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Existence and decay in non linear viscoelasticity

Jaime E. Muñoz Rivera; Félix P. Quispe Gómez — 2003

Bollettino dell'Unione Matematica Italiana

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 $\begin{matrix} u_{t t} + M ([u]) A u - \int_{0}^{t} g (t - τ) N ([u]) A u d τ = 0, in L^{2} (0, T; H) \\ u (0) = u_{0}, u_{t} (0) = u_{1} \end{matrix}$ where by $[u (t)]$ we are denoting $[u (t)] = ((u (t), u_{t} (t), (A u (t), u_{t} (t)), ∥ A^{\frac{1}{2}} u (t) ∥^{2}, ∥ A^{\frac{1}{2}} u_{t} (t) ∥^{2}, ∥ A u (t) ∥^{2}) \in ℝ^{5}$ $A : D (A) \subset H \to H$ is a nonnegative, self-adjoint operator, $M$ , $N : R^{5} \to R$ are $C^{2}$ - functions and $g : R \to R$ 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 $D (A) D (A^{1 / 2})$ provided they are close enough...

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