scholarly journals Radially symmetric solutions of a chemotaxis model in the plane–the supercritical case

2008 ◽  
Author(s):  
Piotr Biler
Keyword(s):  
Chemotaxis Model ◽  
Supercritical Case ◽  
Radially Symmetric Solutions

The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane

2006 ◽  
Vol 29 (13) ◽  
pp. 1563-1583 ◽  
Author(s):  
Piotr Biler ◽  
Grzegorz Karch ◽  
Philippe Laurençot ◽  
Tadeusz Nadzieja
Keyword(s):  
Chemotaxis Model ◽  
Radially Symmetric Solutions

scholarly journals On the decay of the energy for radial solutions in Moore–Gibson–Thompson thermoelasticity

2021 ◽  
pp. 108128652199425
Author(s):  
Noelia Bazarra ◽  
José R Fernández ◽  
Ramón Quintanilla
Keyword(s):  
Numerical Simulations ◽  
Exponential Decay ◽  
Time Variable ◽  
Radial Solutions ◽  
Radially Symmetric Solutions ◽  
Thermoelastic Theory

In this paper, we consider the Moore–Gibson–Thompson thermoelastic theory. We restrict our attention to radially symmetric solutions and we prove the exponential decay with respect to the time variable. We demonstrate this fact with the help of energy arguments. Later, we give some numerical simulations to illustrate this behaviour.

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

scholarly journals Radially symmetric solutions to the Hénon–Lane–Emden system on the critical hyperbola

2014 ◽  
Vol 16 (03) ◽  
pp. 1350030 ◽  
Author(s):  
Roberta Musina ◽  
K. Sreenadh
Keyword(s):  
Variational Methods ◽  
Existence Results ◽  
Qualitative Properties ◽  
Radially Symmetric Solutions ◽  
Qualitative Properties Of Solutions

We use variational methods to study the existence of non-trivial and radially symmetric solutions to the Hénon–Lane–Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative properties of solutions and non-existence results.

scholarly journals ENERGY CRITICAL NLS IN TWO SPACE DIMENSIONS

2009 ◽  
Vol 06 (03) ◽  
pp. 549-575 ◽  
Author(s):  
J. COLLIANDER ◽  
S. IBRAHIM ◽  
M. MAJDOUB ◽  
N. MASMOUDI
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Energy Space ◽  
Well Posedness ◽  
Initial Value ◽  
Exponential Nonlinearity ◽  
Supercritical Case ◽  
Two Space Dimensions

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity [Formula: see text] We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.

scholarly journals Spatiotemporal evolution in a (2+1)-dimensional chemotaxis model

2012 ◽  
Vol 391 (1-2) ◽  
pp. 107-112 ◽  
Author(s):  
Santo Banerjee ◽  
Amar P. Misra ◽  
L. Rondoni
Keyword(s):  
Spatiotemporal Evolution ◽  
Chemotaxis Model

On ring solutions of neural field equations

2021 ◽  
pp. 363-371
Author(s):  
Rachid Atmania ◽  
Evgenii O. Burlakov ◽  
Ivan N. Malkov
Keyword(s):  
Activation Function ◽  
Stationary Solutions ◽  
Neural Field ◽  
Neuronal Activation ◽  
Field Equations ◽  
Broad Ring ◽  
Activation Threshold ◽  
Electrical Potentials ◽  
Radially Symmetric Solutions ◽  
Neural Field Equations

The article is devoted to investigation of integro-differential equation with the Hammerstein integral operator of the following form: ∂_t u(t,x)=-τu(t,x,x_f )+∫_(R^2)▒〖ω(x-y)f(u(t,y) )dy, t≥0, x∈R^2 〗. The equation describes the dynamics of electrical potentials u(t,x) in a planar neural medium and has the name of neural field equation.We study ring solutions that are represented by stationary radially symmetric solutions corresponding to the active state of the neural medium in between two concentric circles and the rest state elsewhere in the neural field. We suggest conditions of existence of ring solutions as well as a method of their numerical approximation. The approach used relies on the replacement of the probabilistic neuronal activation function f that has sigmoidal shape by a Heaviside-type function. The theory is accompanied by an example illustrating the procedure of investigation of ring solutions of a neural field equation containing a typically used in the neuroscience community neuronal connectivity function that allows taking into account both excitatory and inhibitory interneuronal interactions. Similar to the case of bump solutions (i. e. stationary solutions of neural field equations, which correspond to the activated area in the neural field represented by the interior of some circle) at a high values of the neuronal activation threshold there coexist a broad ring and a narrow ring solutions that merge together at the critical value of the activation threshold, above which there are no ring solutions.

scholarly journals Skeletons of Near-Critical Bienaymé-Galton-Watson Branching Processes

2015 ◽  
Vol 47 (2) ◽  
pp. 530-544
Author(s):  
Serik Sagitov ◽  
Maria Conceição Serra
Keyword(s):  
Critical Case ◽  
Branching Processes ◽  
Threshold Value ◽  
High Threshold ◽  
Birth Process ◽  
Birth And Death Processes ◽  
Supercritical Case ◽  
Natural Choice ◽  
Future Reproduction ◽  
Asymptotic Representations

Skeletons of branching processes are defined as trees of lineages characterized by an appropriate signature of future reproduction success. In the supercritical case a natural choice is to look for the lineages that survive forever (O'Connell (1993)). In the critical case it was suggested that the particles with the total number of descendants exceeding a certain threshold could be distinguished (see Sagitov (1997)). These two definitions lead to asymptotic representations of the skeletons as either pure birth process (in the slightly supercritical case) or critical birth-death processes (in the critical case conditioned on the total number of particles exceeding a high threshold value). The limit skeletons reveal typical survival scenarios for the underlying branching processes. In this paper we consider near-critical Bienaymé-Galton-Watson processes and define their skeletons using marking of particles. If marking is rare, such skeletons are approximated by birth and death processes, which can be subcritical, critical, or supercritical. We obtain the limit skeleton for a sequential mutation model (Sagitov and Serra (2009)) and compute the density distribution function for the time to escape from extinction.

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