Asymptotic Decay for Some Differential Systems with Fading Memory

By Mauro Fabrizio, Sergio Polidoro. 2002

136 citations recorded by the source corpus.

We study the large time behavior of the solution u to an initial and boundary value problem related to the following integro-differential equation $$ u_{tt} = G_0 \Delta u + \int_0^t G'(t-s) \Delta u(x, s)\, ds - a u_t \eqno(0.1) $$ where G 0 , a are real constant coefficients, G 0 > 0, a S 0 and $ G\,' \in L^1({{\shadR}}^ + ) \cap L^2({{\shadR}}^ + ), G\,' \le 0 $ . It is known that, when G ' L 0 and a > 0, the solution u of (0.1) exponentially decays. Here we prove that, for any nonnegative a and for any $ G ' \not \equiv 0 $ , the solution u of the Eq. (0.1) exponentially decays only if the relaxation kernel G ' does. In other words, the introduction of the dissipative term related to G ' does not allow the exponential decay due to the presence of the positive coefficient a . We also prove analogous results for the polynomial decay.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/e86d1157acdfa3bbd19c653b736c1b2b77fc123a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Computer Networks and Communications

In Control and Systems Engineering

Citations in the corpus

Energy decay for Timoshenko systems of memory type (2003, background): “In fact, if we consider the memory effect with a dissipative relaxation function together with other stronger dissipative effects, for example the frictional damping, then the resulting dissipation does not produce any rate of decay if the relaxation function does not decay uniformly, see the work o”
Existence and asymptotic stability of a viscoelastic wave equation with a delay (2011, background): “Fabrizio and Polidoro [ 10 ] treated the problem (7) with a(x )= a0 and showed that the solution of the problem (7) decays exponentially only if the relaxation kernel g does.”
Stability and exponential stability in linear viscoelasticity (2009, methodology): “For this reason, (5.1) is still necessary even if the equation contains damping terms, such as ˙ u(t )o rA ˙ u(t), although the corresponding equations without memory generate exponentially stable semigroups on V × H. A similar feature has been devised in [ 9 ], for a Volterra-type equation.”
Existence and sharp decay rate estimates for a von Karman system with long memory (2015, unknown): “This area of research was followed with a considerably large number of papers published on viscoelasticity and their asymptotic behavior quantized by decay rates obtained for the total energy: [25–28, 35–37,29,30,38,39].”
Moore-Gibson-Thompson equation with memory, part II: general decay of energy (2015, unknown): “There is an abundant literature on the topic, see [2, 3, 4, 6, 5, 9, 10, 14, 20, 21, 23, 24], to name a few.”
Intrinsic Decay Rate Estimates for Semilinear Abstract Second Order Equations with Memory (2014, background): “Problems related to long time behavior, including asymptotic behavior have occupied central role in the literature on viscoelastic models [16, 30, 31] including references therein and more recently [1, 2, 5, 17, 25, 26, 28, 32] among others Our goal is to study the decay rates of the energy associat”
General decay rate estimates for viscoelastic dissipative systems (2008, background): “In fact this is wrong, as proved in Fabrizio and Polidoro [10]: roughly speaking, the energy of the viscoelastic system cannot decay faster than the relaxation function.”