Existence and decay in non linear viscoelasticity

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

Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003), p. 1-37 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

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.

Résumé

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.

Détail

Publié le : 2003-02-01

MR 1955694 | Zbl 1177.74082

Bibliographie

[1] Arosio, A.-Spagnolo, S., Global solution of the Cauchy problem for a nonlinear hyperbolic equation, Nonlinear partial differential equations an their applications, College de France Seminar, Edited by H. Brezis & J. L. Lions, Pitman, London, 6 (1984), 1-26. | Zbl 0598.35062

[2] Bernstein, S., Sur une classe d'equations fonctionnelles aux dérivées partielles, Izv. Akad. Nauk SSSR, ser. Mat., 4 (1940), 17-26 (Math. Rev. 2 No. 102). | JFM 66.0471.01 | MR 2699 | Zbl 0026.01901

[3] Dafermos, C. M., An Abstract Volterra Equation with application to linear Viscoelasticity, J. Differential Equations, 7 (1970), 554-589. | MR 259670 | Zbl 0212.45302

[4] Dafermos, C. M.-Nohel, J. A., Energy methods for non linear hyperbolic Volterra integro-differential equation, Comm. PDE, 4 (1979), 219-278. | MR 522712 | Zbl 0464.45009

[5] D'Ancona, P.-Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. | MR 1161092 | Zbl 0785.35067

[6] Dickey, R. W., Infinite systems of nonlinear oscillation equation related to the string, Proc. Amer. Math. Soc., 23 (1969), 459-468. | MR 247189 | Zbl 0218.34015

[7] Dickey, R. W., Infinite systems of nonlinear oscillation equations with linear damping, SIAM Journal of Applied Mathematics, 19, No. 1 (1970), 208-214. | MR 265654 | Zbl 0233.34014

[8] Gelfand, I. M.-Vilenkin, N. Ya., Fonctions Généralisées, IV, Academic Press, 1961.

[9] Greenberg, J. M.-Hu, S. C., The initial value problem for the stretched string, Quarterly of Applied Mathematics (1980), 289-311. | MR 592197 | Zbl 0487.73006

[10] Huet, D., Décomposition spectrale et opérateurs, Presses Universitaires de France, 1977. | MR 473900 | Zbl 0334.47015

[11] Lagnese, J. E., Asymptotic energy estimates for Kirchhoff plates subject to weak viscoelastic damping, International series of Numerical Mathematics, 91, 1989, Birhäuser, Verlag, Bassel. | MR 1033061 | Zbl 0699.93070

[12] Lions, J. L., Quelques méthodes de resolution des problèmes aux limites non lineares, Dunod Gauthier Villars, Paris, 1969. | MR 259693 | Zbl 0189.40603

[13] Lions, J. L.-Dautray, R., Mathematical Analysis and Numerical Methods for science and Tecnology, 3, Spectral Theory and Applications, Springer Verlag1985, Masson, Paris, 1988. | Zbl 0766.47001

[14] Medeiros, L. A.-Milla Miranda, M. A., On a nonlinear wave equation with damping, Revista de Matemática de la Universidad Complutense de Madrid, 3, No. 2 (1990). | MR 1081312 | Zbl 0721.35044

[15] Perla Menzala, G. A., On global classical solution of a nonlinear wave equation with damping, Appl. Anal., 10 (1980), 179-195. | MR 577330 | Zbl 0441.35037

[16] Muñoz Rivera, J. E., Asymptotic behaviour in Linear Viscoelasticity, Quarterly of Applied Mathematics, III, 4, (1994), 629-648. | Zbl 0814.35009

[17] Muñoz Rivera, J. E., Global Solution on a Quasilinear Wave Equation with Memory, Bollettino U.M.I. (7), 8-b (1994), 289-303. | MR 1278337 | Zbl 0802.45006

[18] Nishihara, K., Degenerate quasilinear hyperbolic equation with strong damping, Funkcialaj Ekvacioj, 27 (1984), 125-145. | MR 763940 | Zbl 0555.35094

[19] Nishihara, K., Global existence and Asymptotic behaviour of the solution of some quasilinear hyperbolic equation with linear damping, Funkcialaj Ekvacioj, 32 (1989), 343-355. | MR 1040163 | Zbl 0702.35165

[20] Pohožaev, S. I., On a class of quasilinear hyperbolic equation, Math. USSR-Sb., 25-1 (1975), 145-158.

[21] Renardy, M.-Hrusa, W. J.-Nohel, J. A., Mathematical problems in Viscoelasticity, Pitman monograph in Pure and Applied Mathematics, 35, 1987. | MR 919738 | Zbl 0719.73013

[22] Torrejón, R.-Yong, J., On a Quasilinear Wave Equation with memory, Nonlinear Analysis, Theory and Methods & Applications, 16, 1 (1991), 61-78. | Zbl 0738.35096