3,608 research outputs found
On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay
An initial-boundary value problem for a viscoelastic wave equation subject to
a strong time-localized delay in a Kelvin & Voigt-type material law is
considered. Transforming the equation to an abstract Cauchy problem on the
extended phase space, a global well-posedness theory is established using the
operator semigroup theory both in Sobolev-valued - and BV-spaces. Under
appropriate assumptions on the coefficients, a global exponential decay rate is
obtained and the stability region in the parameter space is further explored
using the Lyapunov's indirect method. The singular limit is
further studied with the aid of the energy method. Finally, a numerical example
from a real-world application in biomechanics is presented.Comment: 34 pages, 4 figures, 1 set of Matlab code
General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback
In this paper we consider a viscoelastic wave equation with a time-varying
delay term, the coefficient of which is not necessarily positive. By
introducing suitable energy and Lyapunov functionals, under suitable
assumptions, we establish a general energy decay result from which the
exponential and polynomial types of decay are only special cases.Comment: 11 page
Moore-Gibson-Thompson equation with memory, part II: general decay of energy
We study a temporally third order (Moore-Gibson-Thompson) equation with a
memory term. Previously it is known that, in non-critical regime, the global
solutions exist and the energy functionals decay to zero. More precisely, it is
known that the energy has exponential decay if the memory kernel decays
exponentially. The current work is a generalization of the previous one (Part
I) in that it allows the memory kernel to be more general and shows that the
energy decays the same way as the memory kernel does, exponentially or not.Comment: 22 page
The response function of a sphere in a viscoelastic two-fluid medium
In order to address basic questions of importance to microrheology, we study
the dynamics of a rigid sphere embedded in a model viscoelastic medium
consisting of an elastic network permeated by a viscous fluid. We calculate the
complete response of a single bead in this medium to an external force and
compare the result to the commonly-accepted, generalized Stokes-Einstein
relation (GSER). We find that our response function is well approximated by the
GSER only within a particular frequency range determined by the material
parameters of both the bead and the network. We then discuss the relevance of
this result to recent experiments. Finally we discuss the approximations made
in our solution of the response function by comparing our results to the exact
solution for the response function of a bead in a viscous (Newtonian) fluid.Comment: 12 pages, 2 figure
On a poroviscoelastic model for cell crawling
In this paper a minimal, one–dimensional, two–phase, viscoelastic, reactive, flow model for a crawling cell is presented. Two–phase models are used with a variety of constitutive assumptions in the literature to model cell motility. We use an upper–convected Maxwell model and demonstrate that even the simplest of two–phase, viscoelastic models displays features relevant to cell motility. We also show care must be exercised in choosing parameters for such models as a poor choice can lead to an ill–posed problem. A stability analysis reveals that the initially stationary, spatially uniform strip of cytoplasm starts to crawl in response to a perturbation which breaks the symmetry of the network volume fraction or network stress. We also demonstrate numerically that there is a steady travelling–wave solution in which the crawling velocity has a bell–shaped dependence on adhesion strength, in agreement with biological observation
Asymptotic profiles and singular limits for the viscoelastic damped wave equation with memory of type I
In this paper, we are interested in the Cauchy problem for the viscoelastic
damped wave equation with memory of type I. By applying WKB analysis and
Fourier analysis, we explain the memory's influence on dissipative structures
and asymptotic profiles of solutions to the model with weighted initial
data. Furthermore, concerning standard energy and the solution itself, we
establish singular limit relations between the Moore-Gibson-Thompson equation
with memory and the viscoelastic damped wave equation with memory
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