Decay rates for viscoelastic plates with memory

Abstract

We consider the viscoelastic plate equation and we prove that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel g satisfies g(t) <~ -cog(t) ~+~/~ and g,g~+~/P ~ Ll(~) withp > 2, then the energy decays as 1/(1 + t) p. (1991): 35B40, 35L05, 35L70. Mathematics Subject Classifications

Stability for the beam equation with memory in non‐cylindrical domains

“…In a ÿxed domain, it is well-known that, the relaxation function g decays to zero implies that the energy of the system also decays to zero; See References [9][10][11][12]. But in a moving boundary setting, the transverse de ection u(x; t) of a beam charges its conÿguration at each instant of time, increasing its deformation and hence increasing its tension.…”

Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term

“…For more related results about the existence, finite time blow-up and asymptotic properties, we refer the reader to [5][6][7][8][9][10][11][12][13][14][15][16].…”

Arbitrary decays for a viscoelastic equation

“…The assumptions (1.2) and (1.3), on g, are frequently encountered in the linear case (r = 0), see [1,2,[4][5][6]13,22,23,[29][30][31]. Lately, these conditions have been weakened by some researchers.…”

“…Later, this result was extended by Messaoudi and Tatar [15] to a situation where a nonlinear source term is competing with the dissipation terms induced by both the viscoelasticity and the viscosity. Recently Messaoudi and Tatar [14] studied problem (1.1) for the case of g = 0, they improved the result in [3] by showing that the solution goes to zero with an exponential or polynomial rate, depending on the decay rate of the relaxation function g.The assumptions (1.2) and (1.3), on g, are frequently encountered in the linear case (r = 0), see [1,2,[4][5][6]13,22,23,[29][30][31]. Lately, these conditions have been weakened by some researchers.…”