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1534-0392

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## Communications on Pure & Applied Analysis

2012 , Volume 11 , Issue 1

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2012, 11(1): i-i
doi: 10.3934/cpaa.2012.11.1i

*+*[Abstract](318)*+*[PDF](72.5KB)**Abstract:**

This volume deals with the mathematical modeling and the analysis of reaction-diffusion systems, as well as their applications in a number of different fields. It grew from a workshop organized by ReaDiLab, a Japan-France research collaboration unit of CNRS (Laboratoire International Associé du CNRS). This workshop took place at the University of Paris-Sud in June, 2009, bringing together many members of ReaDiLab with researchers from other French and Japanese laboratories. ReaDiLab is composed of 33 Japanese and 36 French researchers in the fields of mathematics, biology, medicine, and chemistry. Its goal is to develop mathematical modeling, analysis and numerical methods for reaction-diffusion systems arising in all those fields.

In order to understand the problems occurring in these areas of application, one should not only apply known methods, but also develop novel mathematical tools. Because of this, many results corresponding to new approaches are given in the main topics of this CPAA Special Volume, including demography and travelling waves in epidemics modelling, structured populations growth, propagation in inhomogeneous media, ecology and dry land vegetation, formation of stationary spatio-temporal patterns in reaction-diffusion systems both from a mathematical and an experimental view point, spatio-temporal dynamics of cooperation, cell migration and bacterial suspensions. This issue also includes more mathematically oriented topics such as interface dynamics, stability of non-constant stationary solutions, heterogeneity-induced spot dynamics, boundary spikes, appearance of anomalous singularities in parabolic equations, finite time blow-up, a multi-parameter inverse problem, and the numerical approximation of parabolic equations and chemotactic systems. We hope these advanced results will be useful to the community of researchers working in the domain of partial differential equations, and that they will serve as examples of mathematical modelling to those working in the different areas of application mentioned above.

2012, 11(1): 1-18
doi: 10.3934/cpaa.2012.11.1

*+*[Abstract](237)*+*[PDF](427.2KB)**Abstract:**

We investigate the singular limit, as $\varepsilon\to 0$, of the Fisher equation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $||x|| \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. Moreover, we obtain a new estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.

2012, 11(1): 19-46
doi: 10.3934/cpaa.2012.11.19

*+*[Abstract](291)*+*[PDF](3370.5KB)**Abstract:**

We present a PDE model for dilute suspensions of swimming bacteria in a three-dimensional Stokesian fluid. This model is used to calculate the statistically-stationary bulk deviatoric stress and effective viscosity of the suspension from the microscopic details of the interaction of an elongated body with the background flow. A bacterium is modeled as an impenetrable prolate spheroid with self-propulsion provided by a point force, which appears in the model as an inhomogeneous delta function in the PDE. The bacterium is also subject to a stochastic torque in order to model tumbling (random reorientation). Due to a bacterium's asymmetric shape, interactions with prescribed generic planar background flows, such as a pure straining or planar shear flow, cause the bacterium to preferentially align in certain directions. Due to the stochastic torque, the steady-state distribution of orientations is unique for a given background flow. Under this distribution of orientations, self-propulsion produces a reduction in the effective viscosity. For sufficiently weak background flows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to an effective viscosity of the suspension that is lower than the viscosity of the ambient fluid. This is in qualitative agreement with recent experiments on suspensions of

*Bacillus subtilis*.

2012, 11(1): 47-60
doi: 10.3934/cpaa.2012.11.47

*+*[Abstract](227)*+*[PDF](543.0KB)**Abstract:**

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension $2$, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d\geq 3$.

2012, 11(1): 61-82
doi: 10.3934/cpaa.2012.11.61

*+*[Abstract](378)*+*[PDF](1372.5KB)**Abstract:**

Classical models of epidemics by Ross and McKendrick have to be revisited in order to take into account the demography (fecundity, mortality and migration) both of host and vector populations and also the diffusion and mutation of infectious agents. The classical models are supposing the populations involved in the infectious disease to be constant during the epidemic wave, but the presently observed pandemics show that the duration of their spread during months or years imposes to take into account the host and vector population changes, and also the transient or permanent migration and diffusion of hosts (susceptible or infected), as well as those of vectors and infectious agents. One example is presented concerning the malaria in Mali.

2012, 11(1): 83-96
doi: 10.3934/cpaa.2012.11.83

*+*[Abstract](257)*+*[PDF](528.0KB)**Abstract:**

We prove the existence of nontrivial steady states to reaction-diffusion equations with a continuous parameter appearing in selection/mutation/competition/migration models for populations, which are structured both with respect to space and a continuous trait.

2012, 11(1): 97-113
doi: 10.3934/cpaa.2012.11.97

*+*[Abstract](311)*+*[PDF](393.2KB)**Abstract:**

In this note we analyze a spatially structured SI epidemic model with vertical transmission, a logistic effect on vital dynamics and a density dependent incidence. For a bounded spatial domain we show global stability of the endemic state when it is feasible. Then we look at the existence of travelling wave solutions connecting the endemic and the disease free states.

2012, 11(1): 115-145
doi: 10.3934/cpaa.2012.11.115

*+*[Abstract](219)*+*[PDF](538.2KB)**Abstract:**

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. The authors of [3] showed that if an initial value is close to a spiky pattern and its peak is far away from the boundary, the solution of the shadow Gierer-Meinhardt system, called a

*interior spike solution*, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a

*boundary spike solution*moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.

Approximation of nonlinear parabolic equations using
a family of conformal and non-conformal schemes

2012, 11(1): 147-172
doi: 10.3934/cpaa.2012.11.147

*+*[Abstract](326)*+*[PDF](3082.8KB)**Abstract:**

We consider a family of space discretisations for the approximation of nonlinear parabolic equations, such as the regularised mean curvature flow level set equation, using semi-implicit or fully implicit time schemes. The approximate solution provided by such a scheme is shown to converge thanks to compactness and monotony arguments. Numerical examples show the accuracy of the method.

2012, 11(1): 173-188
doi: 10.3934/cpaa.2012.11.173

*+*[Abstract](312)*+*[PDF](470.3KB)**Abstract:**

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

2012, 11(1): 189-207
doi: 10.3934/cpaa.2012.11.189

*+*[Abstract](260)*+*[PDF](943.3KB)**Abstract:**

We present a semi-empirical experimental design method to produce nontrivial chemical reaction-diffusion patterns in open reactors. We specially focus on the development of stationary patterns. The method is based on autoactivated reactions that produces spatial bistability, the addition of an independent antagonist reaction to produce spatio-temporal oscillations, and the introduction of a low mobility complexing agent that rapidly and reversibly binds the main autoactivatory species. The method is presented in formal way. Actual experimental results are used for illustration. We point out the open problems of the mathematical description: they relate to the boundary conditions, to the dimensionality of the system, and to the coupled time- and space-scale changes induced by the complexing agent.

2012, 11(1): 209-228
doi: 10.3934/cpaa.2012.11.209

*+*[Abstract](221)*+*[PDF](413.3KB)**Abstract:**

We consider a model for haptotaxis with bistable growth and study its singular limit. This yields an interface motion where the normal velocity of the interface depends on the mean curvature and on some nonlocal haptotaxis term. We prove the result for general initial data after establishing a result about generation of interface in a small time.

2012, 11(1): 229-241
doi: 10.3934/cpaa.2012.11.229

*+*[Abstract](326)*+*[PDF](485.6KB)**Abstract:**

In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous cells, whose proliferation is controlled by growth factors diffusing in the extracellular medium and binding to the cell surface. We extend the results on the existence of nonhomogenous stationary solutions obtained in [9] to a general Hill-type production function and full parameter set. Using spectral analysis and perturbation theory we derive conditions for the linearized stability of such spatial patterns.

2012, 11(1): 243-260
doi: 10.3934/cpaa.2012.11.243

*+*[Abstract](345)*+*[PDF](953.9KB)**Abstract:**

This paper deals with a class of macroscopic models for cell migration in a saturated medium for two-species mixtures. Those species tend to achieve some motion according to a desired velocity, and congestion forces them to adapt their velocity. This adaptation is modelled by a correction velocity which is chosen minimal in a least-square sense. We are especially interested in two situations: a single active species moves in a passive matrix (cell migration) with a given desired velocity, and a closed-loop Keller-Segel type model, where the desired velocity is the gradient of a self-emitted chemoattractant.

We propose a theoretical framework for the open-loop model (desired velocities are defined as gradients of given functions) based on a formulation in the form of a gradient flow in the Wasserstein space. We propose a numerical strategy to discretize the model, and illustrate its behaviour in the case of a prescribed velocity, and for the saturated Keller-Segel model.

2012, 11(1): 261-273
doi: 10.3934/cpaa.2012.11.261

*+*[Abstract](233)*+*[PDF](554.3KB)**Abstract:**

Dryland landscapes self-organize to form various patterns of vegetation patchiness. Two major classes of patterns can be distinguished: regular patterns with characteristic length scales and scale-free patterns. The latter form under conditions of global competition over the water resource. In this paper we show that the asymptotic dynamics of scale-free vegetation patterns involve patch coarsening similar to Ostwald ripening in two-phase mixtures. We demonstrate it numerically, using a spatially explicit model for water-limited vegetation, and further study it by drawing an analogy to an activator-inhibitor system that shares many properties with the vegetation system. The ecological implications of patch coarsening may not be highly significant due to the long time scales involved. The reported results, however, raise an interesting pattern formation question associated with the incompatibility of mechanisms that stabilize vegetation spots and the condition of global competition.

2012, 11(1): 275-305
doi: 10.3934/cpaa.2012.11.275

*+*[Abstract](276)*+*[PDF](6044.4KB)**Abstract:**

We are concerned with a reaction diffusion model describing slow smoldering combustion. The process consists of a sheet of paper ignited on one side and in the presence of a flow of air confined in a narrow gap above the paper. It is observed that thermal-diffusion instability generates diverse spatial patterns in combustion front propagation, depending on flow velocity of gas supply. Particularly, if the velocity is rather fast, planar front propagating with almost constant velocity appears. Motivated by this observation, we discuss the existence and stability of $1$ dimensional traveling wave solutions of the model.

2012, 11(1): 307-338
doi: 10.3934/cpaa.2012.11.307

*+*[Abstract](348)*+*[PDF](1388.2KB)**Abstract:**

Spatially localized patterns form a representative class of patterns in dissipative systems. We study how the dynamics of traveling spots in two-dimensional space change when heterogeneities are introduced in the media. The simplest but fundamental one is a line heterogeneity of jump type. When spots encounter the jump, they display various outputs including penetration, rebound, and trapping depending on the incident angle and its height. The system loses translational symmetry by the heterogeneity, but at the same time, it causes the emergence of various types of heterogeneity-induced-ordered-patterns (HIOPs) replacing the homogeneous constant state. We study these issues by using a three-component reaction-diffusion system with one activator and two inhibitors. The above outputs can be obtained through the interaction between the HIOPs and the traveling spots. The global bifurcation and eigenvalue behavior of HISPs are the key to understand the underlying mechanisms for the transitions among those dynamics. A reduction to a finite dimensional system is presented here to extract the model-independent nature of the dynamics. Selected numerical techniques for the bifurcation analysis are also provided.

2012, 11(1): 339-364
doi: 10.3934/cpaa.2012.11.339

*+*[Abstract](515)*+*[PDF](486.1KB)**Abstract:**

We are concerned with the finite-element approximation for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. The scheme makes use of a semi-implicit time discretization with a time-increment control and Baba-Tabata's conservative upwind finite-element approximation in order to realize the positivity and mass conservation properties. The main aim is to present error analysis that is an application of the discrete version of the analytical semigroup theory.

2012, 11(1): 365-373
doi: 10.3934/cpaa.2012.11.365

*+*[Abstract](312)*+*[PDF](354.4KB)**Abstract:**

A linear second order elliptic equation describing heat or mass diffusion and convection on a given velocity field is considered in $R^3$. The corresponding operator $L$ may not satisfy the Fredholm property. In this case, solvability conditions for the equation $L u = f$ are not known. In this work, we derive solvability conditions in $H^2(R^3)$ for the non self-adjoint problem by relating it to a self-adjoint Schrödinger type operator, for which solvability conditions are obtained in our previous work [13].

2012, 11(1): 375-386
doi: 10.3934/cpaa.2012.11.375

*+*[Abstract](216)*+*[PDF](340.8KB)**Abstract:**

In this paper, we propose a model describing the dynamics of human society. Then we show that classic theorems on traveling wave solutions in a reaction diffusion equation can be readily applied and we obtain some mathematical result. Our model considers a human society where people play a two-strategy multi-player game. The key concepts are 1) payoff-dependent update rule of strategy and 2) social learning with conformism. Because of conformism, the system can be bistable and our primary concern is whether one global majority appears or not when multiple societies that initially have different local majorities are spatially connected. Applying the result of this general framework to a public goods game example, we show that cooperation is less likely maintained by conformism and that the spread of irrational spite behavior can occur.

2012, 11(1): 387-405
doi: 10.3934/cpaa.2012.11.387

*+*[Abstract](250)*+*[PDF](398.6KB)**Abstract:**

The Cauchy problem for a parabolic partial differential equation with a power nonlinearity is studied. It is known that in some parameter range, there exists a time-local solution whose singularity has the same asymptotics as that of a singular steady state. In this paper, a sufficient condition for initial data is given for the existence of a solution with a moving singularity that becomes anomalous in finite time.

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