A sharper decay rate for a viscoelastic wave equation with power nonlinearity

We consider a viscoelastic wave equation with power nonlinearity. First, we construct a local solution by the Faedo-Galerkin approximation scheme and contraction mapping theorem. Next, we continue the local solution to the global one by a priori estimates obtained from a decreasing energy. Finally, we discuss the decay rate of the global solution by assuming that the kernel function is convex.     

Stability of Viscoelastic Wave Equation with Structural $\delta$-Evolution in $\mathbb{R}^{n}$

The aim of this paper is to study the Cauchy problem for the viscoelastic wave equation for structural $\delta$-evolution models. By using the energy method in the Fourier spaces, we obtain the decay estimates of the solution to considered problem.     

Well-posedness and general decay for Moore–Gibson–Thompson equation in viscoelasticity with delay term

Ricerche di Matematica - The problem of Moore–Gibson–Thompson equation with infinite memory and time delay terms is considered. Under suitable Lyapunov functionals with an appropriate...     

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[ \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.     

General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity

This work deals with the study of a new class of nonlinear viscoelastic Kirchhoff equation with Balakrishnan‐Taylor damping and logarithmic nonlinearity. A decay result of the energy of solutions for the problem without imposing the usual relation between a certain relaxation function and its derivative is established. This result generalizes earlier ones to an arbitrary rate of decay, which is not necessarily of exponential or polynomial decay.