Existence and decay in non linear viscoelasticity

Muñoz Rivera, Jaime E. ; Quispe Gómez, Félix P.

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 1-37

Résumé

In this work we study the existence, uniqueness and decay of solutions to a class of viscoelastic equations in a separable Hilbert space $H$ given by \begin{gather*} u_{tt} + M([u]) Au - \int_{0}^{t} g(t-\tau) N([u]) Au \, d\tau = 0, \quad \text{ in } L^{2}(0, T; H) \\ u(0)=u_{0}, \quad u_{t}(0)=u_{1} \end{gather*} where by $[u(t)]$ we are denoting \begin{equation*} [u(t)]= \left( ( u(t), u_{t}(t), (Au(t), u_{t}(t)), \|A^{\frac{1}{2}} u(t) \|^{2}, \|A^{\frac{1}{2}} u_{t}(t) \|^{2}, \|A u(t) \|^{2} \right) \in \mathbb{R}^{5} \end{equation*} $A \colon D(A) \subset H \to H$ is a nonnegative, self-adjoint operator, $M$, $N \colon \mathbb{R}^{5} \to \mathbb{R}$ are $C^{2}$- functions and $g \colon \mathbb{R} \to \mathbb{R}$ is a $C^{3}$-function with appropriates conditions. We show that there exists global solution in time for small initial data. When $[u(t)]= \| A^{\frac{1}{2}} u\|^{2}$ and $N=1$, we show the global existence for large initial data $(u_{0}, u_{1})$ taken in the space $D(A) \times D(A^{1/2})$ provided they are close enough to Gevrey data. Uniform rate of decay is also proved.