Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition

We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t), u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.Comment: 26 page

On a viscoelastic heat equation with logarithmic nonlinearity

This work deals with the following viscoelastic heat equations with logarithmic nonlinearity ut − ∆u + Z t 0 g(t − s)∆u(s)ds = |u| p−2u ln |u|. In this paper, we show the effects of the viscoelastic term and the logarithmic nonlinearity to the asymptotic behavior of weak solutions. Our results extend the results of Peng and Zhou [Appl. Anal. 100(2021), 2804–2824] and Messaoudi [Progr. Nonlinear Differential Equations Appl. 64(2005), 351–356.]

The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions

This paper is concerned with the existence and the regularity of global solutions to the linear wave equation associated with two-point type boundary conditions. We also investigate the decay properties of the global solutions to this problem by the construction of a suitable Lyapunov functional.Comment: 18 page

Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil

This paper is concerned with the linear ODE in the form $y'(t)=\lambda\rho(t)y(t)+b(t)$, $\lambda <0$ which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function $\rho(t)$, a linear drift in the coefficient $b(t)$ involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.Comment: 18 page

Stabilization for equal-order polygonal finite element method for high fluid velocity and pressure gradient

This paper presents an adapted stabilisation method for the equal-order mixed scheme of finite elements on convex polygonal meshes to analyse the high velocity and pressure gradient of incompressible fluid flows that are governed by Stokes equations system. This technique is constructed by a local pressure projection which is extremely simple, yet effective, to eliminate the poor or even non-convergence as well as the instability of equal-order mixed polygonal technique. In this research, some numerical examples of incompressible Stokes fluid flow that is coded and programmed by MATLAB will be presented to examine the effectiveness of the proposed stabilised method

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to study a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical resultsComment: 2