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Displaying 101 –
120 of
671
We study a reaction-diffusion equation
with an integral term describing nonlocal consumption of resources
in population dynamics. We show that a homogeneous equilibrium can
lose its stability resulting in appearance of stationary spatial
structures. They can be related to the emergence of biological
species due to the intra-specific competition and random
mutations.
Various types of travelling waves are observed.
We study pattern-forming instabilities in reaction-advection-diffusion systems. We
develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a
spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal
with the flows periodic in space that may have arbitrary time dependence. We propose a
discrete in time model, where reaction, advection, and diffusion act as successive
operators, and show that...
A global feedback control of a system that exhibits a subcritical monotonic instability
at a non-zero wavenumber (short-wave, or Turing instability) in the presence of a zero
mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero
mode. The method based on a variational principle is applied for the derivation of a
low-dimensional evolution model. In the framework of this model the investigation of the
system’s dynamics...
Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on...
This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a...
This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations via impressed current. The optimization problem is to find optimal voltages so that, under certain constraints on the voltages and the temperature, a...
In 1962, Dyson showed that the spectrum of a random Hermitian matrix, whose
entries (real and imaginary) diffuse according to independent Ornstein-Uhlenbeck
processes, evolves as non-colliding Brownian particles held together by a drift term.
When , the largest eigenvalue, with time and space properly
rescaled, tends to the so-called Airy process, which is a non-markovian continuous
stationary process. Similarly the eigenvalues in the bulk, with a different time and
space rescaling, tend...
The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.
We apply Robin penalization to Dirichlet optimal control problems governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.
We apply Robin penalization to Dirichlet optimal control problems
governed by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.
A second order elliptic problem with axisymmetric data is solved in a finite element space, constructed on a triangulation with curved triangles, in such a way, that the (nonhomogeneous) boundary condition is fulfilled in the sense of a penalty. On the basis of two approximate solutions, extrapolates for both the solution and the boundary flux are defined. Some a priori error estimates are derived, provided the exact solution is regular enough. The paper extends some of the results of J.T. King...
The Penrose transform gives an isomorphism between the kernel of the -Dirac operator over an affine subset and the third sheaf cohomology group on the twistor space. In the paper we give an integral formula which realizes the isomorphism and decompose the kernel as a module of the Levi factor of the parabolic subgroup. This gives a new insight into the structure of the kernel of the operator.
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