate dictated by the weaker damping. Our results substantially improve some earlier related results in the literature. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the following weakly damped nonlinear suspensionbridge equation utt(x, y, t) + ∆2u(x, y, t) + α(t)g(ut) = 0, and establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions α and g and obtained without any restriction growth assumption on g at the origin. The multiplier method, the properties of the convex and the dual of the convex functions, Jensen’s inequality and the generalized Young inequality are used to establish the stability results. [ABSTRACT FROM AUTHOR]

In this work we study the well-posedness and the asymptotic stability of a linear thermoelastic Timoshenko system free of second spectrum, where the heat conduction is given by Cattaneo's law. We give a detailed proof of the well-posedness, using the semigroup theory. Then, we prove that the system is exponentially stable irrespective of the coefficients. We also illustrate our theoretical results by performing some numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the general decay rate of the solutions for a class of fractional Laplace viscoelastic equations in the whole space ℝn,n≥1$$ {\mathbb{R}}^n,n\ge 1 $$, given by utt−Δu+u−∫0tg(t−s)(−Δ)αu(s)ds=0,$$ {u}_{tt}-\Delta u+u-{\int}_0^tg\left(t-s\right){\left(-\Delta \right)}^{\alpha }u(s) ds=0, $$with 0<α<1$$ 0<\alpha <1 $$ and g$$ g $$ satisfying g′(t)≤−η(t)g(t)$$ {g}^{\prime }(t)\le -\eta (t)g(t) $$. We prove the existence of a solution formula then use the energy method in the Fourier space to establish a general decay result which depends on α$$ \alpha $$ and η$$ \eta $$. We also present three illustrative examples by the end. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of a Bresse system with memory-type boundary conditions. For a wider class of kernel functions, we establish an optimal explicit energy decay result. Our stability result improves many earlier results in the literature. Finally, we also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following viscoelastic problem with variable exponent nonlinearities: utt−Δu+∫0tg(t−s)Δu(s)ds+a|ut|m(·)−2ut=|u|q(·)−2u,where m(.) and q(.) are two functions satisfying specific conditions. This type of problems appears in fluid dynamics, the electrorheological fluids (smart fluids), which show changing (often dramatically) in the viscosity when an electrical field is applied. The Lebesgue and Sobolev spaces with variable exponents are efficient tools to analyze such problems. In this work, we prove a global existence result using the well‐depth method and establish explicit and general decay results under a very general assumption on the relaxation function. Our results extend and generalize many results in the literature. [ABSTRACT FROM AUTHOR]

In this work, we consider a swelling porous system where the damping terms are on the boundary. We establish an explicit and general decay result, without imposing restrictive growth assumption near the origin on the damping terms. Our result allows a larger class of damping terms, and the usual exponential and polynomial decay estimates are only special cases. We also give some illustrative examples. [ABSTRACT FROM AUTHOR]

In this work, we consider a swelling porous system where the damping terms are on the boundary. We establish an explicit and general decay result, without imposing restrictive growth assumption near the origin on the damping terms. Our result allows a larger class of damping terms, and the usual exponential and polynomial decay estimates are only special cases. We also give some illustrative examples. [ABSTRACT FROM AUTHOR]

Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two "smart" materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state (0 , 0) either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a viscoelastic Moore–Gibson–Thompson equation with a type-II memory term and a relaxation function satisfying g ′ (t) ≤ - η (t) g (t) . By constructing appropriate Lyapunov functionals in the Fourier space, we establish a general decay estimate of the solution under the condition β - γ α - ϱ 2 > 0. We then give the decay rate of the L 2 -norm of the solution. We also give two examples to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we consider a swelling porous-elastic system coupled with a heat system of second sound. We establish the existence of solutions then we prove an exponential decay result. Unlike the "purely" porous system, this result is obtained without the equal-speed requirement. We also perform some numerical tests to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

We study the following wave equation u tt - Δ u + α (t) u t m (·) - 2 u t = 0 with a nonlinear damping having a variable exponent m(x) and a time-dependent coefficient α (t) . We use the multiplier method to establish energy decay results depending on both m and α . We also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme. [ABSTRACT FROM AUTHOR]

In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type. [ABSTRACT FROM AUTHOR]

This work focusses on the well–posedness and the exponential stability of a simplified porous elastic system. By omitting the second time derivative term of the function φ in the porous elastic system, we make the system free of the damaging consequences of the second spectrum. We consider an elastic dissipation and prove the well–posedness by the use of Hille–Yosida therorem and some arguments of elliptic equations. Next, we use the multiplier method and establish an exponential decay result without any restriction on coefficients of the system. [ABSTRACT FROM AUTHOR]

However, with the advancement of sciences and technology, the study and the understanding of several physical and engineering models have required more sophisticated tools and new mathematical functional spaces. The use of functional analysis, in particular the function spaces such as the classical Lebesgue and Sobolev spaces, has been an essential tool to solve and study the qualitative properties of these solutions. The Lebesgue and Sobolev spaces with variable exponents and some other more general spaces such as Orlicz spaces proved to be the right and efficient means to tackle such problems as well as other models like the image processing. [Extracted from the article]

This paper considers a one-dimensional thermoelastic Timoshenko beam system with suspenders, general weak internal damping with time varying coefficient, and time-varying delay terms. Under suitable conditions on the nonlinear terms, we prove a general stability result for the beam model, where exponential and polynomial decay are special cases. We also gave some examples to illustrate our theoretical finding. [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional linear Bresse system in a bounded open interval with one infinite memory acting only on the shear angle equation. First, we establish the well posedness using the semigroup theory. Then, we prove two general (uniform and weak) decay estimates depending on the speeds of wave propagations and the arbitrary growth at infinity of the relaxation function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following viscoelastic swelling porous-elastic system 1 ρ z z tt - a 1 z xx - a 2 u xx + ∫ 0 t g (t - s) z xx (x , s) d s = 0 , in (0 , 1) × (0 , ∞) , ρ u u tt - a 3 u xx - a 2 z xx = 0 , in (0 , 1) × (0 , ∞) , and establish explicit energy decay result with a very general assumption on the behavior of g at infinity; that is 2 g ′ (t) ≤ - k (t) H g (t) , t ≥ 0 , where k and H are two functions satisfying some conditions mentioned in the paper. Our result substantially improves earlier related result in the literature. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the long-time behavior of a viscoelastic equation with a logarithmic source term and a nonlinear feedback localized on a part of the boundary. In the framework of potential well, we first show the global existence. Then, we discuss the asymptotic behavior of the problem with a very general assumption on the behavior of the relaxation function g, namely, g ′ (t) ≤ − ξ (t) G (g (t)) . We establish explicit and general decay results from which we can recover the well-known exponential and polynomial rates when G(s) = sp and p covers the full admissible range [1,2). Our results are obtained without imposing any restrictive growth assumption on the boundary damping term. This work generalizes and improves many earlier results in the literature. [ABSTRACT FROM AUTHOR]