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.

Abstract (VO)

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*} $A:D(A)\subset H\to H$ is a nonnegative, self-adjoint operator, $M$, $N:{\mathbb{R}}^5\to \mathbb{R}$ are $C^2$- functions and $g:\mathbb{R}\to \mathbb{R}$ is a $C^3$-function with appropriates conditions. We show that there exists global solution in time for small initial data. When $[u(t)]=\Vert A^{\frac{1}{2}}u{\Vert }^2$ 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.

Riassunto

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*} $A:D(A)\subset H\to H$ è un operatore autoaggiunto non-negativo, $M$, $N:{\mathbb{R}}^5\to \mathbb{R}$ sono funzioni di classe $C^2$ e $g:\mathbb{R}\to \mathbb{R}$ è una funzione di classe $C^3$ verificante condizioni opportune. Mostriamo che esistono soluzioni globali nel tempo per piccoli dati iniziali. Quando $[u(t)]=\Vert A^{\frac{1}{2}}u{\Vert }^2$, $M:\mathbb{R}\to \mathbb{R}$ 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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