ISSN:

1937-1632

eISSN:

1937-1179

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## Discrete & Continuous Dynamical Systems - S

February 2012 , Volume 5 , Issue 1

Issue on fast reaction - slow diffusion scenarios:

PDE approximations and free boundaries

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2012, 5(1): i-i
doi: 10.3934/dcdss.2012.5.1i

*+*[Abstract](1737)*+*[PDF](81.5KB)**Abstract:**

This issue is focussed on the modeling, analysis and simulation of fast reaction-slow transport scenarios as well as corresponding fast-reaction limits. Within this framework, internal sharp and thin reaction layers form and travel through the spatial domain often producing unexpected effects. Such situations appear in a variety of significant applications; for example flame propagation in combustion, segregation and aggregation of biological individuals, chemical attack on reactive porous materials (such as concrete or natural stone), dissolution and precipitation reactions in minerals, tumor growth, grain boundary motion, and temperature-induced phase transitions in shape-memory alloys represent typical cases in which the fast process is localized within a

*a priori*unknown internal active layer.

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2012, 5(1): 1-13
doi: 10.3934/dcdss.2012.5.1

*+*[Abstract](2418)*+*[PDF](165.5KB)**Abstract:**

In our previous works we proposed and studied the mathematical model for the position of the joint of a shape memory alloy and a bias springs in case the temperature is known. The purpose of this paper is to establish a mathematical model with unknown temperature and to show a local existence of a solution to the model in time.

2012, 5(1): 15-28
doi: 10.3934/dcdss.2012.5.15

*+*[Abstract](2152)*+*[PDF](456.1KB)**Abstract:**

We study a one-dimensional model describing the motion of a shape-memory alloy spring at a small characteristic time scale, called here

*fast-temperature-activation limit*. At this level, the standard Falk's model reduces to a nonlinear elliptic partial differential equation (PDE) with Newton boundary condition. We show existence and uniqueness of a bounded weak solution and approximate this numerically. Interestingly, in spite of the nonlinearity of the model, the approximate solution exhibits nearly a linear profile. Finally, we extend the reduced model to the simplest PDE system for shape memory alloys that can capture oscillations and then damp out these oscillations numerically. The numerical results for both limiting cases show excellent agreement. The graphs show that the valve opens in an instant, which is realistic behavior of the free boundary.

2012, 5(1): 29-48
doi: 10.3934/dcdss.2012.5.29

*+*[Abstract](2387)*+*[PDF](491.7KB)**Abstract:**

We consider the elastic theory of single crystals at constant temperature where the free energy density depends on the local concentration of one or more species of particles in such a way that for a given local concentration vector certain lattice geometries (phases) are preferred. Furthermore we consider possible large deformations of the crystal lattice. After deriving the physical model, we indicate by means of a suitable implicite time discretization an existence result for measure-valued solutions that relies on a new existence theorem for Young measures in infinite settings. This article is an overview of [2].

2012, 5(1): 49-59
doi: 10.3934/dcdss.2012.5.49

*+*[Abstract](2615)*+*[PDF](355.5KB)**Abstract:**

We consider reaction-diffusion systems which, in addition to certain slow reactions, contain a fast irreversible reaction in which chemical components A and B form a product P. In this situation and under natural assumptions on the RD-system we prove the convergence of weak solutions, as the reaction speed of the irreversible reaction tends to infinity, to a weak solution of a limiting system. The limiting system is a Stefan-type problem with a moving interface at which the chemical reaction front is localized.

2012, 5(1): 61-92
doi: 10.3934/dcdss.2012.5.61

*+*[Abstract](2052)*+*[PDF](563.5KB)**Abstract:**

In this paper we introduce a novel generic destabilization mechanism for (reversible) spatially periodic patterns in reaction-diffusion equations in one spatial dimension. This Hopf dance mechanism occurs for long wavelength patterns near the homoclinic tip of the associated Busse balloon ($=$ the region in (wave number, parameter space) for which stable periodic patterns exist). It shows that the boundary of the Busse balloon locally has a fine-structure of two intertwining 'dancing' (or 'snaking') Hopf destabilization curves (or manifolds) that limit on the Hopf bifurcation value of the associated homoclinic limit pulse and that have infinitely many, accumulating, intersections. The Hopf dance is first recovered by a detailed numerical analysis of the full Busse balloon in an explicit Gray-Scott model. The structure, and its generic nature, is confirmed by a rigorous analysis of singular long wave length patterns in a normal form model for pulse-type solutions in two component, singularly perturbed, reaction-diffusion equations.

2012, 5(1): 93-113
doi: 10.3934/dcdss.2012.5.93

*+*[Abstract](2625)*+*[PDF](557.0KB)**Abstract:**

In this paper we consider a two-phase flow problem in porous media and study its singular limit as the viscosity of the air tends to zero; more precisely, we prove the convergence of subsequences to solutions of a generalized Richards model.

2012, 5(1): 115-126
doi: 10.3934/dcdss.2012.5.115

*+*[Abstract](2684)*+*[PDF](316.0KB)**Abstract:**

We formulate a reaction diffusion equation with non-local term as a mean field equation of the master equation where the particle density is defined continuously in space and time. In the case of the constant mean waiting time, this limit equation is associated with the diffusion coefficient of A. Einstein, the reaction rate in phenomenology, and the Debye term under the presence of potential.

2012, 5(1): 127-146
doi: 10.3934/dcdss.2012.5.127

*+*[Abstract](2422)*+*[PDF](419.5KB)**Abstract:**

We consider a model for grain boundary motion with constraint. In composite material science it is very important to investigate the grain boundary formation and its dynamics. In this paper we study a phase-filed model of grain boundaries, which is a modified version of the one proposed by R. Kobayashi, J.A. Warren and W.C. Carter [18]. The model is described as a system of a nonlinear parabolic partial differential equation and a nonlinear parabolic variational inequality. The main objective of this paper is to show the global existence of a solution for our model, employing some subdifferential techniques in the convex analysis.

2012, 5(1): 147-158
doi: 10.3934/dcdss.2012.5.147

*+*[Abstract](2703)*+*[PDF](346.0KB)**Abstract:**

Reaction-diffusion system approximations to a cross-diffusion system are investigated. Iida and Ninomiya~[Recent Advances on Elliptic and Parabolic Issues, 145--164 (2006)] proposed a semilinear reaction-diffusion system with a small parameter and showed that the limit equation takes the form of a weakly coupled cross-diffusion system provided that solutions of both the reaction-diffusion and the cross-diffusion systems are sufficiently smooth. In this paper, the results are extended to a more general cross-diffusion problem involving strongly coupled systems. It is shown that a solution of the problem can be approximated by that of a semilinear reaction-diffusion system without any assumptions on the solutions. This indicates that the mechanism of cross-diffusion might be captured by reaction-diffusion interaction.

2012, 5(1): 159-181
doi: 10.3934/dcdss.2012.5.159

*+*[Abstract](2648)*+*[PDF](530.0KB)**Abstract:**

In this paper we study an optimal control problem for a singular diffusion equation associated with total variation energy. The singular diffusion equation is derived as an Allen-Cahn type equation, and then the observing optimal control problem corresponds to a temperature control problem in the solid-liquid phase transition. We show the existence of an optimal control for our singular diffusion equation by applying the abstract theory. Next we consider our optimal control problem from the view-point of numerical analysis. In fact we consider the approximating problem of our equation, and we show the relationship between the original control problem and its approximating one. Moreover we show the necessary condition of an approximating optimal pair, and give a numerical experiment of our approximating control problem.

2012, 5(1): 183-189
doi: 10.3934/dcdss.2012.5.183

*+*[Abstract](2254)*+*[PDF](155.6KB)**Abstract:**

We study a new formulation for the Eikonal equation $|\nabla u| =1$ on a bounded subset of $\R^2$. Considering a field $P$ of orthogonal projections onto $1$-dimensional subspaces, with div$ P \in L^2$, we prove existence and uniqueness for solutions of the equation $P$ div $P$=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve.

This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals.

2012, 5(1): 191-207
doi: 10.3934/dcdss.2012.5.191

*+*[Abstract](2008)*+*[PDF](557.9KB)**Abstract:**

In this paper we discuss two models involving protein binding. The first model describes a system involving a drug, a receptor and a protein, and the question is to what extent the affinity of the drug to the protein affects the drug-receptor binding and thereby the efficiency of the drug. The second model is the basic model underlying

*Target-Mediated Drug Disposition*, which describes the pharmacokinetics of a drug in the presence of a target, often a receptor.

2012, 5(1): 209-217
doi: 10.3934/dcdss.2012.5.209

*+*[Abstract](2109)*+*[PDF](313.9KB)**Abstract:**

This paper studies a dynamical stability of the steady state for some thermoelastic and thermoviscoelastic systems in multi-dimensional space domain. More general nonlinear term can be taken here than the one in [6] which studied the stability for the one-dimensional system called the Falk model system. We also give applications to thermoviscoelastic systems treated in [8] and [9].

2012, 5(1): 219-234
doi: 10.3934/dcdss.2012.5.219

*+*[Abstract](2064)*+*[PDF](336.7KB)**Abstract:**

Consider the reaction front formed when two initially separated reactants A and B are brought into contact and react at a rate proportional to $A^n B^m$ when the concentrations $A$ and $B$ are positive. Further, suppose that both $n$ and $m$ are less than unity. Then the leading order large time asymptotic reaction rate has compact support, i.e. the reaction zone where the reaction takes place has a finite width and the reaction rate is identically zero outside of this region. In the large time asymptotic limit an analytical approximate solution to the reactant concentrations is constructed in the vicinity of the reaction zone. The approximate solution is found to be in good agreement with numerically obtained solutions. For $n \ne m$ the location of the maximum reaction rate does not coincide with the centre of mass of the reaction, and further for $n>m$ this local maximum is shifted slightly closer to the zone that initially contained species A, with the reverse holding when $m>n$. The three limits $m\rightarrow 0$, $n\rightarrow 1$ and $m,n\rightarrow 1$ are given special attention.

2012, 5(1): 235-244
doi: 10.3934/dcdss.2012.5.235

*+*[Abstract](1896)*+*[PDF](418.7KB)**Abstract:**

We consider a relation between proliferation of solid tumor cells and time-changes of the quantities of heat shock proteins in them. To do so, in the present paper we start to obtain some experimental data of the proliferation curves of solid tumor cells, actually, A549 and HepG2, as well as the time-changes of proteins, especially HSP90 and HSP72, in them. And we propose a mathematical model to re-create the experimental data of the proliferation curves and the time-changes of the quantities of heat shock proteins, which is described by ODE systems. Finally, we discuss a problem which exists between mitosis of solid tumor cells and time-changes of the quantities of heat shock proteins, from the viewpoint of biotechnology.

2019 Impact Factor: 1.233

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