## Differential and Integral Equations

- Differential Integral Equations
- Volume 24, Number 1/2 (2011), 159-176.

### Global attractor for some wave equations of $p-$ and $p(x)-$Laplacian type

Nikolaos M. Stavrakakis and Athanasios N. Stylianou

#### Abstract

We study the existence of solutions for the equation $u_{tt}-\Delta_{p(x)} u - \Delta u_{t} + g(u) = f(x,t), \; x \in \Omega$ (bounded) $ \subset \mathbb R^n, \; t>0$ in both the isotropic case $(p(x) \equiv p$, a constant) and the anisotropic case $(p(x)$ a measurable function). Furthermore, in the isotropic case we obtain results concerning the asymptotic behavior of solutions. Since uniqueness for this type of problem seems rather difficult, a method implementing generalized semiflows is being used to prove the existence of a global attractor in the phase space $W_0^{1,p}(\Omega)\times L^2(\Omega)$, when $p\geq n$.

#### Article information

**Source**

Differential Integral Equations, Volume 24, Number 1/2 (2011), 159-176.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019049

**Mathematical Reviews number (MathSciNet)**

MR2759356

**Zentralblatt MATH identifier**

1240.35056

**Subjects**

Primary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B40: Asymptotic behavior of solutions 35B41: Attractors 35B45: A priori estimates 35L15: Initial value problems for second-order hyperbolic equations 35L20: Initial-boundary value problems for second-order hyperbolic equations 35L70: Nonlinear second-order hyperbolic equations 35L80: Degenerate hyperbolic equations

#### Citation

Stavrakakis, Nikolaos M.; Stylianou, Athanasios N. Global attractor for some wave equations of $p-$ and $p(x)-$Laplacian type. Differential Integral Equations 24 (2011), no. 1/2, 159--176. https://projecteuclid.org/euclid.die/1356019049