We consider the viscoelastic plate equation and we prove that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel g satisfies g(t) <~ -cog(t) ~+~/~ and g,g~+~/P ~ Ll(~) withp > 2, then the energy decays as 1/(1 + t) p. (1991): 35B40, 35L05, 35L70. Mathematics Subject Classifications

En este trabajo estudiamos la existencia y unicidad de la solución global de la ecuación de Kirchoff .... Con una disipación au' y demostraremos el decaimiento exponencial de su energía.

We consider the anisotropic and inhomogeneous viscoelastic equation and we prove that the first and second order energy decay polynomially as time goes to infinity when the relaxation function also decays polynomially to zero. That is, if the kernel G^/ satisfies

In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem \[u_{tt}- M\left(\,\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u - \delta\triangle u_{t}= \mu|u|^{\rho-2}u\quad \text{in } \Omega \times ]0,\infty[,\] where \[M(s)=\begin{cases}a-bs &\text{for } s \in [0,\frac{a}{b}[,\\ 0, &\text{for } s \in [\frac{a}{b}, +\infty[.\end{cases}\] If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay.

The object of this work is to study the existence of solutions for a nonlocal (p1(x), p2(x)) Laplace equation with dependence on the gradient. We establish our results by using the degree theory for operators of (S+) type in the framework of variable exponent Sobolev spaces.