Existence and decay in non linear viscoelasticity

Jaime E. Muñoz Rivera; Félix P. Quispe Gómez

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 1, page 1-37
  • ISSN: 0392-4041

Abstract

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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 u t t + M ( [ u ] ) A u - ∫ 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 where by u ⁢ t we are denoting [ u ( t ) ] = ( u ( t ) , u t ( t ) , ( A u ( t ) , u t ( t ) ) , ∥ A 1 2 u ( t ) ∥ 2 , ∥ A 1 2 u t ( t ) ∥ 2 , ∥ A u ( t ) ∥ 2 ∈ ℝ 5 A : D ⁢ A ⊂ H → H is a nonnegative, self-adjoint operator, M , N : R 5 → R are C 2 - functions and g : R → 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 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 to Gevrey data. Uniform rate of decay is also proved.

How to cite

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Muñoz Rivera, Jaime E., and Quispe Gómez, Félix P.. "Existence and decay in non linear viscoelasticity." Bollettino dell'Unione Matematica Italiana 6-B.1 (2003): 1-37. <http://eudml.org/doc/196152>.

@article{MuñozRivera2003,
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)), \Vert A^\{\frac\{1\}\{2\}\} u(t) \Vert ^\{2\}, \Vert A^\{\frac\{1\}\{2\}\} u\_\{t\}(t) \Vert ^\{2\}, \Vert A u(t) \Vert ^\{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.},
author = {Muñoz Rivera, Jaime E., Quispe Gómez, Félix P.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {1-37},
publisher = {Unione Matematica Italiana},
title = {Existence and decay in non linear viscoelasticity},
url = {http://eudml.org/doc/196152},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Muñoz Rivera, Jaime E.
AU - Quispe Gómez, Félix P.
TI - Existence and decay in non linear viscoelasticity
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/2//
PB - Unione Matematica Italiana
VL - 6-B
IS - 1
SP - 1
EP - 37
AB - 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)), \Vert A^{\frac{1}{2}} u(t) \Vert ^{2}, \Vert A^{\frac{1}{2}} u_{t}(t) \Vert ^{2}, \Vert A u(t) \Vert ^{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.
LA - eng
UR - http://eudml.org/doc/196152
ER -

References

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