## Differential and Integral Equations

- Differential Integral Equations
- Volume 27, Number 11/12 (2014), 1147-1170.

### On a viscoelastic plate equation with strong damping and $\overrightarrow{p}(x,t)-$ Laplacian. Existence and uniqueness

#### Abstract

In this paper, we study the class of non-linear viscoelastic equation with lower order perturbation of $\overrightarrow{p}(x,t)-$ Laplacian type and a memory term \[ u_{tt}+\Delta^{2}u-\Delta_{\overrightarrow{p}(x,t)}u+\int_{0}^{t}g(t-s)\Delta u(s)ds -\epsilon\Delta u_{t}+f(u) =0\text{, } \] \[ (x,t)\in Q_{T}=\Omega\times(0,T),\quad\Omega\in\mathbb{R}^{n}. \] We prove local and global existence and uniqueness of weak solutions. These results are obtained assuming a strong damping $\epsilon\Delta u_{t}$ $(\epsilon>0)$ acting in the domain and provided the memory term decays exponentially and $f(u)$ is a nonlinear perturbation.

#### Article information

**Source**

Differential Integral Equations Volume 27, Number 11/12 (2014), 1147-1170.

**Dates**

First available in Project Euclid: 18 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1408366787

**Mathematical Reviews number (MathSciNet)**

MR3250757

**Zentralblatt MATH identifier**

1340.35342

**Subjects**

Primary: 35B40: Asymptotic behavior of solutions 35L45: Initial value problems for first-order hyperbolic systems 335L70

#### Citation

Antontsev, S.; Ferreira, J. On a viscoelastic plate equation with strong damping and $\overrightarrow{p}(x,t)-$ Laplacian. Existence and uniqueness. Differential Integral Equations 27 (2014), no. 11/12, 1147--1170. https://projecteuclid.org/euclid.die/1408366787