Global convergence of a kinetic model of chemotaxis to a perturbed Keller–Segel model

We consider a class of kinetic models of chemotaxis with two positive non-dimensional parameters coupled to a parabolic equation of the chemo-attractant. If both parameters are set equal zero, we have the classical Keller–Segel model for chemotaxis. We prove global existence of solutions of this two-parameters kinetic model and prove convergence of this model to models of chemotaxis with global existence when one of these two parameters is set equal zero. In one case, we find as a limit model a kinetic model of chemotaxis while in the other case we find a perturbed Keller–Segel model with global existence of solutions.     

Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.     

Existence,uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller–Segel system in the plane

In the present article we consider several issues concerning the doubly parabolic Keller–Segel system  and  in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. In particular, we prove that there exist integral solutions of any mass, provided that ε>0$\epsilon >0$ is sufficiently large. With those results at hand, we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term (α=0)$(\alpha =0)$ the solutions behave like self-similar solutions, while in the presence of the degradation term (α>0)$(\alpha >0)$ the global solutions behave like the heat kernel.     

A finite volume scheme for the Patlak–Keller–Segel chemotaxis model

A finite volume method is presented to discretize the Patlak–Keller–Segel (PKS) modeling chemosensitive movements. First, we prove existence and uniqueness of a numerical solution to the proposed scheme. Then, we give a priori estimates and establish a threshold on the initial mass, for which we show that the numerical approximation converges to the solution to the PKS system when the initial mass is lower than this threshold. Numerical simulations are performed to verify accuracy and the properties of the scheme. Finally, in the last section we investigate blow-up of the solution for large mass.     

Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect

We consider the elliptic-parabolic PDE system

     

Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains

We study the initial-boundary-value problems for multidimensional scalar conservation laws in noncylindrical domains with Lipschitz boundary. We show the existence-uniqueness of this problem for initial-boundary data in L^∞ and the flux-function in the class C¹. In fact, first considering smooth boundary, we obtain the L¹-contraction property, discuss the existence problem and prove it by the Young measures theory. In the end we show how to pass the existence-uniqueness results on to some domains with Lipschitz boundary.     

A direct identification method for current dipoles from electromagnetic fields

We discuss the identification problem for current dipoles in a spherically symmetric conductor. This mathematical model is used for a biomedical inverse problem such as the source current identification for the human brain activity. We have already proposed a direct identification method for this inverse source problem using observations of the magnetic fields outside of the conductor. One of the difficulties of current dipole identification using the magnetic fields is caused by the fact that magnetic field does not include any information about the radial component of dipole moments. In this paper, we consider an improvement of the direct method to identify both radial and tangential components of current dipole moments by combining electric and magnetic observation data. Furthermore, our approach is effective in the case where the number of dipoles is unknown.     

Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annulus

Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.     

Nilpotent Lie algebras with 2-dimensional commutator ideals

We classify all (finitely dimensional) nilpotent Lie k-algebras h with 2-dimensional commutator ideals h^′, extending a known result to the case where h^′ is non-central and k is an arbitrary field. It turns out that, while the structure of h depends on the field k if h^′ is central, it is independent of k if h^′ is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h^′ and dim_kh?11.     

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

D-optimal designs embedded in Hadamard matrices and their effect on the pivot patterns

In this paper we develop a new approach for detecting if specific D-optimal designs exist embedded in Sylvester-Hadamard matrices. Specifically, we investigate the existence of the D-optimal designs of orders 5, 6, 7 and 8. The problem is motivated to explaining why specific values appear as pivot elements when Gaussian elimination with complete pivoting is applied to Hadamard matrices. Using this method and a complete search algorithm we explain, for the first time, the appearance of concrete pivot values for equivalence classes of Hadamard matrices of orders n = 12, 16 and 20.     

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology

In this paper we will focus on a parabolic degenerate system with respect to unknown functions u and w on a bounded domain of the two dimensional Euclidean space. This system appears as a mathematical model for some biological processes. Global existence and uniqueness of a nonnegative classical Hölder continuous solution are proved. The last part of the paper is devoted to the study of the asymptotic behavior of the solutions.     

Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces

In this paper, we study Keller-Segel systems with fractional diffusion and a nonlocal term. We establish the global existence, uniqueness and stability of solutions for systems with small initial data in critical Besov spaces. Our main tools are the L^p−L^q estimates for in Besov spaces and the perturbation of linearization.     

Realization of prescribed patterns in the competition model

We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D₁,…,D_m of a given spatial domain Ω in R^N, if d₁,d₂,a₁,a₂,c,d,e are positive continuous functions on Ω and b(x) is identically zero on D?D₁∪?∪D_m and positive in the rest of Ω, then for suitable choices of the parameters λ, μ and all small ε>0, the competition model

     

Existence and asymptotic behaviour for the time‐fractional Keller–Segel model for chemotaxis

One of the most important systems for understanding chemotactic aggregation is the Keller–Segel system. We consider the time‐fractional Keller–Segel system of order . We prove an existence result with small initial data in a class of Besov–Morrey spaces. Self‐similar solutions are obtained and we also show an asymptotic behaviour result.     

An asymptotic expansion in wavelet analysis and its application to accurate numerical wavelet decomposition

We consider a dilation operatorT admitting a scaling function with compact support as fixed point. It is shown that the adjoint operatorT*admits a sequence of polynomial eigenfunctions and that a smooth functionf admits an expansion in these eigenfunctions, which reveals the asymptotic behavior ofT* forn.Due to this asymptotic expansion, an extrapolation technique can be applied for the accurate numerical computation of the integrals appearing in the wavelet decomposition of a smooth function. This extrapolation technique fits well in a multiresolution scheme.