Project Euclid

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 $p\geq n$.