On Some Viscoelastic Strongly Damped Nonlinear Wave Equations

Abstract

We study the problem of existence, uniqueness end asymptotic behaviour for t → ∞ of (weak or strong) solutions of equation in the form

$${u_u} - \lambda \Delta {u_t} - \sum\nolimits_{i = 1}^N {\partial /\partial {x_i}{\sigma _i}\left( {{u_{x1}}} \right) + f\left( {u,{u_i}} \right)} = 0,\left( {x,t} \right) \in \Omega \times \left( {0,T} \right)$$

((1))

$$u = 0\quad on\quad \partial \Omega$$

((2))

$$\begin{gathered} u\left( {x,0} \right) = {u_o}\left( x \right)\quad ,\quad {u_t}\left( {x,0} \right) = {u_1}\left( x \right) \hfill \\ where\;\lambda \geqslant 0\;,\;{u_t} = \partial u/\partial t\;and\;{u_{xi}} = \partial u/\partial {x_i} \hfill \\ \end{gathered}$$

((3))

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