Initial Value Problems in Viscoelasticity

1988 ◽  
Vol 41 (10) ◽  
pp. 371-378 ◽  
Author(s):  
W. J. Hrusa ◽  
J. A. Nohel ◽  
M. Renardy
Keyword(s):  
Classical Solutions ◽  
Linear Wave ◽  
Time Behavior ◽  
Initial Value ◽  
One Dimensional ◽  
Nonlinear Viscoelastic ◽  
Linear Wave Propagation ◽  
Long Time ◽  
Nonlinear Viscoelastic Materials ◽  
Time Existence

We review some recent mathematical results concerning integrodiff erential equations that model the motion of one-dimensional nonlinear viscoelastic materials. In particular, we discuss global (in time) existence and long-time behavior of classical solutions, as well as the formation of singularities in finite time from smooth initial data. Although the mathematical theory is comparatively incomplete, we make some remarks concerning the existence of weak solutions (i e, solutions with shocks). Some relevant results from linear wave propagation will also be discussed.

Long-time existence of classical solutions to a one-dimensional swelling gel

2014 ◽  
Vol 25 (01) ◽  
pp. 165-194 ◽  
Author(s):  
M. Carme Calderer ◽  
Robin Ming Chen
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Lagrangian Formulation ◽  
Classical Solutions ◽  
Governing Equations ◽  
Fixed Boundary ◽  
One Dimensional ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

In this paper, we derived a model which describes the swelling dynamics of a gel and study the system in one-dimensional geometry with a free boundary. The governing equations are hyperbolic with a weakly dissipative source. Using a mass-Lagrangian formulation, the free boundary is transformed into a fixed boundary. We prove the existence of long-time C1-solutions to the transformed fixed boundary problem.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

Long-time behavior for the full one-dimensional Frémond model for shape memory alloys

2000 ◽  
Vol 12 (6) ◽  
pp. 423-433 ◽  
Author(s):  
Pierluigi Colli ◽  
Philippe Laurençot ◽  
Ulisse Stefanelli
Keyword(s):  
Shape Memory Alloys ◽  
Shape Memory ◽  
Time Behavior ◽  
Long Time Behavior ◽  
One Dimensional ◽  
Long Time

scholarly journals Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis–haptotaxis

2019 ◽  
Vol 29 (07) ◽  
pp. 1387-1412 ◽  
Author(s):  
Peter Y. H. Pang ◽  
Yifu Wang
Keyword(s):  
Tumor Angiogenesis ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
System Of Differential Equations ◽  
One Dimensional ◽  
Long Time ◽  
Bounded Smooth ◽  
The One ◽  
Global Boundedness ◽  
Angiogenesis Model

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain [Formula: see text] ([Formula: see text]): [Formula: see text] where [Formula: see text] and [Formula: see text] are positive parameters. For any reasonably regular initial data [Formula: see text], we prove the global boundedness ([Formula: see text]-norm) of [Formula: see text] via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution [Formula: see text] converges to [Formula: see text] with an explicit exponential rate as time tends to infinity.

FROM BOUNDARY VALUE PROBLEMS TO DIFFERENCE EQUATIONS: A METHOD OF INVESTIGATION OF CHAOTIC VIBRATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1285-1306 ◽  
Author(s):  
E. YU. ROMANENKO ◽  
A. N. SHARKOVSKY
Keyword(s):  
Difference Equation ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Dynamical Behavior ◽  
Time Behavior ◽  
One Dimensional ◽  
Long Time ◽  
The Difference ◽  
The One

Among evolutionary boundary value problems for partial differential equations, there is a wide class of problems reducible to difference, differential-difference and other relevant equations. Of especial promise for investigation are problems that reduce to difference equations with continuous argument. Such problems, even in their simplest form, may exhibit solutions with extremely complicated long-time behavior to the extent of possessing evolutions that are indistinguishable from random ones when time is large. It is owing to the reduction to a difference equation followed by the employment of the properties of the one-dimensional map associated with the difference equation, that, it is in many cases possible to establish mathematical mechanisms for one or other type of dynamical behavior of solutions. The paper presents the overall picture in the study of boundary value problems reducible to difference equations (on which the authors have a direct bearing over the last ten years) and demonstrates with several simplest examples the potentialities that such a reduction opens up.

Non-linear wave propagation in a relaxing gas

1969 ◽  
Vol 37 (1) ◽  
pp. 31-50 ◽  
Author(s):  
P. A. Blythe
Keyword(s):  
Near Field ◽  
Linear Wave ◽  
Steady Flows ◽  
Energy Limit ◽  
High Frequency Region ◽  
One Dimensional ◽  
Non Linear ◽  
Linear Wave Propagation ◽  
Matching Techniques ◽  
Relaxing Gas

An outline of the classical far-and near-field solutions for small-amplitude one-dimensional unsteady flows in a general inviscid relaxing gas is given. The structure of the complete flow field, including a non-linear near-frozen (high frequency) region at the front, is obtained by matching techniques when the relaxation time is ‘large’.If the energy in the relaxing mode is small compared with the total internal energy, the solution in the far field is, in general, more complex than that predicted by classical theory. In this case the rate process is not necessarily able to diffuse all convective steepening. An equation valid in this limit is derived and discussed. In particular, a sufficient condition for the flow to be shock-free is established. For an impulsively withdrawn piston it is shown that the solution is single-valued both within and downstream of the fan. Some useful similarity rules are pointed out.The corresponding formulation for two-dimensional steady flows is also noted in the small energy limit.

scholarly journals Towards the solution of creep problems of thin-shelled tubular elements in isotropic nonlinear viscoelastic materials

2019 ◽  
pp. 42-45
Author(s):  
V. P. Golub
Keyword(s):  
Creep Strain ◽  
Viscoelastic Properties ◽  
Volumetric Strain ◽  
Combined Loading ◽  
Strain Intensity ◽  
Viscoelastic Materials ◽  
One Dimensional ◽  
Nonlinear Viscoelastic ◽  
Nonlinear Viscoelastic Materials ◽  
Transverse Strains

A new approach to the creep strains analysis of thin-shelled tubular elements in isotropic nonlinear viscoelastic materials under combined loading with uniaxial tension and torsion has been proposed. The system of equations that is constructed according to the deviators proportionality hypothesis has been chosen as the creep constitutive equations the nonlinearity of viscoelastic properties in which is given with respect to the creep strain intensity and volumetric strain by the Rabotnov type models. The kernels of creep strain intensity and volumetric strain are given by the relations that establish the relationships between these kernels and one-dimensional creep kernels determined from a system of base experiments. One-dimensional tension with the measurement of longitudinal and transverse strains as well as one-dimensional tension and pure torsion with the measurement of longitudinal and shearing strains have been considered as base experiments. The functions of nonlinearity of viscoelastic properties are given by smoothing cubic splines. The problems of the analysis of longitudinal, transverse and shearing strains of thin-shelled tubular specimens made of “high density polyethylene PEHD” have been solved and experimentally approved.

ON A HYPERBOLIC–PARABOLIC SYSTEM MODELING CHEMOTAXIS

2011 ◽  
Vol 21 (08) ◽  
pp. 1631-1650 ◽  
Author(s):  
DONG LI ◽  
TONG LI ◽  
KUN ZHAO
Keyword(s):  
Steady State ◽  
Parabolic System ◽  
Smooth Solution ◽  
System Modeling ◽  
Fourier Method ◽  
Classical Solutions ◽  
Time Behavior ◽  
Time Dynamics ◽  
Blowup Criterion ◽  
Long Time

We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic–parabolic system derived from the Keller–Segel model describing chemotaxis. It is shown that local smooth solution blows up if and only if the accumulation of the L∞ norm of the solution reaches infinity within the lifespan. Our blowup criteria are consistent with the chemotaxis phenomenon that the movement of cells (bacteria) is driven by the gradient of the chemical concentration. Furthermore, we study the long-time dynamics when the initial data is sufficiently close to a constant positive steady state. By using a new Fourier method adapted to the linear flow, it is shown that the smooth solution exists for all time and converges exponentially to the constant steady state with a frequency-dependent decay rate as time goes to infinity.

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