Muñoz Rivera, Jaime E. and Quispe Gómez, Félix P.:
Existence and decay in non linear viscoelasticity
Bollettino dell'Unione Matematica Italiana Serie 8 6-B (2003), fasc. n.1, p. 1-37, Unione Matematica Italiana (English)
pdf (365 Kb), djvu (384 Kb). | MR1955694 | Zbl 1177.74082
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In questo lavoro si studia l'esistenza, l'unicità e il decadimento di soluzioni a una classe di equazioni viscoelastiche in uno spazio di Hilbert $H$ separabile, dato da: \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*} dove con $[u(t)]$ si denota \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*} $A \colon D(A)\subset H \to H$ è un operatore autoaggiunto non-negativo, $M$, $N \colon \mathbb{R}^{5} \to \mathbb{R}$ sono funzioni di classe $C^{2}$ e $g \colon \mathbb{R} \to \mathbb{R}$ è una funzione di classe $C^{3}$ verificante condizioni opportune. Mostriamo che esistono soluzioni globali nel tempo per piccoli dati iniziali. Quando $[u(t)]= \| A^{\frac{1}{2}} u\|^{2}$, $M \colon \mathbb{R} \to \mathbb{R}$ e $N=1$, si mostra l'esistenza globale per grandi dati iniziali $(u_{0}, u_{1})$ presi negli spazi $D(A) \times D(A^{1/2})$ a condizione che siano abbastanza prossimi a dati analitici. È anche dimostrato un tasso uniforme di decadimento.
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