Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation

Marcelo Moreira Cavalcanti

doi:10.3934/dcds.2002.8.675

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2002, Volume 8, Issue 3: 675-695. Doi: 10.3934/dcds.2002.8.675

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Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation

Marcelo Moreira Cavalcanti^1,

1.
Departamento de Matemática - Universidade Estadual de Maringá, 87020-900 Maringá - PR

Received: June 2001

Revised: October 2001

Published: July 2002

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Abstract

The linear Euler-Bernoulli viscoelastic equation

$u_{t t} +\Delta^2 u-\int_0^t g(t-\tau) \Delta^2 u(\tau)d\tau = 0\quad$ in $\Omega \times (0,\infty)$

subject to nonlinear boundary conditions is considered. We prove existence and uniform decay rates of the energy by assuming a nonlinear and nonlocal feedback acting on the boundary and provided that the kernel of the memory decays exponentially.

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Mathematics Subject Classification: 74K20, 74D10, 34B15, 93D20.

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Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation

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